Physical science {physics, science} can be about atoms, dynamics, electromagnetism, entropy, fluids, heat, kinetics, matter, measurement, quantum mechanics, relativity, space, time, and waves. Entropy includes information and free energy. Matter, includes energy, momentum, and conservation laws. Measurement includes standards, instruments, and precision. Quantum mechanics includes quanta, uncertainty, statistical processes, waves, particles, and wave-particle duality. Relativity includes invariance, constancy, simultaneity, and reference frames. Space includes distance, length, direction, discontinuity, and continuity.
mathematics and physics
Most mathematics tools find applications in physics: vectors, matrices, calculus, topology, statistics, geometry, algebra, trigonometry, group theory, information theory, chaos theory, and complexity theory.
Measurement {measurement} requires starting point or baseline value and ending point. Measurements use unit measures. Measurement counts number of units between starting and ending points. Measurement requires measurement units. Measurements use length, mass, and time units {fundamental units}.
Measurement depends on standard values {measurement unit} {unit of measurement}.
All other units combine length, time, and mass units {fundamental unit}| {absolute unit}. The seven fundamental SI units are length {meter, length unit}, time {second, time unit}, mass {kilogram, mass}, electric current {ampere}, thermodynamic temperature {Kelvin, thermodynamic temperature}, substance amount {mole, particle number}, and light intensity {candela, light intensity}.
derived units
The 22 derived SI units are plane angle {radian, plane angle}, solid angle {steradian, angle}, force {newton}, pressure {pascal}, energy {joule}, power {watt}, temperature {Celsius, temperature}, electric charge {coulomb, electric charge}, electric potential {volt}, electric capacitance {farad}, electric resistance {ohm}, electric conductance {siemens}, magnetic flux {weber}, magnetic flux density {tesla}, magnetic inductance {henry}, light flux {lumen, light flux}, illuminance {lux}, frequency {hertz}, rates and radioactivity {becquerel, radioactivity rate}, radiation dose {gray} {sievert}, and catalytic activity {katal}.
If constants c, G, k are set equal to 1, so c = G = k = 1, all measurements are in centimeters or inverse centimeters {geometrized unit}|. Length uses cm, time uses cm^-1, and mass uses cm^5. The basis for geometrized units is that time and distance relate by constant light speed. Mass, time, and length relate, because photons have different energies at different frequencies, and time relates to frequency. Fundamental units can be velocity, expressed as centimeters per second, or time, expressed as seconds.
To convert unit to another unit {conversion of units} {unit conversion}, multiply original measurement by new-unit to original-unit ratio, obtained from reference books, to cancel original unit and leave new unit. For example, to convert meters to centimeters, (15 meters) * (100 centimeters/meters) = 1500 centimeters.
Unit systems {metric system}| can use meter for length, kilogram for mass, and second for time {meter-kilogram-second system} (MKS), or centimeter for length, gram for mass, and second for time {centimeter-gram-second system} (CGS).
World standard measurement system {Systeme Internationale} (SI) is metric system.
English measurement system {Imperial measure} is in USA.
Prefixes {prefix, measurement} represent powers. 10^24 {yotta} (Y). 10^21 {zetta} (Z). 10^18 {exa} (E). 10^15 {peta} (P). 10^12 {tera} (T). 10^9 {giga} (G). 10^6 {mega} (M). 10^3 {kilo} (kg). 10^2 {hecto} (h). 10^1 {deca} (da). 10^-1 {deci} (d). 10^-2 {centi} (c). 10^-3 {milli} (m). 10^-6 {micro} (µ). 10^-9 {nano} (n). 10^-12 {pico, measurement} (p). 10^-15 {femto} (f). 10^-18 {atto} (a). 10^-21 {zepto} (z). 10^-24 {yocto} (y).
The iso- prefix {iso-, measurement} means the same at different locations. Isobar is for pressure. Isobath is for depth. Isocline is for magnetic dip. Isogon is for magnetic declination. Isohaline is for ocean-surface salinity. Isohel is for hours of sunshine. Isohyct is for rainfall. Isoseismal is for earthquake shock. Isotherm is for temperature. Isochronous is for time.
Measurement digits have different importance {significant figure}| {significant digit}. Zeroes to left of left-most non-zero digit are not meaningful. If number has no decimal point, zeroes to right of right-most non-zero digit are not meaningful. If number has decimal point, zeroes to right of right-most non-zero digit are meaningful. Zeroes between non-zero digits are meaningful.
instruments
Number of significant digits is number of measured digits, including estimated digit.
products
For calculation results, number of significant digits is smallest measured number of significant digits. Multiplying or dividing measurements must maintain number of meaningful digits.
Analyses have number of significant figures {accuracy}|.
Analog-instrument sensitivity {precision}| is tenth of distance between marks. For example, ruler with millimeter markings has precision tenth of millimeter. Digital-instrument precision is displayed numbers.
Quantitative analyses have variability range. Repeated measurements must have similar results {reproducibility}|. If repeated measurement results in significantly different values, results are not reproducible.
Quantitative analyses can detect substance from among other materials {selectivity, measurement}|. For example, instruments detect one frequency or smaller or larger frequency range.
Quantitative analyses have minimum detectable amount {sensitivity, measurement}|. For example, instruments detect intensity threshold.
Precise-measurement technique {nulling}| cancels unknown values using known standard or sensed values using expected values. Nulling methods do not require apparatus to be linear.
Quantities {extensive quantity}| can be sums or differences over time or space ranges. They are total amounts: distance, area, volume, time interval, moles, energy, heat, entropy, charge, and mass. They have magnitude but no direction. Measuring extensive quantities requires accumulation. Extensive quantities are intensive-quantity integrals.
Quantities {intensive quantity}| can be values at times and places. They are instantaneous or local amounts, amount changes, or measurement ratios. Temperature, time, rate, concentration, and chemical potential are scalar magnitudes, with no direction. Distance, direction, velocity, acceleration, force, pressure, momentum, and intensity are vector quantities, with magnitude and direction. Intensive quantities can be matrices. Measuring intensive quantities requires comparing two points over time or distance, to make an extensive quantity. Intensive quantities are extensive-quantity differentials. For example, thermometers measure temperature by linearly expanding mass.
People {people measurements} are 1 meter tall, weigh 10^5 grams, and live 10^10 seconds. Head volume is 1 liter. Heartbeat force is 1 newton. Heartbeat work is 1 joule. Heartbeat power is 1 watt. Walking speed is 2 meters per second. Every day, people make 8000 J or 2000 Calories. Human internal temperature is 37 C. Room temperature or human surface temperature is 21 C. After walking across wool rug, charge on finger is 1 Coulomb.
Number measures {number measurement} include 12 {dozen}, 144 {gross}, and Avogadro's number 6.02 x 10^23 {mole, number} (mol).
Cloth threads {cloth measurement} have units {denier}. Denier range is 100 to 350.
Cycles divided by time in seconds {cycles per second} (Hertz) (Hz) is frequency {frequency measurement}. Frequency is always a positive number. Alternating current is 50 Hz or 60 Hz.
Mechanical-object frequencies measure revolutions per minute (rpm). Car internal-combustion engines operate at 700 rpm to 6500 rpm.
Lowest frequency is one cycle during age of universe, 1/(10^21 s) = 10^-21 Hz. Lowest practical electromagnetic-wave frequency is 1 Hz (with wavelength 10^8 meters, one light-year).
By quantum mechanics, highest frequency {Planck frequency} is inverse Planck time 1/(10^-43 s) = 10^43 Hz. Highest practical electromagnetic-wave frequency is 10^23 Hz (with wavelength 10^-15 meters, electron diameter).
500 sheets {ream} is paper-quantity unit {paper measurement}.
Many different units {becquerel, radioactivity} {roentgen} {rad} {dose equivalent} {rem} {curie} measure radioactivity {radioactivity measurement}.
Electric-charge units {charge measurement} are one mole of electrons {faraday} (F) or total charge {coulomb, charge} (C) moved one meter by one newton. 1 F equals 96494 C.
Electric charge, in coulombs, divided by time, in seconds is electric current {electric current measurement} in amperes (A). Currents are typically 0.1 A to 5 A. Radio current is 1 A.
Electric energy, in joules, divided by charge, in coulombs, is electric potential {potential measurement} in volts (V). Small-battery voltage is 1.5 V. Car-battery voltage is 12 V. Magnetic field has unit {gauss}.
Electric potential, in volts, divided by current, in amperes, is electrical resistance {resistance measurement} in ohms, which has upper-case omega symbol.
Electric-potential change divided by current change is electric capacitance {capacitance measurement} in farads (F). Capacitors have farad range from 1 F to 500 F.
Electric-current change divided by electric-potential change is electrical inductance {inductance measurement} in henrys.
In metric system, fundamental mass units {mass measurement} are gram {gram} (g) or kilogram {kilogram, mass unit} (kg). A derived mass measure is 1000 kilograms {tonne}. Atom mass unit is 10^-24 grams {atomic mass unit}, 1/12 of carbon-12 atom mass. Mass unit in English units is slug {slug, mass}.
Mass, in kilograms, times acceleration, in meters per second squared, is force unit {force measurement} in newtons (N), in MKS system. Mass, in grams, times acceleration, in centimeters per second squared, is force unit {dyne} in CGS system.
Mass times gravity acceleration is weight. In English measure, weight measurements use different systems, depending on mass range.
avoirdupois
In avoirdupois units {avoirdupois}, 28.35 grams is 1 ounce {avoirdupois ounce}. Avoirdupois weights include 16 avoirdupois ounces {avoirdupois pound} {pound, weight}, 100 avoirdupois pounds {hundredweight}, 2000 avoirdupois pounds {ton, weight}, 1/16 avoirdupois ounces {dram}, 1/7000 avoirdupois pounds {grain, mass unit}, 100 avoirdupois pounds {cental}, 100 avoirdupois pounds or 112 avoirdupois pounds or 100 kilograms {quintal}, and 2240 avoirdupois pounds {long ton}.
apothecary
Apothecary weights {apothecaries measure} depend on 1 grain {apothecary grain}. Apothecary weights include 20 apothecary grains {apothecary scruple}, 3 apothecary scruples {apothecary dram}, 8 apothecary drams {apothecary ounce}, 12 apothecary ounces {apothecary pound}. Apothecary pound has same weight as troy pound.
troy
Troy weights, based on 1 grain {troy grain}, are for weighing gold, silver, and jewels. Troy weights include 24 troy grains {troy pennyweight}, 20 troy pennyweight {troy ounce}, and 12 troy ounce {troy pound}.
electromagnetism
Electric force is in newtons. Magnetic force has units {unit pole}.
Force, in newtons, times distance, in meters, is work and energy {energy measurement} in joules (J), in MKS system. Force, in dynes, times distance, in centimeters, is energy unit {erg} in CGS system. 10000 erg equals 1 joule.
In English measure, force is weight, in pounds, times distance, in feet {foot-pound}.
electric
Voltage, in volts, times charge, in coulombs, is electric energy in joules. For subatomic particles, electric energy is voltage, in volts, times electron charge {electron-volt} (eV). Electric energy is power, in kilowatts, times time, in hours {kilowatt-hour}. USA houses use hundreds of kilowatts each month.
heat
Heat energy is energy needed to heat one gram of water one degree Celsius {calorie} (cal) or one kilogram of water one degree Celsius {Calorie} (Cal). 1 Cal = 1000 cal. In English measure, heat energy uses larger units {British thermal unit} (BTU).
Temperature units {temperature measurement} relate to water freezing and boiling temperatures. In Kelvin (K), freezing point is 273 degrees and boiling temperature is 373 degrees. In Celsius (C), freezing point is 0 degrees and boiling temperature is 100 degrees. In English measure {Fahrenheit} (F), freezing point is 32 degrees and boiling temperature is 212 degrees. An English temperature measure {Rankine} is Fahrenheit plus 459.69 degrees.
Force, in dynes, times time, in seconds {dyne-second}, is impulse, impact, or action {impulse measurement}.
In SI system, power per area {intensity measurement} can be power per steradian. Informally, intensity is flux per steradian, pressure divided by time, or force per second per unit distance.
Sound intensity uses logarithmic scale {bel}, or its tenths {decibel} (db) (dB), so 10 db equals 1 bel. Whispers are 30 db. Siren loudness is 100 dB. Larger units {neper} are 6.686 db.
Light intensity compares to standard candle {candela, intensity} (cd) {international candle} {candle, light} {lumen, light}. One candela is light intensity 1/683 watts per steradian, at frequency 540 * 10^12 hertz.
Energy, in joules, divided by time, in seconds, is power {power measurement} in watts (W). In refrigeration, 3517 watts are one ton {ton, refrigeration}. In English measure, power compares to standard horse power {horsepower}.
Mixing alkanes and benzenes makes gasoline {gasoline measurement}, which has engine power {octane number}. Octane is 87 for regular gas, 89 for intermediate gas and 91 for premium gas.
Force, in newtons, divided by area, in square meters, is pressure {pressure measurement} in pascals. Atmospheric pressure has pressure units {bar, pressure unit} {atmosphere, pressure unit} (atm). Pressure is how high mercury rises in columns {torr} {mm Hg} {inches Hg}. 1 atm equals 760 mm Hg. Standard temperature and pressure (STP) is 25 C and 1 atm. In English measure, pressure is force, in pounds, divided by area, in square inches {pounds per square inch} (psi).
Mass, in grams, divided by volume, in cubic centimeters, is density {density measurement}: d = g/cm^3. Water density is 1 g/cm^3.
Metals {gold measurement} can have purity {karat} (K). 28 K is pure. 24 K is high purity. 18 K is good purity. 14 K is strong. 10 K is strongest.
Dynamic-viscosity unit {poise, viscosity unit} differs from kinetic-viscosity unit {myriastoke} {viscosity measurement}.
Space measurement {length measurement} {distance measurement} uses distance units. The MKS system has a fundamental length unit {meter, length} (m). Derived length units include 10^-2 meters {centimeter}, 10^-6 meters {micron}, and 10^-10 meters {Angstrom}.
English measure
0.001 inches (mil) measures wire diameter. 2.54 centimeters {inch} (in). 39.37 inch = 1 meter. 0.66 feet {li, length unit}. 3 inches {palm, length}. 4 inches {hand}. 7 to 9 inches {span}. 12 inches {foot, length} (ft). {board foot}. 18 inches to 22 inches {cubit}. 2.5 feet {pace, length}. 3 feet {yard, length} (yd). 6 feet {fathom}. 16.5 feet {rod, measure}. 100 links or 66 feet {chain, length} {Gunter's chain} {surveyor's chain}. 100 feet or 100 links {chain} {engineer's chain}. 625 Greek feet or 0.1196 mile {stade}. 220 yards {furlong}. 5280 ft, 1760 yards, 8 furlongs, or 1.609 kilometers {mile} {statute mile}. 1 arc-minute on great circle of sphere with area equal to Earth area is 6080 feet {nautical mile}. 1 arc-minute at equator is 6087 feet {geographic mile}. 3 miles, or 2.4 to 4.6 miles {league}.
railroad
Railroad track {railroad measurement} has width {standard gauge} {narrow gauge} {HO guage}.
wire gauge
Wires have diameters {wire gauge}, such as American or Brown and Sharpe system.
minimum
By quantum mechanics, minimum length is Planck length, 1.6 * 10^-35 meters, where length becomes quantized.
maximum
Maximum length is circumference of universe, 10^30 meters.
Length, in meters, divided by time, in seconds {meters/second}, is velocity {speed measurement}. In English measure, velocity is miles per hour (mph) or nautical miles per hour {knot, speed}. Velocity divided by time, in seconds {meters/second squared}, is acceleration. Earth-gravity acceleration is 9.8 m/s^2. Car acceleration is 3 m/s^2 or 1/3 gravity.
Surface measurement {area measurement} is length squared or square meters (m^2). Area measure can be 1000 m^2 {hectare}, which is 2.5 acres. In English measure, area can be 10 square chains, 160 square rods, or 0.4047 hectare {acre} or can be 36 square miles {township}. 640 acres equals 1 square mile. For cross-sections {cross-sectional area}, 1 barn is 10^-28 cm^2.
Space measurement {volume measurement} depends on length measurement. Length cubed is in cubic meters. Volume measure is 1000 cm^3 or 0.001 m^3 {liter} (L).
English measure
1/360 fluid ounce {minim}. 1/6 fluid ounce {teaspoon} (tsp). 3 teaspoons {tablespoon} (tbsp). 2 tbsp {fluid ounce} (fl oz). 1 1/2 fl oz {jigger}. 0.25 pint or 4 fluid ounces {gill, volume}. 8 fluid ounces {cup}. 2 cups or 16 fluid ounces {pint}. 1/16 peck {dry pint}. 2 pints or 1.1 liter {quart, volume}. 1/8 peck {dry quart}. 4 quarts {gallon}. gallon slightly larger than regular gallon {Imperial gallon}. {stere}. 0.25 bushel {peck}. 0.3524 hektaliter {bushel}. 1/4 barrel or 9 gallons {firkin}. 31.5 gallon {barrel, volume unit}. {demijohn}. 2 barrels {hogshead}. 3000 ml or four-fifths gallon {double-magnum} {jeroboam}, for wine. two-wine-bottle or 1500 ml {magnum, champagne}. {split}. {trencher}. 128 ft^3 {cord, volume}. hay {bale}.
Two intersecting lines have opening amount {angle measurement} at intersection. If lines are the same, angle between them is 0 degrees, and angle around them is 360 degrees, two times pi radians. If lines are parallel and meet at center, angle between them is 180 degrees, pi radians. If lines are perpendicular, angle between them is 90 degrees, pi/2 radians. Angle measure can use larger unit {rhumb}, which is 11.25 degrees, 1/16 of reverse turn, or one point on mariner's compass.
Metal wires have diameters {gauge, measurement}. Smaller numbers are larger diameters.
Aperture size {opening measurement}, as for camera lens, has units {f number} {f stop}. f stops are numbers 2 for wide open, 2.8, 4, 5.6, 8, 11, 16, 22, and 32 for almost closed.
Earth locations have distances and elevations {surveying measurements}. Measured great-circle arc {level, survey} can be surveying basis.
Finding elevation difference uses instrument {level, instrument} {spirit level}| that measures angles.
High objects have angle {acclivity} to horizontal.
Low objects have angle {declivity} to horizontal.
Surveying instruments {theodolite}| can measure horizontal and vertical angles.
Observers above ground can see Earth surface out to distance {horizon, Earth}. If Earth is spherical, horizon relates to height above ground: r = 1.23 * h^0.5, where r is radius from observation point to horizon, and h is height above ground.
Surveyors place metal plates {benchmark, surveying}| at locations with precise elevation.
Measuring time {time measurement} requires standard vibration or rotation frequency. Fundamental time unit is one second {standard second} {SI second} (s), which is 9,192,631,770 periods of unperturbed microwave transition between two cesium-133 ground-state hyperfine levels, as measured by atomic clocks {Coordinated Universal Time} (UTC).
year
Standard second can depend on Earth motion around Sun. Tropical year 1900 has 31,556,925,974 seconds {ephemeris second} {ephemeris time} {Newtonian time}. Gravity predicts solar-system object positions at distinct times. Ephemeris time uses actual number of days {universal time} (UT0) or universal time corrected for Earth-axis wobble {navigator's time} (UT1) (UT2).
derived units
60 seconds {minute, time} (min). 60 minutes {hour} (hr). 24 hours {day}. 7 days {week}. 14 days and nights {fortnight}. 4 1/3 week or 28 to 31 days {month}. 365 days {year} (yr). 366 days every fourth year {leap year}. 10 years {decade}. 100 years {century, time}. 1000 years {millennium}. 1,000,000 years {eon}.
minimum
By quantum mechanics, minimum time is Planck time, 10^-43 seconds, where time becomes quantized.
maximum
Maximum time is age of universe, 10^21 seconds.
Earth time measurement can compare to Sun {solar time}. Varying orbital speeds and varying seasons cause Sun-referenced Earth-rotation period to average four minutes longer than star-referenced rotation period. Sun-referenced Earth-rotation period can be ahead 16 minutes in early November or behind 14 minutes in early January.
Years {anomalistic year} can be time from Earth perihelion to next perihelion, which is five minutes longer than sidereal years, because major axis of Earth elliptical orbit moves around Sun in same direction as Earth moves.
Years {sidereal year} can be time between same positions with respect to fixed stars. Time {sidereal time} can be relative to fixed stars.
Years {tropical year} can be time from first point of Aries to next first point of Aries, which is 20 minutes shorter than sidereal year, because first point moves around Sun in opposite direction to Earth motion.
Causes {cause, physics} {causation, physics} change particles or particle spatial relations. Causes are forces. Cause precedes effect. Mind activities are indirect causes. Causes are desired states. Human thought has purposes. Physical forces put thoughts into action.
Space-like separated fields either commute or anticommute {microcausality}, because otherwise effects travel faster than light speed.
Previous causes require later effects. Perhaps, physical-law asymmetry can mediate causal temporal asymmetry {third arrow strategy, causation}. However, physical laws have no asymmetry that can mediate temporal asymmetry.
Matter amount is mass {matter}.
Matter amount {mass, physics}| is sum of elementary-particle amounts. Matter has spatial extension and divisibility. Mass has inertia. Mass is scalar quantity. Object interaction with universal field, such as Higgs field or technicolor field, causes mass.
Object mass has balance point {mass center} {center of mass} {center of gravity}. In calculations, total object mass can be at mass center.
balance
To find object mass center, hold object at a point off center, allow object to swing freely under gravity until it stops, draw vertical line through point, and repeat for two more points. The three lines intersect at mass center.
spin
Spinning objects spin around mass center.
example
People cannot pick up chairs while standing against walls, because chair-person combination has mass center beyond toes, and so they must tip over.
Objects tend to keep same direction and speed {inertia}| {matter, inertia}. Mass resists motion change. Inertia is resistance to acceleration.
cause
All universe masses exert gravity on object masses. All universe masses contribute to space-time curvature at space-time points. Objects take the geodesic shortest path between two space-time points. Geodesics are paths that keep same direction and speed. Inertia is following geodesics. Space-time curvature depends on mass, so inertia depends on mass. The dependence is the same. Mass and inertia are the same.
examples
It is harder to pull or push more massive object. If someone puts a big rock on your head and hits it with hammer, you do not feel the hit, because big mass changes motion slowly. People can rapidly pull smooth tablecloths out from under dishes on tables, without moving dishes.
Substances have mass per volume {density, physics}|: density = mass / volume. For example, boxes can have different masses inside, while volume stays the same, so densities can differ.
Particles have paired particles {antiparticle} {antimatter}| with same properties but with positive energy and positive pressure. Negative-energy states fill space and are undetectable because they are uniform. Photons can interact with negative-energy particles, to make missing negative-energy states, which are like positive-energy antiparticles. Because negative-energy state is missing, antiparticles are like particles going backward through time.
properties
Antiparticles have same mass, spin, and angular momentum as particles. Antiparticles have opposite charge and opposite of every other property.
energy
If particle and antiparticle meet, they create energy 100 times more than energy in same mass in nuclear reactions.
sources
Cosmic rays hitting high atmosphere can make antiparticles. Particle accelerators can make proton {antiproton}, neutron {antineutron}, and electron {positron, antiparticle} antiparticles.
mass
Antimatter has to have positive mass, not negative mass. All fields have positive mass. If they have negative mass, vacuum polarization around nuclei makes negative mass and shields nuclei from gravity. There can be no gravity. Antimatter is negative-energy absence and so is positive energy.
universe
By symmetry, universe has equal matter and antimatter, but instruments do not observe antimatter. Perhaps, if space has short, circular, higher dimensions, antimatter hides in them.
In early universe, matter phase was like hot fluid and was symmetric in all directions, as in gas or liquid, not as in asymmetric crystals. Later cooling changed matter phase differently at different places, and different space regions had different ground-state energies. Boundaries formed between different-phase regions. As universe expanded, boundaries stretched into long strings {cosmic string}. Cosmic strings have large mass, positive energy, and positive pressure.
Space can contain negative energy {exotic matter}. Energy density and pressure can be negative.
cause
Energy fluctuations, required by uncertainty principle, cause negative energy in zero-average-energy-density regions, such as space vacuum. If non-linear destructive quantum interference can damp energy fluctuations {squeezed vacuum}, energy in alternating regions can be less than zero.
amount
Negative-energy magnitude is inversely proportional to space-time volume. Larger negative energies must pair with more and closer positive energies.
total energy
System total energy is always greater than zero.
examples
Moving mirror can make negative-energy flux.
effects
Negative energy makes light rays diverge. Negative-energy regions allow travel faster than light, because they contract space-time.
The three elementary-particle families have similar interactions among family members, but Standard Model and SSMs cannot explain this {family problem}. String theory allows many families.
Particle-interaction graphs {Feynman diagram} can show all interaction paths. In Feynman diagrams, exchanges with antiparticles go backward in time.
Phase transition back to solid is twice as fast as predicted {HBT puzzle}, as measured by interferometry {Hanberry-Brown-Twiss interferometry} using model {nuclear optical model}.
A vacuum-like layer {Leidenfrost layer} is 0.001 to 0.1 light-year thick, has gamma and shortwave radiation, and is between matter and antimatter plasmas.
One positron and one electron can make a particle {positronium}. Two positroniums can make a particle {di-positronium}.
Potential-energy field is complex-number field with different same-energy phases. Energies have many phases. Higgs field starts at high energy and goes to lowest energy {spontaneous symmetry breaking}. Higgs field can go to complex-number-field values below zero {non-zero Higgs field vacuum expectation value}, though lowest energy is zero.
Space regions {vacuum, space}| can have no mass or energy. However, uncertainty principle requires that vacuums have energy fluctuations. Energy fluctuates to positive and negative energies. If non-linear destructive quantum interference can damp energy fluctuations to squeeze vacuum, energy in alternating regions can be less than zero.
Not-directly-observable particles {virtual particle}| can exist for times shorter than force-interaction time. According to quantum-mechanics, virtual particles can spontaneously appear in space and then interact before observable time or space.
cause
In vacuum, mass, energy, and electric charge average zero. In space and in baryons, according to quantum-mechanics, mass, energy, and electric charge have random fluctuations above and below zero over time and space.
pairs
Virtual particles arise as pairs: quark-antiquark pairs, electron-positron pairs, photon pairs, and graviton pairs. To make average momentum be zero, zero-rest-mass virtual particles must form in same-particle pairs that travel in opposite directions. To make average charge be zero, virtual particles with mass must form in pairs, one with positive charge and one with negative charge. To make average energy be zero, virtual particles with mass must form in pairs, one with positive energy and one with negative energy.
antiparticles
Uncertainty principle allows virtual particles to move faster than light, and relativity requires that particles that move faster than light must go backward in time. By relativity and quantum mechanics, charge moving backward in time is equivalent to opposite charge moving forward in time. Particles and antiparticles must have opposite charges. Virtual particles go forward in time. Virtual antiparticles move backward in time.
As an electron moves through space-time, moving observers see different electron velocities, and some see electron moving backward in space-time.
As an electron moves through space-time, it can change velocity twice. This is equivalent to an electron-positron virtual-particle pair arising at the second velocity-change point, the virtual positron going backward in time to annihilate the original electron at the first velocity-change point, and the virtual electron continuing on from the second velocity-change point as a real electron.
objects
All protons, neutrons, atoms, and molecules have virtual particles at all times. In atoms, virtual negative charges stay closer to positively-charged nucleus, and virtual positive charges stay closer to negatively-charged electron orbits. Because virtual particles continually spontaneously appear and then annihilate, all objects always have virtual-particle distributions and complicated mass, energy, and/or charge distributions. Dirac's relativistic-quantum-mechanics equations can account for all virtual-particle distributions, to any accuracy degree, by including primary, secondary, tertiary, and/or higher levels of virtual-particle creation and annihilation. The virtual-particle distribution accounts for most proton, neutron, atom, molecule, and object mass, so quarks and electrons are only a small part of object mass.
energy
By uncertainty principle, more-energetic virtual particles have shorter times. Long-lived-virtual-particle energies have lifetimes of 10^-8 seconds, so long-lived-virtual-particle energies are 1.22 * 10^-16 GeV, or 6 x 10^-13 electron masses. Times can be as short as Planck time, 10^-43 seconds, so short-lived-virtual-particle energies are 1.22 * 10^19 GeV, or 6 x 10^22 electron masses. Higher energies are more infrequent. By uncertainty principle, total energy is finite but very high, equal to 10^120 times universe mass-energy. To make vacuum energy average zero, space vacuum must have negative energy (dark energy) almost equally high, to cancel. This space energy makes space expand.
conservation laws
Because, by uncertainty principle, short times and small spaces have high energies and momenta, virtual particles do not necessarily conserve energy and momentum.
experiments
Exciting hydrogen atoms with microwaves moves electrons from s to p orbitals (Lamb shift), and electromagnetic-field quantum fluctuations make virtual particles. In mass-173-atom nucleus, strong electric field can produce real particle from virtual particle.
theory
Perhaps, real particles are detectable parts of virtual-particle clouds around particles.
theory: infinities
In quantum mechanics, because virtual particles can arise spontaneously at any point and time, for any particle process the number of possible particle paths is infinite. In particular, electron mass and charge become infinite. In quantum electrodynamics, quantum chromodynamics, and electroweak theory {Weinberg-Salam theory}, renormalization cancels infinities. In quantum-gravity theories, masses and their secondary interactions can be large, so renormalization is not always possible. (In string theories, strings have vibration states, with no infinities.)
Most physical laws do not change if antiparticle, which has opposite charge, replaces particle {charge conjugation}|.
Universe particles can have handedness {chirality, universe}|. Zero-rest-mass particles conserve chiral symmetry. Neutrinos, pions, and kaons have handedness. Other particles conserve chiral symmetry. Particles with mass can change handedness by losing mass by symmetry transformations other than chirality.
Most physical laws do not change if coordinates invert through origin or reflect through plane {parity, physics}|, to change right-handed into left-handed, because most particles do not have handedness. Parity conserves in electromagnetic and strong nuclear forces but not in weak nuclear force, because neutrinos, pions, and kaons have odd intrinsic parity. Other particles, including all zero-rest-mass particles, have even intrinsic parity and no handedness. Parity violation is greater for charged particles, compared to uncharged ones.
Quanta {quantum number}|, such as electric charge, can be additive integers {additive quantum number}. Antiparticles have negative of additive-particle quanta. Mass is not additive. Quanta, such as parity and g-parity, can be multiplicative {multiplicative quantum number}, based on nth roots of unity. Fermions have -1 parity. Bosons have +1 parity. Two interacting fermions make boson: -1 * -1 = +1. Two interacting bosons make boson: +1 * +1 = +1. Three quarks make hadron, with parity -1: -1 * -1 * -1 = -1.
Perhaps, underlying field is particles {techniquark}, bound by force {technicolor}|.
Particles {subatomic particle}| are like field singularities, vortexes, or discontinuities. Higher-mass particles have excited particle states. Perhaps, fundamental particles are statistical entities, with charge, mass, and so on, distributions. Quantum-wave equations arise from particle statistical nature.
Light nuclei have one to three clusters {alpha particle, atom}| with two protons and two neutrons.
Perhaps, particles {exotic particle} can be more fundamental than quarks and leptons. Perhaps, three prequark bosons {preon}, from three families, make higher particles. Family has two flavors, with four of one {chromon} and three of the other {somon}. However, no method makes masses come out right for quarks and leptons using preons. Hypercolor binds preons together. Perhaps, three particles {rishon} have charge 0 or 1/3, have a color or its anti-color, and combine to give particles.
Baryons and mesons {hadron}| share properties. Baryons, such as protons and neutrons, have three quarks. Mesons, such as pions, have two quarks. Both strong and weak nuclear forces affect hadrons. Hadrons and leptons account for all particles. Photons with 10^9 more energy than average act like hadrons and have strong nuclear force interactions. Electrons at high energy act like hadrons.
Perhaps, particles {magnetic monopole}| can be one-pole magnets. Magnetic monopoles can combine bosons to make fermions.
Possible particles {tachyon}| can travel faster than light. Tachyons go backward in time. Tachyons have imaginary mass. Tachyon energy increases as it slows.
Particles {boson}| {messenger particle}, such as photons, gluons, W and Z bosons, and gravitons, can carry force fields. Gravitons, photons, mesons, gluons, W particles, Z particles, and all exchange particles have integer spins and follow Bose-Einstein statistics. Unlike fermions, two bosons can have same quantum numbers. Rather than always having same units, boson quanta can vary in energy. Fermions and bosons account for all particles.
Spin
Some bosons {scalar boson}, such as Higgs particle and W particle, have zero spin. Some bosons {vector boson}, such as photon, graviton, and Z particle, have non-zero integer spin.
states
Bosons in same state tend to cluster together. Identical particles with same spin can interfere constructively if their waves are in phase. Identical particles with same spin can interfere destructively if their waves are in opposite phase. Therefore, if boson is present, another same-type-boson probability is greater.
fields
Interacting particles use field to store energy and momentum while they send signals between particles and cause interaction. Field preserves conservation laws. Fields carry signals as bosons, which carry energy and momentum to distant objects. Local interactions caused by boson exchanges mediate all action-at-a-distance.
statistics
Bosons and fermions with the same quantum numbers are exactly the same, so two different photons or electrons with the same quantum numbers are exactly the same. Because they have no relativistic effects on each other, bosons have symmetric wave functions: f(b+) = f(b-), where b+ has spin +1 and b- has spin -1. Different bosons can have the same state, because bosons do not attract or repel each other by relativistic effects. Their changing fields are symmetrical and cancel. Because they have relativistic effects on each other, fermions have anti-symmetric wave functions: f(e+) = -f(e-), where e+ has spin +1/2 and e- has spin -1/2. For two fermions, wavefunction is anti-symmetric for fermion exchange: f(e+,e-) = -f(e-,e+). For helium atoms (with two electrons in lowest orbital), with no time changes, the ground-state wavefunction is anti-symmetric, but the main (zero-order) wavefunction is symmetric, so the spin wavefunction is anti-symmetric. Electrons with same spin cannot be in same state (Pauli exclusion principle), because f(e+,e+) = -f(e+,e+) can be true only if f(e+,e+) = 0. Different fermions have different states, because fermions repel each other by relativistic effects. Changing electric fields induce magnetic fields that affect moving electric charges. Their changing fields are anti-symmetrical and do not cancel.
Strong-nuclear-force-exchange bosons {gluon}| have eight types, mass 0, spin 1, and charge 0. They do not feel electromagnetism or weak force. They affect gluons and quarks.
Gravity-exchange bosons {graviton}| have mass 0, spin 2, and charge 0. Perhaps, gravitons differ over time, as space phase changes. Perhaps, at high energies, space and time decouple.
Stress-energy density makes virtual gravitons. By tidal-force induction, those gravitons make adjacent virtual gravitons and then become zero again, so virtual gravitons propagate through space at light speed. General-relativity gravity fields are virtual-graviton streams.
When masses have tidal forces, tidal-force accelerations make real gravitons that travel outward in that direction as gravitational waves. Real-graviton tidal-force accelerations induce adjacent virtual gravitons that go back to zero and make adjacent real gravitons, so propagating gravitons through space. Tidal-force accelerations push existing virtual-graviton streams sideways, putting a kink in them.
A weak-force field {Higgs field} is evenly distributed throughout space and interacts with W bosons, Z bosons [1983], Higgs bosons, quarks, and leptons and so associates mass with them. Without Higgs field, particles affected by the weak force have no mass. Even in empty space, the Higgs field has non-zero negative value {vacuum expectation value}. The Higgs field interacts with particles affected by the weak force, differently for right-handed and left-handed particles, and so its existence causes, below critical temperature, weak-force spontaneous symmetry breakdown. Without Higgs field, particles affected by the weak force have the same physics for right-handed and left-handed particles. Stronger Higgs field interactions make higher-mass particles. Stronger Higgs field interactions are over shorter distances.
The Higgs field interacts with fermions to make a small part of their mass, which is mostly due to gluons and 1% to quarks. Photons, gluons, and gravitons do not interact with Higgs field and have no mass.
Standard Model requires only one Higgs field and one Higgs particle. Standard Model gives correct mass ratio between W and Z bosons and all particle masses. Supersymmetric Standard Models have two Higgs fields and five Higgs particles, three neutral and two charged. Supersymmetric Standard Models have non-zero energy minimum and give mass to superpartners, as Higgs fields interact. Perhaps, neutrino masses come from Higgs-field interactions or from third Higgs field.
Higgs boson
Higgs-field perturbations make bosons {Higgs particle} {Higgs boson} that may be elementary or composite. Higgs bosons are their own antiparticle. Higgs bosons are CP-even.
By Standard Model, smallest mass is 114 to 192 GeV. By measurement [2010], Higgs-boson mass is 115 to 156 GeV or 183 to 185 GeV (200 GeV is same as tau particle and slightly more than charm quark). By measurement [2011], Higgs-boson mass is 115 to 140 GeV. If quantum effects cause smallest Higgs-boson mass to be higher, other-particle masses are too high. By minimal supersymmetry, there are five Higgs bosons at 114 to 192 GeV, 300 GeV (similar to top quark), 370 GeV, and 420 GeV.
Higgs bosons are unstable and quickly decay, and so are not directly observable. If elementary, Higgs bosons can decay to bottom quark and bottom antiquark, photons, and/or tau particle and antitau particle, which are observable.
Higgs bosons have no spin and so are scalar bosons, not vector bosons.
Higgs bosons have no charge and so do not affect electromagnetism, and electromagnetism does not affect them.
Higgs bosons have no color and so do not affect strong force, and strong force does not affect them.
interactions
Particle attraction to Higgs field vibrates Higgs field and makes Higgs field denser at particle, causing (otherwise zero-rest-mass) particle to slow from light speed. Higgs-field interactions with matter cause mass, inertia, and space curvature, because Higgs bosons form as particles acquire mass. Mass is proportional to Higgs-field strength and interaction strength. Different particles have different interactions and different masses. For example, zero-rest-mass photons do not interact with Higgs field and maintain zero mass and light speed.
Higgs field resists accelerations, not velocities.
space
Higgs field is everywhere in space, so particle masses are constant throughout space. Higgs field started at universe origin and fills space-time.
field strength and self-interaction
Standard-Model Higgs particles can interact with themselves, and supersymmetry different Higgs-particle types can interact with other Higgs-particle types. Self-interaction causes negative field strength at lowest energy in universe, so Higgs field at lowest energy is negative energy.
temperature
High temperature makes Higgs field fluctuate. In zero-rest-mass empty space, Higgs field fluctuates above and below zero energy. Above 10^15 K, average energy was zero, and all fermions and bosons had zero mass. Universe was symmetric. At 10^15 K, 10^-11 seconds after universe origin, average Higgs field reached lowest negative value. Some particles acquired mass from Higgs field. Universe was not symmetric (spontaneous symmetry breaking).
In grand unified theory, electromagnetic, weak, and nuclear forces unify before 10^-35 seconds after universe origin, above 10^28 K, under SU(3) x SU(2) x U(1) Lie symmetry group, where SU(3) is for strong-force quark color, SU(2) is for weak-force W and Z bosons, and U(1) is for electromagnetic charge, making grand unified Higgs field. Grand unified theory allows proton decay.
Above 10^15 K, electroweak symmetry is unbroken, and W and Z particles have zero rest mass. Above 10^15 K, electromagnetic and weak forces unify under SU(2) x U(1) Lie symmetry group, making electroweak Higgs field. SU(2) is for the Higgs-field spinor with two complex components: SU(2) doublet. The Standard Model U(1) charge is -1.
At cooler temperature, electromagnetism and weak force do not unify. The W and Z gauge bosons have mass after electroweak symmetry breaking below 10^15 K, by interaction with the Higgs field {Higgs mechanism} {Englert-Brout-Higgs-Guralnik-Hagen-Kibble mechanism} [1964] (François Englert and Robert Brout; Peter Higgs, from ideas of Philip Anderson; Gerald Guralnik, C. R. Hagen, and Tom Kibble). The Higgs field, an SU(2) doublet, has four degrees of freedom. Three degrees of freedom make non-physical Goldstone bosons. One degree of freedom makes one Higgs boson, in the Standard Model. The Minimal Supersymmetric Standard Model requires a series of Higgs bosons. The Technicolor models or Higgsless models have no Higgs bosons but do have Higgs mechanism.
Electromagnetic-force-exchange particle {photon, particle}| has mass 0, spin 1, and charge 0. Range is infinite. It has light speed. All zero-mass particles have spin axis in motion direction or in opposite direction.
W particle and Z particle [1973] {intermediate vector boson}| {weak gauge boson} have speed 1000 meters per second and range 10^-18 meters.
Weak-nuclear-force exchange bosons {W particle}| can have mass 80.4 GeV, spin 1, and charge +1 or -1 [found in 1973].
Weak-nuclear-force exchange bosons {Z particle}| can have mass 91 GeV, spin 1, and charge 0 [found in 1973].
Possible exchange bosons {Z-prime particle}| can indicate a new force type.
Hadron bosons include exchange particles {meson}| for nuclear force.
properties
Mesons have masses between one-seventh proton mass and four times proton mass. Mesons have charge -1, 0, or +1. Mesons have spin 0 or 1. Mesons have lifetime from 10^-23 to 10^-8 seconds.
examples
More than 20 mesons include pi meson (pion), K meson (kaon), and eta meson. Rho meson, phi meson, and omega meson are vector mesons with negative intrinsic parity.
quarks
Mesons have quark and antiquark. Pion has up or down quark. Kaon has strange quark. Upsilon particle meson has top quark.
Psi particle or J particle meson {charmonium} has charmed quark.
Mesons {pion}| can have masses one-seventh proton mass. Pion has up quark and down antiquark, so charge is -2/3 + -1/3 = -1, and color and complementary color add to white.
charm quark-antiquark pairs {psi particle}.
Baryons, hadrons, and non-zero-mass leptons {fermion}| have half-integer spins, with Fermi-Dirac statistics. No two fermions can have same quantum numbers. Fermion energy quanta always have same units. Same-type fermions are indistinguishable. For example, all electrons are exactly alike. Fermions and bosons account for all particles.
statistics
Bosons and fermions with the same quantum numbers are exactly the same, so two different photons or electrons with the same quantum numbers are exactly the same. Because they have no relativistic effects on each other, bosons have symmetric wave functions: f(b+) = f(b-), where b+ has spin +1 and b- has spin -1. Different bosons can have the same state, because bosons do not attract or repel each other by relativistic effects. Their changing fields are symmetrical and cancel. Because they have relativistic effects on each other, fermions have anti-symmetric wave functions: f(e+) = -f(e-), where e+ has spin +1/2 and e- has spin -1/2. For two fermions, wavefunction is anti-symmetric for fermion exchange: f(e+,e-) = -f(e-,e+). For helium atoms (with two electrons in lowest orbital), with no time changes, the ground-state wavefunction is anti-symmetric, but the main (zero-order) wavefunction is symmetric, so the spin wavefunction is anti-symmetric. Electrons with same spin cannot be in same state (Pauli exclusion principle), because f(e+,e+) = -f(e+,e+) can be true only if f(e+,e+) = 0. Different fermions have different states, because fermions repel each other by relativistic effects. Changing electric fields induce magnetic fields that affect moving electric charges. Their changing fields are anti-symmetrical and do not cancel.
Protons, neutrons, and over 100 other particles {baryon}|, such as lambda, sigma, delta, cascade, omega, and upsilon, share properties. Baryons have baryon number 1, while other particles have baryon number 0. Baryons have three quarks.
Particles {hyperon} similar to protons and neutrons can have higher masses. Hyperons have masses 2 to 10 times proton mass. Hyperons have charge -1, 0, +1, or +2. Hyperons have spin 1/2 or 3/2. Hyperons have lifetime 10^-23 to 10^-10 seconds. Hyperons have three quarks and are baryons.
Particles {neutron}| similar to protons in mass have no charge. Neutron has three quarks, two down and one up. Free neutrons have lifetime 1000 seconds before they decay to proton. Neutrons in atoms are stable, because, in nuclei, strong nuclear force lowers neutron energy, so neutrons do not decay.
The main and lowest-energy baryon {proton}| is mainly in atomic nuclei. Proton has three quarks, two up and one down. Proton mass is 10^-24 grams. Protons have infinite lifetime. However, if superweak nuclear force exists, lifetime is 10^31 years.
Electrons and similar fermions {lepton}| share properties.
size
Leptons have diameter 10^-15 centimeter. Leptons have no internal structure, at least down to 10^-16 centimeter. Quantum electrodynamics requires leptons to be points.
forces
Weak nuclear force affects leptons, but strong nuclear force does not affect them. They have no color charge. Weak nuclear force causes one-quarter of lepton mass.
charge
Electron, muon, and tau particle leptons have charge -1 unit. Neutrinos have charge 0 units. Charge causes part of lepton mass. Lepton charge is sum of infinite negative charge, surrounded by positive-charge cloud induced by negative charge.
lifetime
Electrons cannot decay to smaller particles, so electrons have infinite lifetime.
isospin
Electrons, muons, and taus have weak-isospin third component -1/2, while all neutrinos have +1/2.
quarks
Quarks and leptons are similar. Both are point-like, pair, and have six types.
Negatively charged particles {electron}| rapidly orbit atomic nuclei at varying distances. Electron mass is 10^-27 grams or 0.511 MeV. Electron charge is -1. Lifetime is infinite. Protons equal electrons in neutral atoms. Electrons travel 10^-14 meters in 10^-8 seconds in one orbit.
Leptons {muon}| can be more massive than electrons. They can be in particles caused by cosmic rays hitting upper atmosphere. Muons have masses 204 times electron mass or 106 MeV. Lifetime is 2.2 x 10^-6 seconds, because muon can decay to electron. Muons have electric charge -1. Muon has associated neutrino. Muon has weak-isospin third component -1/2.
Atoms can have muons instead of electrons. Collisions can make two muons {dimuon event} or three muons {trimuon event}. These collisions demonstrate charmed particles and heavy leptons.
Leptons {neutrino}| can have almost no mass, zero charge, and half-integer spin.
types
Electrons {electron neutrino}, muons {muon neutrino}, and taus {tau neutrino} have neutrinos {flavor, neutrino}. Electron neutrinos have masses less than 54000 times electron mass. Muon neutrinos have masses less than 367 times muon mass. Tau neutrinos have masses less than 58 times tau mass. Neutrinos can change into each other, if neutrino mass is greater than 1 eV. Interaction with surrounding matter and energy causes neutrino masses to oscillate from electron to muon to tau neutrinos as they travel.
mass
Fewer neutrinos than expected come from Sun, because they have mass.
forces
Neutrinos do not feel strong force or electromagnetic force, only weak force and gravity. Neutrinos have two orthogonal linear-polarization states at 180-degree angle. Perhaps, weak force does not affect a possible fourth neutrino type {sterile neutrino}.
interactions
Because they have little mass and no charge, neutrinos pass through matter with few interactions. 10^12 neutrinos pass through people each second, because Sun radiation is 10% neutrinos.
antineutrino
Antineutrinos have one-third neutrino cross-sectional area.
Electron antiparticles {positron}| have +1 charge.
Leptons {tau particle}| {tauon} [found in 1975] can be heavier than muons. Tau particles have masses 3519 times electron mass or 1.78 GeV. Electric charge is -1. Lifetime is 0.3 x 10^-12 seconds, because tau can decay to electron. Tau has associated neutrino.
Baryons have units {quark}|. Quarks have no internal structure, have diameter 10^-15 meters, and feel all forces.
types
Up quark has lowest mass, 2 MeV, one-ninth proton mass and nine times electron mass.
Down quark is slightly heavier, 5 MeV, 14 times electron mass.
Strange quark has one-third proton mass, 95 MeV, 1.5 times muon mass. Strange quark is 20 times bigger than up or down quark. Strange quarks are in kaons.
Charmed quark has 1.5 times proton mass, 1.25 GeV, 15 times muon mass. Charmed quarks are in J (psi) particles.
Bottom quark has one-third proton mass, 4.2 GeV, 2.7 times tau mass. Bottom quark is 600 times bigger than up or down quark. Bottom quarks are in B mesons.
Top quark [1995] has 1.5 times proton mass, 171 GeV, 99 times tau mass. Top quark has same mass as osmium.
flavor
Quarks have six flavors: upness, downness, strangeness, charm, topness, and bottomness.
charge
Up, charmed, and top quarks have charge +2/3. Down, strange, and bottom quarks have charge -1/3. The weak interaction has a quantum number T (weak isopin), which has three components. The third component T3 is conserved in all weak interactions (weak isospin conservation law) and in all interactions.
Fermions have spin 1/2. If spin direction and the direction of motion are the same, fermion helicity is right-handed, and spin is counterclockwise +1/2. If spin direction and the direction of motion are the opposite, fermion helicity is left-handed, and spin is clockwise -1/2. Massless particles move at light speed, so all observers see the same helicity. Observers can move faster than massive particles, so such observers see helicity change.
Particles have transformations, some of which (chiral transformations) can be different for left-handed or right-handed particle properties. For example, left-handed fermions have weak interactions, but right-handed fermions do not. Most transformations (vector transformations) are the same for both left-handed and right-handed properties.
Transformations can be symmetric or anti-symmetric, with parity even or odd, respectively. Most transformations involving left-handed and right-handed conserve parity (chiral symmetry), but weak interactions do not.
Left-handed fermions have spin -1/2, have negative chirality, have T = 1/2, are doublets with T3 = +1/2 or -1/2, and so have weak interactions. Right-handed fermions have spin +1/2, have positive chirality, have T = 0, are singlets with T3 = 0, and so never have weak interactions.
Electromagnetism and the weak interaction interact (electroweak). Electromagnetism has electric charges. The weak interaction has gauge bosons W+, W-, and W0. The electroweak interaction has a weak hypercharge Yw that generates the U(1) group of the electroweak gauge group SU(2)xU(1). The (unobservable) gauge boson W0 interacts with weak hypercharge Yw to make (observable) Z gauge boson and photon. For left-handed quarks, Yw = +1/3 or -1/3. [In grand unified theories, weak hypercharge depends on the conserved X-charge and on baryon number minus lepton number: Yw = (5 * (B - L) - X) / 2.]
To make interactions renormalizable, a group of interactions must cancel all asymmetries (anomaly cancellation). The weak interaction has both charge and parity asymmetry, and does not conserve charge or parity, but the electroweak interaction cancels all asymmetries and conserves charge-parity-time (CPT conservation) together ['t Hooft and Veltman, 1972]. This requires that electric charge Q be related to weak isospin T3 and weak hypercharge Yw: Q = T3 + Yw / 2. For left-handed quarks, T3 = +1/2 or -1/2, and Yw = +1/3 or -1/3, so Q = +2/3 or -1/3.
isospin
Quarks are fermions. Up, charmed, and top quarks have weak-isospin third component +1/2. Down, strange, and bottom quarks have weak-isospin third component -1/2. Quarks have no right-handed weak-isospin components.
pairs
Six quarks have three pairs: up and down {up quark} {down quark}, strange and charmed {strange quark} {charmed quark}, and top and bottom {top quark} {bottom quark}.
lifetime
Up quark has infinite lifetime, because it cannot decay to anything. Other quarks can decay to lower-mass quarks. For quark pairs, one can change to the other by emitting W particle or Z particle.
distance
After distance, strong nuclear force stays constant with distance. Inside distance, quarks move freely. After distance, quarks have constant force between them, so they cannot separate. Quarks must be in mesons or baryons. Strong nuclear force makes quarks orbit in shells at relativistic speed.
Perhaps, empty space superconducts color charge and can contain color-charge flux as discrete quanta. Strings between particles are fundamental with derived fields, as in string theory, or color-charge field is fundamental with derived space structure, as in quantum chromodynamics.
leptons
Quarks and leptons are similar. Both are point-like, pair, and have six types.
neutron magnetic moment
Quarks can explain the magnetic moment that zero-charge neutrons have, because quarks have charges.
Quarks have property that uses red, green, or blue {color charge}| to show how they combine to make baryons and mesons, which have no color.
Quark types have one of six properties {flavor, quark}|: upness, downness, strangeness, charm, topness, and bottomness.
Quarks and gluons forced together {color glass condensate} can all have same quantum state, similar to Bose-Einstein condensates, because gluons interact, unlike photons. Color fields randomly orient, to have more stability. Very hot quarks and gluons can form quark-gluon plasma.
Atoms can pair {fermionic condensate} at very cold temperatures, to make superconductors.
Particle accelerators {particle accelerator} are linear accelerator, cyclotron, or synchrotron. Ions colliding with metal can make neutral particles.
Particle accelerators {linear accelerator} can accelerate ions along line, using voltage increments.
Particle accelerators {cyclotron} can accelerate spiral ions in magnetic fields, using oscillating electric fields.
Particle accelerators {synchrotron} can circle ions in timed electric and magnetic fields.
Matter units {atom, matter}| are small and have chemical properties. Atoms have same properties as larger amounts of same element.
types
Most atoms are metals. There are 22 non-metal elemental solids, liquids, and gases.
number
Nature has 90 atoms, and particle accelerators can make more than 13 heavy atoms.
mass
Hydrogen atom has mass 10^-24 grams. Heaviest atom is 250 times more massive.
size
Atoms are 99.99% empty space. Atoms have diameter 10^-8 centimeters. Largest-atom volume is 10 times hydrogen-atom volume.
parts: nucleus
Atoms have positively charged protons and neutral neutrons in orbits at central atom nucleus. Number of protons determines atom properties. Nuclei have diameter 10^-12 centimeters. Protons and neutrons have diameter 10^-13 centimeter.
parts: electrons
Electrons rapidly orbit nucleus at varying distances. Electron mass is 10^-27 grams. In neutral atom, protons equal electrons.
energy
Average kinetic energy equals binding energy. If electromagnetic force is same as now, too-small atoms fly apart, because electron velocities are greater. Too-large atoms cannot exist, because electron velocities are too slow to stay in orbit.
magnetism
Atoms have magnetism, because charges move at relativistic speeds. Most atoms have symmetrical electron and proton arrangements, so magnetic effects cancel. Atoms can have odd numbers of protons and/or neutrons and have net magnetism.
large elements
Carbon nucleus can form from three helium nuclei. Elements higher than carbon can form, because carbon atoms have resonance energy at which three helium nuclei are stable and can add more protons and neutrons.
model
Atom models can have infinite number of linear vibrators, which represent all atom frequencies, momenta, and positions.
Atomic nucleus has mass less than sum of proton and neutron masses {mass defect}, because some mass has become energy.
Atoms have number {atomic number}| of protons.
Atom masses {atomic weight}| {atomic mass} are in atomic mass units.
Atoms have number {mass number}| of protons and neutrons.
Atom centers {nucleus, atom}| have protons and neutrons.
ratio
In the most-massive atoms, neutron number can be up to 1.5 times proton number. In light atoms, neutron number equals proton number.
alpha particles
Light nuclei have alpha particles.
layers
Nuclei lighter than aluminum have no interior and no special surface. Heavier nuclei have surface neutron layer.
shape
Most atomic nuclei are spherical, but some are ellipsoids. If outer shell fills, nucleus is spherical. If outer shell is half-filled, nucleus is ellipsoidal. Spherical and ellipsoidal nuclei can rotate, but other shapes oscillate.
force
Strong nuclear force holds protons and neutrons in nuclear orbits, against electric force repulsions.
force: particle speed
Protons and neutrons have speed 6 x 10^7 meters per second.
force: orbit
Protons have orbits, and neutrons have orbits. Orbits have shells, angular momenta, orientations, and spins.
models
Atomic nuclei can be like charged drops {liquid drop model}, with charge spread evenly throughout. Nuclei can be like radial fields from nucleus center {shell model}.
Nuclei with odd number of protons and odd number of neutrons can break apart {radioactivity}|. Nuclei with even numbers of both protons and neutrons are stable, because orbits are full. Bigger nuclei are less stable, because neutron number is more than proton number. Radioactive decay happens randomly. Temperature, pressure, and other substances do not affect it. However, it can increase above 10^6 K.
Radioactive material takes time {half-life, radioactivity}| to become half as radioactive. Half-life can be several hours to billions of years. Short-half-life isotopes emit high-velocity alpha particles. Long-half-life atoms emit low-velocity alpha particles.
Radioactive nuclei can lose clusters {alpha particle, radiation} with two protons and two neutrons. Paper can stop alpha particles.
Radioactive nuclei can lose electron {beta particle}|. Neutron to proton and electron conversion makes beta particles. Aluminum foil can stop beta particles.
Radioactive nuclei can lose high-energy radiation {gamma particle}|. Five meters of concrete can stop gamma particles.
Devices {Geiger counter} can measure inert-gas ionization in 2000-V potential. Ionization causes current cascade. Current is proportional to ionization.
Devices {proportional counter} can measure gas ionization in 1000-V potential. Current is sensitive to voltage change.
Devices {scintillation counter} can measure sodium-iodide, anthracene, or naphthalene fluorescence. Photomultiplier detects visible light.
Radioactivity detection can use tiny bubbles in saturated fluid {bubble chamber}.
Radioactivity detection can use condensation trails in saturated vapor {cloud chamber}.
Radioactivity detection can combine bubble and spark chamber {streamer chamber}.
Atoms {isotope}| can have same number of protons but different numbers of neutrons. Element isotopes have same physical properties, except for mass differences.
Most isotopes are not radioactive, such as 2H [2 is superscript] {deuterium}.
Isotopes {radioactive isotope}| can be radioactive. Tritium is 3H [3 is superscript]. Carbon-14 is 14C [14 is superscript]. Nitrogen-15 is 15N [15 is superscript]. Phosphorus-32 is 32P [32 is superscript]. Sulfur-35 is 35S [35 is superscript]. Strontium-90 is 90Sr [90 is superscript]. Uranium-235 is 235U [235 is superscript]. Plutonium is 239Pu [239 is superscript].
Because electrons are wave-like, they do not have trajectories but have cloud-like or blurry orbits {orbit, electron} {electron orbit}|. Electron repulsions also spread orbits.
energy
Electron energy has quanta, so electrons have minimum energy. Uncertainty principle requires that energy cannot be zero. Shell, orbital, spin-orbit interaction, and spin angular momentum contribute angular momentum and energy quanta to orbital electrons. Energy levels depend on angular momentum squared.
rotation
Rotations can be spins or orbits. Spins have orientation, frequency, and angular momentum. Orbits have orientation, frequency, angular momentum, and spin-orbit angular-momentum interactions. Spins and orbits have no net linear momentum, because motion is in all directions equally. Rotation is around point or line. Rotation defines plane perpendicular to rotation axis. Rotation axes have orientations in space.
rotation: compared to vibration
Vibrations are oscillations or waves. Vibration is between two extremes. Vibration along length has spatial orientation. Vibration around angle is in plane. Vibrations have frequency. Waves have motion direction. Wave vibration can be transverse to, or longitudinal with, motion direction.
angular momentum
Spins and orbits have angular momentum, because motion is around rotation axis. Orbitals have axis orientation.
width
Orbit width is same as atom diameter, by uncertainty principle. Electrons move all over orbit, by uncertainty principle, but most motion is near shell radius.
independent
Orbitals are orthogonal to all others, with no overlap or interaction, because electrons are fermions and cannot be together in same place (Pauli exclusion principle).
time
Orbitals do not change with time.
speed
Electron orbital speed is 600,000 meters per second and so is not relativistic.
large atoms
For large atoms, inner electrons shield outer electrons from atomic nucleus, so outer electrons have orbits farther from nucleus and have less kinetic energy than with no shielding.
Electrostatic force between nucleus and electron causes electrons to orbit atomic nuclei in main regions {shell, atom}| {atomic shell} at specific distances. Atoms have up to seven shells, from one to seven unit distances from nucleus.
energy
Electron kinetic energy E depends on reciprocal of shell number n squared: E = 1 / n^2. For first shell, n = 1 and E = 1/1 = 1 unit. For second shell, n = 2 and E = 1/4 = 0.25 unit. For third shell, n = 3 and E = 1/9 = 0.11 unit. For fourth shell, n = 4 and E = 1/16 = 0.07 unit, For fifth shell, n = 5 and E = 1/25 = 0.04 unit. For sixth shell, n = 6 and E = 1/36 = 0.03 unit. For seventh shell, n = 7 and E = 1/49 = 0.02 unit. Energy levels are closer together at higher shells, because force depends directly on reciprocal of radius squared.
K shell is 10^4 times atomic-nucleus radius. L shell is 1.5 times farther from nucleus than K shell. M shell is 1.67 times farther from nucleus than K shell. N shell is 1.75 times farther from nucleus than K shell.
electrons
Shells farther from nucleus can hold more electrons, because they allow more quanta combinations. Shells can hold 2 * n^2 electrons, where n is shell number. First shell {K shell} can hold two electrons. Second shell {L shell} can hold eight electrons. Third shell {M shell} can hold 18 electrons. Fourth shell {N shell} can hold 32 electrons. Fifth shell {O shell} can hold 50 electrons. Sixth shell {P shell} can hold 72 electrons. Seventh shell {Q shell} can hold 98 electrons.
shell
Atomic-electron orbits have different radii and energy levels {shell, orbital}. From lowest to highest potential energy, and highest to lowest kinetic energy, radius is 1, 2, 3, 4, 5, 6, and 7 units. Orbit radii increase linearly. Units differ for different atoms. Potential energy depends on radius, so quantum energy changes between shells are equal.
wavelength
Smallest orbit has circumference equal to one wavelength. Wavelength depends on radial force and resistance to force. Smallest orbit has highest frequency. Second-smallest orbit has circumference with wavelength equal to two original wavelengths. Second-smallest orbit has half original frequency. Third-smallest orbit has circumference with wavelength equal to three original wavelengths. Third-smallest orbit has one-third original frequency, and so on.
Atom electrons are in shells with orbit types {orbital}|. Orbital can have zero, one, or two electrons.
energy level
Electron orbitals have different energy levels. From lowest to highest, they are one 1s, one 2s, three 2p, one 3s, three 3p, one 4s, five 3d, three 4p, one 5s, five 4d, three 5p, one 6s, seven 4f, five 5d, three 6p, one 7s, seven 5f, five 6d, and three 7p. Number in parentheses is number of possible orbits with that energy. Before using f orbitals, orbital hybridization causes one electron to go into a d orbital.
electronic transitions
Electrons can jump from orbital to higher or lower orbital. Both orbitals must be anti-symmetric to allow angular-momentum conservation. Angular-momentum units are the same for orbiting and spinning.
angular momentum
Same-shell electrons can have different orbital angular momenta {orbital angular momentum, atom}. Angular momentum adds centrifugal force to electrostatic force. Orbital angular momentum has units h / (2 * pi), where h is Planck constant. First shell allows only 0 units. Second shell allows 0 and 1 units. Third shell allows 0, 1, and 2 units. Fourth shell allows 0, 1, 2, and 3 units, and so on.
shape
In shells, orbit shape determines orbital angular momentum. Spherical s orbital allows zero angular momentum. Double-ellipsoid p orbital allows zero or one angular-momentum unit. Quadruple-ellipsoid or double-ellipsoid/torus d orbital allows zero, one, or two angular-momentum units. Octuple-ellipsoid f orbital allows zero, one, two, or three angular momentum units, and so on.
First shell can only have spherical orbital, because it has minimum potential energy and cannot alter. Second shell can have spherical orbital and three oriented orbitals. Shells above first shell can have spherical orbital, three oriented orbitals, and five, seven, and so on, multiply oriented orbitals.
interactions
Orbital orientation and spin orientation interaction changes angular momentum by precession. Spin-axis orientation is always along z-axis. If orbital-axis orientation is along z-axis, no interaction happens, and total angular momentum does not change. If orbital-axis orientation is perpendicular to z-axis, torque interaction {spin-orbit interaction} effects add or subtract angular momentum units. Electric coupling forces cause torque that causes orbital to precess around orbital vertical axis. Spin-orientation interaction can change angular momentum by -3, -2, -1, 0, +1, +2, or +3 units.
Atom electrons are in orbitals {electron configuration}|. Orbitals {degenerate orbital} can have same energy levels.
Electrons fill orbitals from lowest energy to highest energy {Aufbau principle}. Before using f orbitals, orbital hybridization causes one electron to go into a d orbital.
Electrons tend to enter all shell orbitals before they fill any orbital with two opposite-spin electrons {Hund's rule} {Hund rule}. Hund's rule is true for small atoms, because it takes more energy to put two electrons into one orbital than into two different orbitals.
Fermions are electrons, neutrons, protons, and the like. Because fermions have half-unit spins, when identical fermions interchange, their wavefunctions become the negative of the other. Therefore, no two fermions can have same energy quanta {Pauli exclusion principle, fermion}|.
Hund's rule is true for small atoms. Other rules {Slater's rules} {Slater rules} apply for large atoms.
Orbital shape can be spherical {s orbital}, with zero crossing points. Because spheres are radially symmetric, with electron orbits in all directions and so filling space, spherical orbits have no net orientation, so no interaction with spin makes added angular momentum 0. There can only be one kind of spherical orbital, because it must have radial symmetry.
Orbital shape can be double ellipsoidal along straight line {p orbital}, with one crossing point and one rotation axis. p orbital has two elongated lobes along line with one crossing in middle. Double-ellipsoidal orbit can orient in three spatial directions. If axis is along z-axis, aligned with spin, added angular momentum is 0. If axis is along x-axis or y-axis, perpendicular to spin, added angular momentum is -1 or +1. There can only be three kinds of double-ellipsoidal orbital, because one axis can have only three independent spatial orientations, which fill space. For same shell and same orbital angular momentum, all orientations are equally probable and have equal energy. All orientations add to make spherical orbital with zero net angular momentum.
Orbital shape can be quadruple-ellipsoidal four-leaf clover {d orbital}, with two crossing points and two rotation axes. d orbitals have four elongated lobes, two each along both orthogonal lines, with two crossings in middle. Four-leaf-clover quadruple ellipsoidal orbit can align with x-axis and y-axis; between xy-axis, xz-axis, or yz-axis; or with z-axis, as double ellipsoid and torus. If with x and y or between xy, added angular momentum is -2 or +2, because both axes are perpendicular to z-axis. If between xz or yz, added angular momentum is -1 or +1, because one axis is perpendicular to z-axis. If with z, added angular momentum is 0, because axis aligns with spin axis. There can only be five kinds of quadruple-ellipsoidal orbital, because axes can have only five independent spatial orientations, which fill space. For same shell and same orbital angular momentum, all orientations are equally probable and have equal energy. All orientations add to make spherical orbital with zero net angular momentum.
Orbital shape can be octuple-ellipsoidal eight-lobed clover {f orbital}, with three crossing points and three rotation axes. f orbitals have six elongated lobes, with two each along three orthogonal lines, with three crossings in middle. Successive and more complex clover-leaf-shaped orbits can have 7, 9, or 11 distinct orientations. For same shell and same orbital angular momentum, all orientations are equally probable and have equal energy. All orientations add to make spherical orbital with zero net angular momentum.
Elementary particles have intrinsic angular momentum {spin, particle}| {particle, spin} {intrinsic angular momentum}. Spin conserves energy, momentum, and angular momentum.
axis
Particles always travel at light speed along a space-time motion line. Spin axis is parallel to motion line and is either counter-clockwise or clockwise around that space-time momentum vector.
classical mechanics
In classical mechanics, spin has linear continuous projections onto other axes (and orthogonal axes have no spin components). For example, if object spins around z-axis, observers can measure spin around xz-axis and yz-axis, but spin around x-axis and y-axis (both orthogonal to z-axis) is zero.
Fundamental particles are points (or strings or loops with negligible radius), and some have no mass, so fundamental-particle intrinsic angular momentum is not due to mass rotating at a distance around an axis. Classical mechanics cannot account for elementary-particle spin.
quantum mechanics
Elementary-particle spin is quantum-mechanical and special relativistic. To reconcile quantum mechanics and special relativity, quantum-mechanical-wavefunction components are matrices, not just numbers. Matrices have transformations that are equivalent to spin angular momentum. Reconciling quantum mechanics and general relativity requires that momentum (energy) and position (time) affect each other, so matrices have complex-number elements.
In quantum mechanics, observers can measure spin around any axis. Measurement of elementary-particle spin around any axis finds that spin is an angular-momentum quantum unit, either clockwise or counterclockwise around axis. For example, measuring independent-electron intrinsic angular momentum finds spin equals (0.5 * h) / (2 * pi), where h is Planck constant, which is 1/2 angular-momentum quantum unit. (Electron spin cannot be zero, because electrons have mass.) Spin counterclockwise around motion axis adds 1/2 angular momentum unit, so spin is +1/2. Spin clockwise around motion axis subtracts 1/2 angular-momentum unit, so spin is -1/2.
Measuring independent-photon intrinsic angular momentum finds spin equals (0.5 * h) / pi, where h is Planck constant, which is 1 angular-momentum quantum unit. (Photon spin cannot be zero, because photons have energy.) Spin counterclockwise around motion axis adds 1 angular-momentum unit, so spin is +1. Spin clockwise around motion axis subtracts 1 angular-momentum unit, so spin is -1.
spin: vectors and spinors
Real-number vectors have magnitude, one direction (component), and one orientation (in that direction): (a). Rotating real-number vectors 360 degrees makes the same vector, because vector direction and orientation return to original direction and orientation. Spinning real-number vectors any number of degrees makes the same vector, because vectors have no extensions in perpendicular directions. For example, turning a straight line around its axis keeps the same shape.
Complex-number vectors have magnitude, one direction (in local two-dimensional space), and one orientation (in that direction): (a + b*i). Rotating complex-number vectors 360 degrees makes the same vector, because vector direction and orientation return to original direction and orientation. Spinning complex-number vectors any number of degrees makes the same vector, because vectors have no extensions in perpendicular directions.
Spinors have two complex-number (or quaternion) components: (a + b*i, c + d*i). Spinors have magnitude, two directions, and one orientation that depends on which component goes first. Rotating spinors 360 degrees makes original direction but opposite orientation, like rotating around a Möbius strip, because parity changes. Spinor rotation differs from vector rotation because spinor rotation has phase effects. Spinning spinors any number of degrees makes a different spinor, because spinors have extensions in perpendicular directions.
spin: rotation
Fermion odd-half-integer-spin particles have different statistics than boson integer-spin particles. For bosons, spin and rotation are independent and add. For fermions, spin and rotation are dependent and multiply.
spin: symmetries
Elementary-particle intrinsic angular momentum is about wavefunction symmetries.
Spin-0 particles are scalars (not vectors). Scalars have no direction and so have same physics under any rotation. Because intrinsic angular momentum is zero, clockwise and counterclockwise have no meaning. Spheres have all symmetries: any-degree rotational symmetry, mirror symmetry, radial symmetry, and inversion symmetry. Turning a sphere through any angle, reflecting it through any plane through any diameter, and spinning around any axis results in the same shape and behavior. Around any axis and orientation, observers see no net spin, so spin-rotation interaction is zero. See Figure 1.
Spin-1 particles are vectors, with one symmetry axis. Spin-rotation interaction is non-zero, so observers see opposite spin (anti-symmetry) after 180-degree rotation. Turning a clockwise spinning sphere upside down reverses its orientation and changes clockwise to counterclockwise. Vectors have same physics under 360-degree (and 720-degree, 1080-degree, and so on) rotation (360-degree rotational symmetry). Turning the sphere upside down again puts it back to original orientation and clockwise spin. See Figure 2.
Spin-2 particles are tensors, with two symmetry axes. Spin 2 particles have mirror symmetry. Spin 2 has 90-degree anti-symmetry. Turning the sphere to right angle interchanges axes, so one axis keeps clockwise motion and one axis changes from clockwise to counter-clockwise, reversing the orientation. Two spin-rotation interactions are non-zero but symmetric, so flipping plane over returns system to same spin-rotation interactions. Spin-2 particles have same physics under 180-degree (and 360-degree, 540-degree, 720-degree, and so on) rotation. Turning a sphere spinning clockwise around an axis and clockwise around a perpendicular axis upside down changes clockwise to counterclockwise around both axes but also reverses both axes, so the sphere returns to its original state. See Figure 3.
Spin-1/2 particles are vectors, with two axes sharing one symmetry. Because they share one symmetry, spin-1/2 particles have different spin-rotation interactions than vector bosons, which have no shared symmetry and so spin 1. Spin-rotation interaction is perpendicular at 180-degree rotation, reversed at 360-degree rotation, and opposite perpendicular at 540-degree rotation, and original at 720-degree rotation. Spin 1/2 particles have 360-degree anti-symmetry, like rotating around a Möbius strip, changing parity. Turning a sphere spinning clockwise around an axis, clockwise around a perpendicular axis, and clockwise around a second perpendicular axis completely around changes clockwise to counterclockwise around two axes and reverses both axes, but changes clockwise to counterclockwise around the third axis, which has the same orientation, so the sphere reverses orientation. Spin 1/2 has 720-degree rotational symmetry. Turning the sphere completely around again changes clockwise to counterclockwise around two axes and reverses both axes, but changes counterclockwise to clockwise around the third axis, which has the same orientation, so the sphere returns to original state. See Figure 4.
spin: speculation
Perhaps, elementary-particle intrinsic angular momentum is imaginary-number mass rotating at imaginary-number radius around particle axis, through imaginary-number angle with imaginary-number angular velocity, perhaps through imaginary-number time. Multiplying imaginary numbers results in positive real-number momentum and energy. Hyperbolas have imaginary-number radii, because they have negative curvature. Hyperbolic-curve angles are imaginary-number angles: cos(i*A) = cosh(A) and e^A = cosh(A) + sinh(A), where A is real-number angle. Higgs field has imaginary mass. Imaginary-number time rotations make special-relativity Lorentz transformations. Using imaginary-number time can establish absolute general-relativity space-time.
spin: bosons and fermions
At high concentration and/or low temperature, with Heisenberg uncertainty, for thermal-equilibrium non-interacting bosons, exchange of two particles does not change wavefunction (Bose-Einstein statistics), because particle wavefunction product is commutative (symmetric rank-two tensor): f(a) * f(b) - f(b) * f(a). Combining two spins returns the system to original orientation: f(a) * f(b) = ((-1)^(2*spin)) * (f(b) * f(a)), where spin = +1 or -1. Relativistically applying a rotation operator in imaginary time to integer spin particles results in no Pauli exclusion principle. Bosons are indistinguishable. Only system states matter. It is incorrect to talk about first one and second one, or particle 1 and particle 2. Many bosons can have same energy, momentum, and angular momentum.
At high concentration and/or low temperature, with Heisenberg uncertainty, for thermal-equilibrium non-interacting fermions, exchange of two particles changes wavefunction (Fermi-Dirac statistics), because particle wavefunction product is anti-commutative (anti-symmetric rank-two tensor): f(a) * f(b) + f(b) * f(a). Combining two spins takes the system to opposite orientation: f(a) * f(b) = ((-1)^(2*spin)) * (f(b) * f(a)), where spin = +1/2 or -1/2. Relativistically applying a rotation operator in imaginary time to half-integer spin particles results in Pauli exclusion principle. Fermions are distinguishable. Only system states matter. It is correct to talk about first one and second one, or particle 1 and particle 2. Two particles can have same energy but must have different momentum and/or angular momentum.
Note: At low concentration and/or high temperature, without Heisenberg uncertainty, thermal-equilibrium non-interacting particles have Maxwell-Boltzmann statistics. Exchange of two particles does not matter, because wavefunction has no effect. Particles can have same energy and same or different momentum and angular momentum.
spin: measurement
To measure spin, experimenters must establish a spatial axis, and then measure angular momentum around that axis. (Experimenters cannot know electron trajectories, because electrons have wavefunctions.) Around any chosen axis, instruments measure spin as exactly +1/2 unit or exactly -1/2 unit. By uncertainty principle, instruments measuring spin simultaneously around axes perpendicular to that axis get +1/2 unit or -1/2 unit with equal probability, meaning that those spin measurements have 100% uncertainty.
Instruments cannot measure spin when two electrons are interacting, because system then includes measuring apparatus. Instruments measure after particle creation or interaction. After particle creation or interaction, instruments decohere wavefunction and so destroy particle system and make particles independent.
spin: measurement angle
For electrons (spin 1/2), if measuring axis is at angle A to a clockwise spin-vector (spin -1/2), the probability that the measurement will be spin -1/2 is (cos(A/2))^2. Perhaps, because spin-vector has two axes but shares one symmetry, it is like the spin-vector projects onto an angle A/2 axis as cos(A/2), and the angle A/2 axis vector projects onto the angle A measuring axis as cos(A/2), so the net projection is (cos(A/2))^2.
If a zero-spin state emits entangled electrons in opposite directions (conserving momentum and angular momentum), and one direction is measured at angle A and the other at angle B (with angle difference C), the both-same-spin probability is (sin(C/2))^2, and the each-opposite-spin probability is (cos(C/2))^2.
For photons (spin 1), if measuring axis is at angle A to a clockwise spin-vector (spin -1), the probability that the measurement will be spin -1 is (cos(A))^2. Perhaps, because spin-vector has one axis, it is like the spin-vector projects onto an angle A axis as cos(A) twice, so the net projection is (cos(A))^2.
If a zero-spin state emits entangled photons in opposite directions (conserving momentum and angular momentum), and one direction is measured at angle A and the other at angle B (with angle difference C), the both-same-spin probability is (sin(C))^2, and the each-opposite-spin probability is (cos(C))^2.
orbitals
Orbitals with two electrons typically have one electron with positive spin and one electron with negative spin {anti-symmetric spin state}, so net spin angular momentum is zero, and ground-state orbital is symmetric. In orbitals, paired electron spins {spin pair} cancel magnetic fields.
Outside energy can add spin angular momentum. The first excited orbital state has two electrons with positive spin or two electrons with negative spin {symmetric spin state}. Net spin angular momentum is 1, and excited-state orbital is anti-symmetric.
In orbitals, two electrons have probability 0.25 to have total spin 0 and 0.75 to have total spin 1.
In different orbitals, electrons can have same lower-energy spins. Two electrons enter two different orbitals before going into same orbital, because electrostatic repulsions are greater in energy than magnetic interactions, energy differences between orbitals are small, and repulsions between electrons in different orbitals are smaller than repulsions in same orbital.
Electron has spin and can precess {spin dragging}| or move in electric fields.
Low-temperature materials can behave like ice {spin ice}|. Magnetic poles can become unaligned.
Atom electrons have coupling {spin-orbit coupling} {Russell-Sanders coupling} {jj coupling} between orbit and spin magnetic fields.
Elements have unique electron configurations around atomic nucleus. Element electron configurations have groups and sequences {periodic table}|, from smallest to largest.
columns
Named columns are alkali metal, alkaline earth metal, chalcogen, halogen, and noble gas.
rows
First row has lightest elements, with electrons in first electron shell, 1s: elements 1 and 2.
Second row has common light elements with electrons in second electron shell, 2s and 2p: elements 3 to 10.
Third row has less common elements with electrons in third electron shell, 3s and 3p: elements 11 to 18.
Fourth row has elements with electrons in third and fourth electron shells, from 19 to 36.
Fifth row has elements with electrons in fourth and fifth electron shells, from 37 to 54.
Sixth row has elements with electrons in fifth and sixth electron shells, from 55 to 86.
Seventh row has elements with electrons in sixth and seventh electron shells, from 87 to 118.
large atoms
Uranium is element 92 and is the largest natural element. Manmade elements go up to 116, but as of 2011 people have not yet made elements 113 and 115. Neptunium is element 93. Plutonium is element 94. Americium is element 95. Curium is element 96. Berkelium is element 97. Californium is element 98. Einsteinium is element 99. Fermium is element 100. Mendelevium is element 101. Nobelium is element 102. Lawrencium is element 103.
6d orbital
Rutherfordium is element 104. Dubnium is element 105. Seaborgium is element 106. Bohrium is element 107. Hassium is element 108. Meitnerium is element 109. Darmstadtium is element 110. Roentgenium is element 111. Copernicium is element 112.
7p orbital
Ununtrium (not made as of 2011) is element 113. Ununquadium is element 114. Ununpentium (not made as of 2011) is element 115. Ununhexium is element 116. Ununseptium (not made as of 2011) is element 117. Ununoctium (not made as of 2011) is element 118.
orbitals
1s orbital has H and He.
2s orbital has Li and Be.
2p orbital has B, C, N, O, F, and Ne.
3s orbital has Na and Mg.
3p orbital has Al, Si, P, S, Cl, and Ar.
4s orbital has K and Ca.
3d orbital has Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, and Zn.
4p orbital has Ga, Ge, As, Se, Br, and Kr.
5s orbital has Rb and Sr.
4d orbital has Y, Zr, Nb, Mo, Tc, Ru, Rh, Pd, Ag, and Cd. Tc 43 is not in nature.
5p orbital has In, Sn, Sb, Te, I, and Xe.
6s orbital has Cs and Ba. 5d orbital has La.
4f orbital has Ce, Pr, Nd, Pm, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, and Lu. Pm 61 is not in nature.
5d orbital has Hf, Ta, W, Re, Os, Ir, Pt, Au, and Hg. 6p orbital has Tl, Pb, Bi, Po, At, and Rn. At 85 is not in nature.
7s orbital has Fr and Ra. Fr 87 is not in nature.
6d orbital has Ac.
5f orbital has Th, Pa, U, Np, Pu, Am, Cm, Bk, Cf, Es, Fm, Md, No, and Lr. Np, Pu, Am, Cm, Bk, Cf, Es, Fm, Md, No, and Lr {actinoid} are not in nature.
6d orbital has Rf, Db, Sg, Bh, Hs, Mt, Ds, Rg, and Cn and are not in nature.
7p orbital has Uut, Uuq, Uup, Uuh, Uus, and Uuo {transactinide element} {super heavy element} and are not in nature.
Stable artificial elements have a number {magic number} of protons or neutrons. Some {doubly magic isotope} have special numbers of both protons and neutrons. Lead-208 has 82 protons and 126 neutrons and is doubly magic. Elements 114, 120, or 126 can be doubly magic, with 184 neutrons.
Periodic table has columns {chemical group}|. Periodic table has 18 columns, one for each orbital.
Leftmost column or 1A column has soft metals {alkali metal} with low densities and melting points.
Second-from-left column or 2A column has harder, higher-density, and higher-melting-point metals {alkaline earth metal}.
Third-from-right column or 6A column has reactive elements {chalcogen} with slight colors.
Second-from-right column or 7A column has colorful and highly reactive gases, liquids, and solids {halogen}|.
Rightmost column or 8A column has inert colorless gases {noble gas}|.
Fourth and fifth rows have reactive elements {transition metal}, with many ionic forms, whose outermost electrons are in d orbitals, not in higher p orbitals. Metals in columns 4 to 16 have 10 d electrons.
Sixth row has elements {inner transition metal} with one or two electrons in d orbital and outermost electrons in f orbitals. Lanthanides and actinides {rare earth}, as well as scandium and yttrium, are solids.
First inner-transition-metal row {lanthanide series}, from element 58 to 71, is solids.
Second inner-transition-metal row {actinide series}, from element 90/91 to 103, is solids.
Large nucleus can split into two nuclei {fission, physics}| {nuclear fission}. Fission releases million times more energy per mass than burning. In nuclear reactions, neutrons collide with uranium or plutonium nuclei to cause fission.
Neutron can decay into proton, electron, and anti-neutrino {beta decay}| {beta radiation}. Beta decay causes nucleus to lose neutron and gain proton.
Nuclear reactors {breeder reactor} can use neutrons from fission to form plutonium from uranium.
Electron and positron collision {electron-positron collision} makes two real photons, positive pion and negative pion, proton and anti-proton, or virtual photon that becomes rho vector meson that makes two pions. Process must make two particles to conserve energy and momentum.
High-energy photon and atomic nucleus can collide to make electron and positron {pair production}. Protons and neutrons absorb photons 200 times less than hyperons.
Particle decays {decay, particle} {particle decay}| always make two particles, to conserve energy and momentum.
Proton and proton collisions {proton-proton collision} at high energies make larger subatomic particles. Scattering happens if both protons have same spin, but not if protons have opposite spins.
Particles can collide and rebound {scattering, collision}|.
path
In gas, particles go average distance, through mean free path, before they hit another particle.
elastic
Both particles can collide, bounce off, and remain intact, with no new particles {elastic scattering}.
inelastic
Both particles can collide to make new particles {inelastic scattering}. Created particles go off in pairs in jets perpendicular to colliding-particle paths. Increased amplitude at collision resonance energy indicates particle creation at that mass.
Small particles scatter through wider angles than larger particles, because cross-sectional area is less. Cross-sectional area increases with energy.
particle size
Particles have minimum diameter at 70 to 300 MeV. Particles grow rapidly in diameter up to at least 1500 MeV. At collision energy 2 GeV, particles reach maximum diameter.
Crystals exposed to radioactivity trap electrons in crystal faults. By heating material, luminescence {thermo-luminescence} measures number trapped. Thermo-luminescence can date from recent times to hundreds of thousands of years ago. Electron-spin resonance also measures number trapped.
Two small nuclei can merge into one nucleus {fusion, physics} {nuclear fusion}|. Fusion releases million times more energy per mass than burning.
products
Nuclear fusion makes all atoms up to and including iron.
efficiency
Nuclear hydrogen fusion to helium makes 0.007 of mass into energy, so efficiency is 0.007. Other fusions make 0.017 of mass into energy. If efficiency is less, universe has no or less helium and heavy atoms. If efficiency is more, universe has more helium and heavy atoms, but no hydrogen. Carbon production also depends on ratio, because it involves resonance energy.
Main fusion reaction {proton-proton cycle} unites two protons. In stars, hydrogen fusion to helium requires 10^6 K. Two protons change to deuterium and proton. These two nuclei combine to make helium 3. Two helium 3 make helium 4 and two protons.
The second-most-important fusion reaction {carbon-nitrogen cycle} makes helium starting from protons and carbon. Carbon acts like catalyst to make lithium, beryllium, and boron, which combine or decay to helium. Carbon-nitrogen cycle is not chain reaction.
Reactions {chain reaction, fusion}| that have proton reactants and make protons can be self-sustaining. Chain reaction continues until limiting reactant amount becomes zero or system disrupts physically.
Minimum mass {critical mass} starts chain reactions. Below minimum mass, too many proton initiators do not collide and escape to outside.
Absorbing protons {damping} slows fusion reactions. In nuclear reactors, metal rods absorb proton initiators to slow reaction.
Non-reactive gases {inert gas}| can have full electron shells.
Common metal atoms {metal atom}, in order of increasing mass, are lithium, sodium, magnesium, aluminum, potassium, calcium, titanium, chromium, manganese, iron, cobalt, nickel, copper, zinc, molybdenum, silver, cadmium, tin, cesium, barium, tungsten, platinum, gold, mercury, lead, radium, and uranium. Metals are shiny, crystalline, and conductive.
In order of increasing mass, non-metallic atoms {non-metal atom} in first two periodic-table rows are hydrogen gas, helium non-reactive gas, boron solid, carbon solid, nitrogen gas, oxygen gas, fluorine gas, and neon inert gas. Heavier ones are silicon solid, sulfur solid, phosphorus solid, chlorine gas, argon inert gas, germanium solid, arsenic solid, selenium solid, antimony solid, bromine solid, krypton inert gas, iodine solid, and zenon inert gas. Non-metal solids are crystals with various properties.
Increased amplitudes {resonance energy} at frequencies indicate particle masses, which are energy concentrations.
Hydrogen emits light in frequency series {spectra, atomic} {atomic spectra} {line spectrum}.
series
Frequencies 82000 cm^-1 to 110000 cm^-1 {Lyman series} are ultraviolet and start from ground state in shell 1. Frequencies 15000 cm^-1 to 28000 cm^-1 {Balmer series} are visible and start from ground state in shell 2. Frequencies 5000 cm^-1 to 12500 cm^-1 {Paschen series} are infrared and start from ground state in shell 3. Frequencies {Brackett series} can start from ground state in shell 4. Frequencies {Pfund series} can start from ground state in shell 5.
Rydberg formula
Hydrogen spectra, and similar electron-transition energy series, are regular {Rydberg formula}.
cause
Heat energy can put electrons into higher orbitals. Materials emit electromagnetic radiation when electrons fall back to lower orbitals.
temperature
In low-density gas, temperature change changes intensities but not frequencies. Intensity E at frequency is proportional to temperature T to fourth power: E = k * T^4.
density
Dense matter emits continuous frequency spectrum, because molecules interact. Dense-matter spectra depend only on temperature, because temperature determines interactions.
radiation temperature
Light at definite wavelength has definite temperature, because light is kinetic energy. Radiation temperature depends on beam solid angle and intensity, as well as wavelength.
Elements absorb light frequencies {absorption spectra}|.
Absorption lines {Fraunhofer line} of Sun elements make absorption spectrum.
Elements emit light frequencies {emission spectra}|.
Moving charges in atoms make magnetic fields that split spectrum peaks {fine structure}| {fine spectra}. Bigger nuclei make bigger magnetic fields and so make larger fine structure. Spin-orbit coupling and Zeeman effect also contribute to fine structure.
Atoms and molecules have temperature-caused random movements, so emission frequencies shift by Doppler effect {Doppler broadening}. Higher temperature makes more Doppler broadening. Higher mass makes less Doppler broadening. Higher frequency makes more Doppler broadening. Microwaves have lower frequencies than optical waves and so have lower Doppler broadening.
Hydrogen-atom electrons can be in 1s orbital or 1p orbital. Hydrogen-atom 1s-to-1p electronic transition has the smallest electronic-transition energy, equivalent to microwave photons. Microwaves have lower frequencies than optical waves and so have smaller Doppler broadening. This system is optimum to measure the fine-structure constant. Microwaves excite hydrogen-atom same-spin electrons from 1s to 1p orbitals {Lamb shift, electron} [1947] (Willis E. Lamb, Jr., and Robert Retherford) (Hans Bethe) to measure the fine-structure constant, which indicates virtual photons.
Electrons in outermost atom orbitals can jump to orbital with higher or lower energy level {electron transition}| {electronic transition} {transition, electron}, if new orbital is not full. Lower-energy orbital electron acquires energy from photon to go to higher-energy orbital. Higher-energy orbital electron loses energy to photon to fall to lower-energy orbital.
time
Collision, radiation, and other energies can send electron to higher-energy orbital in atom in 10^-12 seconds. Electron takes 10^-8 seconds to return to lower-energy orbital, emitting photon. Electronic transitions are random.
channel
Transition from one energy level to another emits or absorbs photons with quanta. Electronic transition can be direct and take one step {direct channel, transition} or go through intermediate steps {cross channel, transition}.
Electronic transitions naturally happen between orbitals differing by one angular-momentum unit {allowed state}, because photon carries that amount.
Transitions take longer to happen between certain orbits {forbidden state}|, because they differ by several angular-momentum units and one photon can carry only one unit.
Strong electric field can shift rotational-frequency lines {Stark effect}.
External magnetic field causes atom electrons to align and splits electron-energy level into slightly higher and slightly lower levels {Zeeman effect}. Magnetic field displaces spectral lines.
In space regions, points can have variable values and directions {vector field, physics}. For example, points have force and momentum. Scalar-field gradients are vector fields, because gradients have direction and magnitude. Vector fields have gradients, flows, constancies, covariances, contravariances, divergences, curls, and Laplace operators. Total effect of variable over region is vector sum. For example, force-vector sum gives total force.
In space regions, points can have variable values {scalar field, physics}. For example, points have mass density, temperature, and position. Total effect of variable over region is scalar sum. For example, summing mass densities gives total mass.
Vector fields can have complex numbers, instead of real numbers, for vector-component coefficients {spinor field}. Spinor fields require twice the dimension number of corresponding vector fields, because complex numbers have real and imaginary components. Spinor spaces have even number of dimensions.
Moving vector fields can expand outward from points to make waves that superimpose {wave front}|. Wave-front component sums indicate net direction and amplitude.
Differential vector gauge field {Yang-Mills field} for strong and weak nuclear forces can be invariant under transformation {Yang-Mills gauge theory}. Energy increases when reference frame carried around loop does not return to original orientation. Gauge fields can have more than one dimension. Unified field theories require Yang-Mills fields [Yang and Mills, 1954].
Motion physics {kinetics} is about distance, time, speed, and acceleration.
Objects can move from one space position to another {displacement, motion}|. Displacement has direction and amount and so is vector.
velocity
Displacement s equals average velocity v times time t: s = v*t. Distance change ds equals constant velocity v times time change dt: ds = v * dt.
Displacement s equals initial velocity vi times time t plus one-half acceleration a times time t squared: s = vi * t + 0.5 * a * t^2. Distance change ds equals initial velocity vi times time change dt plus half acceleration a times square of time change dt during acceleration: ds = vi * dt + 0.5 * a * dt^2.
graph
If acceleration is zero, displacement versus time is straight line. If acceleration is positive, displacement versus time curves up.
If vertical velocity {escape velocity}| is great enough, objects can overcome gravity and go into orbit or keep moving away from Earth.
Curved motion {trajectory, motion}| results if object velocity has components in different directions. Thrown balls have trajectories, because one velocity is from throwing and one velocity is from gravity.
angle
Maximum horizontal distance results from throwing ball at 45-degree angle to horizontal. Ball thrown at an angle, and ball thrown same speed at complementary angle, travel same distance horizontally. Balls thrown at 30 degrees and 60 degrees travel same distance horizontally.
speed
If air resistance is zero, speed that ball has when it comes down is same speed that it had when it starts up.
top
Under gravity, at trajectory top, horizontal acceleration and vertical velocity are zero.
Linear motion can be sum of two circular motions, as in devices {Tusi-couple}.
Motion {velocity}| involves going from one space position to another, over time. Velocity has speed and direction and so is vector. Instantaneous velocity v is distance change ds divided by time change dt: v = ds / dt. Average velocity v is position change s divided by time t: v = s/t.
Final velocity vf equals initial velocity vi plus constant acceleration a times time change dt during acceleration: vf = vi + a * dt.
Final velocity vf squared equals initial velocity vi squared plus two times constant acceleration a times distance over which acceleration applies ds: vf^2 = vi^2 + 2 * a * ds.
types
Translation is in straight lines. Oscillation is back and forth. Spin is around object axis or point. Orbit is around point or axis outside object. Spins and orbits are rotations. Electronic transition is from one orbit to another, in atoms or molecules.
examples
When it is raining, to be less wet, run through rain instead of walking, to hit more drops per second but for fewer seconds.
graph
If acceleration is zero, graph of velocity versus time is a horizontal line. If constant acceleration is positive, velocity versus time is a rising straight line. If constant acceleration is negative, velocity versus time is a falling straight line.
Gas volume is inversely proportional to pressure {Boyle's law} {Boyle law}, if temperature is constant.
Gas volume is directly proportional to temperature {Charles' law} {Charles law}, if pressure is constant.
Abstract gases {ideal gas}| have infinitely small particles, with no interactions except for elastic collisions. Gas density is directly proportional to pressure. Real gases have lower pressure than ideal gas at low pressure, because atomic attractions are more. Real gases have higher pressure than ideal gas at high pressure, because atomic repulsions are more.
Work kinetic energy equals heat kinetic energy {ideal gas law, kinetics}|: P*V = n*R*T, where P = pressure, V = volume, n = moles, R = gas constant, and T = absolute temperature.
Motions {translation, motion}| can be in straight lines.
Motions {uniform motion}| can cover equal distances in equal times or over equal values.
Motions {rectilinear motion}| can be in straight lines.
Two motion components, in different directions, can change motion direction {curved motion}|. Typically, one component is tangential to curve, and one component is normal to curve. Distance {arc length, motion} traveled along curve depends on curvature, which depends on curvature radius r and angle A subtended by arc: arc length = r * A.
Motion can involve speed and/or direction change {acceleration}|. Acceleration has amount and direction, so acceleration is vector. Acceleration a is velocity change dv divided by time change dt: a = dv / dt. Gravity acceleration g is 9.8 meters per second per second: g = 9.8 m/s^2.
Acceleration can change over time {jerk}|: j = da/dt.
Jerk can change over time {jounce} {snap}: dj/dt.
Motion can be through angle around point or axis {rotation, motion}.
comparison
All linear distance, velocity, acceleration, and time relations are true for angular counterparts.
vectors
Angular quantities are vectors, perpendicular to curve plane. If right-hand fingers point in motion direction, vector points in thumb direction.
examples
Top, gyroscope, wheel, gears, banked track, and airplane dive and turn illustrate angular motion.
universe
Rotation is not relative but is absolute against distant-galaxy and universe reference frame.
Motions {spin, object}| can be around object axis or point.
Objects can move around points or axes {revolution, physics}| {orbit, revolution}. Object comes back to starting point after angle 360 degrees (2*pi radians), after traveling circumference distance.
Speed can be constant around circumference {uniform circular motion}|.
Angular speed w and/or direction can change over time t {angular acceleration}|: a = dw / dt. Angular acceleration a depends on angle A passed per second per second: a = (d^2)A / dt^2, where (d^2) is second derivative, and d is derivative.
Angle distance {angular distance}| {total angle} A equals current angular distance A0 plus current angular velocity w times time t plus one-half times angular acceleration aa times time t squared: A = A0 + w * t + 0.5 * aa * t^2, whish is analogous to linear distance equation.
Rotation velocity {angular velocity}| w, in radians per second, is angle change A per time unit t: w = dA / dt. Average angular velocity w equals 360 degrees (2*pi radians) divided by period T: w = 2 * pi / T. Average angular velocity w equals 360 degrees (2*pi radians) times frequency f: w = 2 * pi * f.
A number of orbits or revolutions happens over time {frequency, physics}|. Frequency f is period-T reciprocal: f = 1/T. For example, electric current alternates at 60 cycles per second in USA.
Acceleration {radial acceleration}| can be along perpendicular to curve. For circular motion, object pulls back toward center to make circle, and radial acceleration ar equals tangential velocity vt squared divided by radius r: ar = vt^2 / r. If radial acceleration is more, orbit is ellipse. If radial acceleration is less, orbit is spiral.
Motions {radial velocity}| can be along perpendiculars to curves. Radial velocity equals zero for circular motion, because distance from circle center is constant.
If object rotates around point or axis, object makes one complete revolution during time {period, rotation} {rotation period}|. Complete revolution sweeps through angle of 360 degrees (2*pi radians) and travels circumference distance. Period T is frequency f reciprocal: T = 1/f.
Acceleration {tangential acceleration}| can be along tangent to curve. Tangential acceleration at equals angular acceleration a times curvature radius r: at = a * r.
Motions {tangential velocity}| can be along tangents to curves. Tangential velocity vt equals angular velocity w times curvature radius r: vt = w * r.
Objects can rotate around horizontal axis perpendicular to motion axis {pitch, motion}|. Airplanes can pitch around wings, horizontal to body.
Objects can rotate around motion axis {roll, rotation}|. Airplanes can roll around airplane body.
Objects can rotate around vertical axis perpendicular to motion axis {yaw}|. Airplanes can yaw around tail, vertical to body.
Motion can be back and forth {vibration, motion}| {oscillation}.
period
Vibrations take time to complete one vibration.
frequency
Vibrations have number of vibrations per time unit. Period T relates to frequency f: f = 1/T.
wavelength
Moving vibration travels distance during one period.
velocity
Movement velocity v equals wavelength l times frequency f: v = l*f. Vibration velocity maximizes at center. Vibration velocity is zero at maximum displacement.
acceleration
Acceleration is zero at center. Acceleration maximizes at maximum displacement.
displacement
During vibration, object is at distance from equilibrium or center point. Amplitude is maximum displacement. Period does not depend on amplitude. Large amplitudes have large acceleration, and small amplitudes have small acceleration, so period stays the same.
phase
Two vibrations can have same angle for same displacement {in phase} or not {out of phase}.
rotations
Vibrations are similar to rotations but are back and forth, instead of around axis. Rotation looks like vibration if viewed from orbital plane.
trigonometric function
Sine or cosine functions can model vibration. Sines and cosines have varying displacement, which has maximum amplitude. Sines and cosines have varying phase angle. Angle A equals frequency f times time t times 360 degrees expressed in radians 2*pi: A = 2 * pi * f * t. If time is zero, angle is zero. If time is period, angle is zero. If time is half period, angle is 180 degrees, and sine is zero. If time is one-quarter period, angle is 90 degrees, and sine is one. Sine equals zero if angle is zero. Sine maximizes if angle is 90 degrees.
Angle A equals displacement x divided by wavelength l times 360 degrees expressed in radians 2*pi: A = 2 * pi * x / l. If displacement is zero, angle is zero. If displacement is wavelength, angle is zero. If displacement is half wavelength, angle is 180 degrees, and sine is zero. If displacement is one-quarter wavelength, angle is 90 degrees, and sine is one.
Displacement x equals amplitude A times sine: x = A * sin(2 * pi * f * t) or A * sin(2 * pi * x / l). Vibrations can shift angle: x = A * sin(2 * pi * f * t + Ao), where Ao is starting angle.
string vibration
Vibrating strings are stationary waves and have partial differential equations. Second partial derivative of function y with respect to time t equals constant (a^2) times second partial derivative of function y with respect to distance x. (D^2)y / Dt = (a^2) * (D^2)y / Dx, where (D^2) is second partial derivative, D is partial derivative, a is constant, t is time, x is distance, and y is function of time and distance. To be dimensionless, constant a equals period T in seconds divided by seconds: a = T / second. Because endpoints are stationary, function y at x = zero equals zero: y(t,0) = 0. Function y at x = one wavelength equals zero: y(t,1) = 0. For stationary waves, partial derivative of y with respect to time t, at t equals zero, equals zero: Dy(0,x) / Dt = 0. Function y at t = zero equals function of x: y(0,x) = f(x), which is odd and periodic.
Sine or cosine function has varying magnitude, which can have maximum {amplitude, vibration}|.
Like rotation, vibration has time {period, vibration}| to complete one vibration.
Sine or cosine function has varying angle {phase, vibration}|.
study of forces {dynamics}.
In a constant-force field, particles take shortest time and shortest distance, to minimize force F times distance change ds times time change dt {action, physics}|: F * ds * dt. Action tends to minimize {principle of least action, physics} {least-action principle}. Particles take shortest space-time path. Particle trajectories follow geodesics. Particles take least-resistance path.
In quantum mechanics, action has quanta, which have size Planck constant h. Action values are multiples of h.
Energy is force times distance change, so F * ds * dt = dE * dt. For energy, action is energy change times time change: dE * dt. By the least-action principle, particle trajectories take the shortest time with least energy change. Total action is sum of kinetic-energy KE to potential-energy PE difference over time dt: integral of (KE - PE) * dt. Least action over time makes conservation of energy.
Energy change dE is force F times distance change ds: dE = F * ds. Force F is momentum change dp divided by time change dt: F = dp / dt. Therefore, energy change dE times time change dt equals momentum change dp times distance change ds: dE * dt = F * ds * dt = dp * ds. For momentum, action is momentum change times distance change. By the least-action principle, particle trajectories take the shortest path length with least momentum change. Least action over translation makes conservation of momentum.
Tangential momentum p is angular momentum L divided by radius r: p = L / r, so dp = dL / r. Distance s is radius times angle A in radians, so ds = r * dA. Therefore, momentum change dp times distance change ds equals angular-momentum change dL times angle change dA in radians: dp * ds = (dL / r) * (r * dA) = dL * dA. For angular momentum, action is angular-momentum change times angle change. By the least-action principle, particle trajectories take the shortest rotation with least angular momentum change. Least action over rotation makes conservation of angular momentum.
If force varies, action minimizes dF * ds * dt. For force, action is force change times space-time change. By the least-action principle, particle trajectories take the geodesic with least force change. Least action over space-time makes force zero or flat space-time.
Over an instant or at a point or through an infinitesimal angle, action is zero or a limiting value.
Two colliding objects {collision}| have compressive force. Initial positions and velocities cannot be precise, but deviations are small, so future behavior mostly predictable. For collisions between three objects simultaneously, future behavior can deviate far from predicted behavior. If first two objects collide first, followed by third object, resulting motions can be much different than if last two objects collide first, followed by first object.
If two dense hard objects collide, all energy stays in motion, no heat is made, and objects bounce off each other {elastic collision}|. In elastic collisions, forces are equal and opposite, and momentum (m*v) before and after collision is constant: m1 * v1 = m2 * v2. In elastic collisions, with no heat, total energy E is kinetic energy KE plus potential energy PE and stays constant: E = KE1 + PE1 = 0.5 * m1 * v^2 + m1 * g * h1 = 0.5 * m2 * v^2 + m2 * g * h2 = KE2 + PE2.
examples
Superballs are denser and harder than regular balls. Karate experts try to make elastic collisions, rather than inelastic collisions, to break objects. Atomic-particle collisions are elastic, because they do not deform.
If either colliding object is soft or has low density, collision {inelastic collision}| permanently deforms surface, some collision energy becomes heat, and objects can stick together.
If physical-system coordinates transform, some physical properties remain unchanged {conservation laws}|.
fermions
all same-type fermions are identical. For example, all electrons are identical. Physical laws are symmetric for fermion replacement with same-type fermion.
mass
For non-relativistic conditions, mass stays constant. For example, mass does not change in chemical reactions. However, physical laws are not symmetric with respect to matter-antimatter for weak force.
baryon number
Baryon number stays constant
lepton number
Lepton number stays constant.
parity
Parity conserves, except for weak force. Physical laws are not symmetric with respect to reflection in space for weak force.
strangeness
Strangeness conserves, except for weak force.
no conservation
Physical laws are not symmetric with respect to scale. Physical laws are not symmetric with respect to uniform angular velocity.
symmetries
Conservation laws are about minimizations and symmetries. Symmetries require reference point, feature, and reference frame. Symmetry types depend on feature types. For example, rotating spheres with no features have no detectable spin. Particles with dipoles have detectable spin, which can be right or left. Particles must have mass, spin, or other feature to be detectable. Featureless objects or spaces have no symmetries. Symmetries can cancel large physical quantities. Physical theories have one symmetry for each conserved quantity (Noether) [1915].
energy
Energy conservation requires time symmetry: forward and backward in time are usually the same physically. By observing a physical process, observers cannot tell if time flows backwards or forwards.
Total closed-system energy is constant. However, energy can exchange between potential and kinetic energy. Kinetic energy minus potential energy {Lagrangian} measures energy exchange. The path integral of Lagrangian over time is the physical action. For cyclic processes, the system periodically returns to the same Lagrangian value, Lagrangian change is zero, and action is zero. For cyclic processes, the wave equations of motion are path integrals of Lagrangians over time set equal to zero.
momentum
Momentum conservation requires special-relativity constant-velocity reference-frame equivalence. When observing a physical process, observers have no preferred reference frame. The distance metric is the same for all constant-velocity observers (Lorentz invariance).
angular momentum
Angular momentum conservation requires right-left (parity) symmetry. When observing a physical process, observers cannot tell if it is right-handed or left-handed. Clockwise and counterclockwise rotations have same physics.
electric charge
Electric charge stays constant. Electric-charge conservation requires electromagnetism gauge invariance.
Basic space-time symmetries keep physical laws the same under various conditions {invariance, physics}. Baryon number, spatial rotation, and space-time translation are always invariant.
Charge conjugation, parity, and time reversal combined are invariant for all physical laws. Charge conjugation and parity together are invariant, except for strange-particle decays in weak nuclear forces. Mass-strength and strong-force-strength differences in up and down and other quarks cause charge-conjugation symmetry breaking. Parity breaks down in weak nuclear forces. Time reversal breaks down in weak nuclear forces.
Heat and work are kinetic energy. Force fields cause potential energy. Total energy is sum of kinetic and potential energies, which can interconvert. Isolated-system total energy is constant {energy conservation, dynamics}| {conservation of energy, dynamics}. Energy is invariant through time-coordinate translations. Physical laws are symmetric with respect to time dimension, so physics does not change if time reverses direction. Physical laws remain true at all times. All physical interactions are the same if time reverses, charges reverse, and positions reverse. However, weak-force physical laws are not symmetric with respect to time.
cause
Isolated systems have no added forces and so no added potential energy. Isolated systems have no volume changes and so no added distances or potential energy. Object movements interchange potential energies and kinetic energies, no matter which space-time path objects take.
vacuum energy
Kinetic energy and potential energy exert pressure on background vacuum energy. Kinetic energy has particle motions that make internal pressure. Potential energy has fields that make pressure by causing particle self-energy. Motions and fields pressure space-time points through which they pass. Space-time points have energy flux. Kinetic energy and potential energy both contribute to vacuum energy in the same way. Only energy amount counts. As masses move, vacuum adjusts to keep potential constant. Potential, flux, or pressure is constant at all vacuum points, making a new conservation law.
relativity
Mass and energy can interchange in space-time. By equipartition, all partition kinetic energies must be equivalent. Energy conservation remains true under relativistic conditions.
In general relativity, accelerations are equivalent to forces, which cause accelerations. Accelerations are velocity changes. Velocity changes change kinetic energy. Objects change velocity as they change field position and potential energy. Kinetic and potential energies are equivalent in general relativity.
quantum mechanics
Quantum mechanically, mass and energy states are the same. Energy conservation remains true under quantum mechanics.
dark energy
Energy conservation remains true for dark energy, which is symmetric in time.
Momentum conservation {conservation of momentum} means that total momentum is constant, no matter which direction objects take through space. Momentum is invariant under spatial-coordinate translations. All directions are equivalent. Physical laws are symmetric with respect to space dimensions. Physics does not change if space directions reverse, rotate, or translate. Physical laws remain true at all space points.
System total momentum stays constant. For interacting objects, one object's momentum change balances other object's momentum change, because both objects interact over same time. For non-interacting objects, motion states and object masses do not change, so momenta do not change. Fields and their bosons carry or contain momentum and inertia.
Angular momentum conservation {conservation of angular momentum} means that total angular momentum is constant, no matter what rotations (spins or orbits) objects take, at any orientation. Angular momentum is invariant under rotations. Angular momentum is invariant under handedness change, right-handed or left-handed. All rotations are equivalent. Physical laws are symmetric with respect to rotation. Physics typically does not change if spin directions reverse or change orientation. Physical laws remain true for right-handed or left-handed arrangements.
System total angular momentum, from spins, orbits, and curved trajectories, stays constant. Using infinite radius, angular-momentum conservation is equivalent to momentum conservation. Angular-momentum conservation implies that system mass center stands still, so universe either does not rotate or does not move. Angular-momentum conservation implies that forces have equal and opposite reaction-forces and that masses have inertia.
Force can exert over distance {energy}|. Energy is scalar, because motion component does not matter since force vector and distance vector have same direction. Net force can act over distance in direction {work, energy}. Forces can act over distances in all random directions {heat}. Heat has no net force. Forces must act for some time over some distance, so energy can exchange and motion can change.
Energy {kinetic energy}| can involve motion. Kinetic energy equals one-half times mass m times velocity v squared: KE = 0.5 * m * v^2. If object does work and loses speed, object loses kinetic energy. If object receives work and increases speed, object gains kinetic energy.
force
Collision, pushing, pulling, or other contact force F acting over distance ds makes mass m accelerate a to velocity v: dKE = F * ds = m * a * ds = m * (dv / dt) * ds = m * dv * ds / dt = m * v * dv, where t is time. Integral of dv = v/2, so KE = 0.5 * m * v^2.
comparison
Energy is either kinetic expressed energy or potential stored energy, because force times distance makes kinetic energy, and kinetic energy can act against force over distance to make potential energy.
Energy {potential energy}| can depend on force exerted over distance against field. Potential energy PE equals m times acceleration a times distance moved in field h: PE = m * a * h. If object moves to position with less force, object gains potential energy. If object moves to position with more force, object loses potential energy.
field
Field is gravitational, electric, or nuclear force field.
position
Potential energy depends on field force and object field position. If object moves farther away from attraction center, object gains potential energy. If object moves farther away from repulsion center, object loses potential energy. Small movements in strong fields can equal large movements in weak fields.
Exerted force can work against field force, and object gains potential energy. Field force can move object to do work, and object loses potential energy.
action
Going from one point to another point in potential field has only one path, with least average difference between kinetic and potential energy over time.
potential
Fields have measures {electric potential} {potential, electricity} of potential energy that depend on only source mass or charge, not on test-object mass or charge. Gravitational potential V is gravitational constant G times mass m at field center, divided by distance r from center: V = G * m / r. Electrical potential V is electric constant k times charge q at field center, divided by distance r from center: V = k * q / r.
Energy can flow per unit time {power, physics}|. Time t divides into energy E: P = (Ef - Ei) / (tf - ti). Power is constant force F times constant velocity v: P = F*v = F * (ds / dt) = (F * ds) / dt = dE / dt. Power is scalar, because energy is scalar.
Spinning or orbiting objects have rotation energy {rotational kinetic energy}. Because tangential velocity v equals angular velocity w times radius r, rotational kinetic energy KE equals half moment of inertia I times angular velocity w squared: KE = 0.5 * m * v^2 = 0.5 * m * (w*r)^2 = 0.5 * m * r^2 * w^2 = 0.5 * (m * r^2) * w^2 = 0.5 * I * w^2, where moment of inertia I = m * r^2.
work
Energy can be force over distance around axis {rotational work}. Rotational work W equals torque T times angle A in radians: W = F*s = (T/r) * (r*A) = T*A.
power
Energy can be force over time around axis {rotational power}. Rotational power P equals torque T times angular velocity w: P = E/t = (F*s) / t = ((T/r) * (r*A)) / t = T * (A/t)= T*w.
Force can act over distance in direction to transfer energy {work, physics}|. Net force F acts over distance change (sf - si) to perform work E: E = F * (sf - si). Small force Fs exerted over large distance sl can do same work as large force Fl exerted over short distance ss: Fs * sl = Fl * ss.
Objects in fluids have forces and motions {fluid dynamics}|.
thrust
Forward force {thrust, fluid} pushes objects through fluid.
drag
Friction {drag} retards moving objects in fluid. Drag rises as velocity increases.
velocity
If thrust stays constant, velocity rises and drag increases, until force balance makes no more acceleration, at terminal velocity. Example is feather falling through air under gravity.
energies
At pipe points, energies are kinetic energy from fluid flow, potential energy from liquid standing in open pipes, and/or energy from outside forces and pressures. Pipe fluids have energy conservation, by Bernoulli's theorem. For streamline flow, sum of pressure P and kinetic energy KE per volume V is constant: P + (KE / V) = constant. P * V = PE, so (PE / V) + (KE / V) = KE + PE = constant total energy.
pressure
Outside force can exert pressure on fluid. Force moves fluid small distance, and kinetic energy distributes throughout fluid, increasing fluid pressure. Outside pressure P is kinetic energy per volume and is force F times distance s divided by volume V: P = KE / V = (F * s) / V.
depth
At fluid depth, gravity causes pressure. Stationary pressure P is potential energy per volume and is density d times gravity acceleration g times depth h: P = PE / V = (m*g*h) / V = (m/V) * g * h = d * g * h.
flow
Fluid flow causes kinetic energy, which exerts pressure in flow direction. Directed pressure P is kinetic energy divided by volume and is half density d times velocity v squared: P = KE / V = (0.5 * m * v^2) / V = 0.5 * (m/V) * v^2 = 0.5 * d * v^2.
Rotating one cylinder inside another causes intervening liquid to flow {Couette-Taylor flow}. First, flow streamlines. At faster speed, fluid cylinder separates into separate layers along cylinder axis, so fluid goes up and down in cylinder. At higher frequency, flow is chaotic, with no defined frequencies.
At fluid boundaries {no-slip condition}, fluid does not slip.
Mixtures of large organic positive ions and inorganic negative ions {ionic liquid}| can be liquid at room temperature, because large charge is spread over large space, so crystal is loose. Liquid has polar and non-polar parts, so it can dissolve organic materials.
non-turbulent flow {laminar flow}|.
Material can become fluid {liquefaction}|.
After torpedo goes 50 meters per second in water, water pressure is low enough to allow water vapor to make vapor cavity around object {supercavitation, fluid}|, allowing high speed.
Wax surfaces repel water very well {superhydrophobicity}|.
City water pressure {water pressure}| is 30 to 50 pounds per square inch, which can lift water 25 to 30 meters.
Friction {drag, fluid}| slows objects moving through fluid. Drag increases if velocity increases. Pipe walls retard fluid flow by friction one millimeter into fluid.
Wing induces drag as it lifts {lifting line theory}.
Drag rises as velocity increases, while forward force stays constant, until forces balance with no more acceleration {terminal velocity}|. Feathers fall through air under gravity with terminal velocity.
Material density and water density have ratio {specific gravity}|. Specific gravity is one for water. Metals have higher specific gravities and sink in water. Wood has lower specific gravity and floats in water. Density D multiplied by gravity acceleration g is weight m*g per volume V {specific weight}: d*g = (m / V) * g = m*g / V.
Instruments {hydrometer}| can measure specific gravity.
Forces between molecules make fluid stick together {viscosity}|.
causes
In liquids, van der Waals forces cause viscosity. In gases, non-ideal molecular collisions cause viscosity.
pressure
Gas viscosity increases if pressure increases.
temperature
Temperature increase increases gas viscosity and decreases liquid viscosity.
factors
Fluid viscosity depends on fluid density, pressure, temperature, and velocity. In pipe, pipe-opening size affects viscosity. Intermolecular forces tend to pull fluid sideways in pipes and contribute to turbulence. Fluid sideways pressure P equals viscosity V times velocity change dv divided by length change dl: P = v * dV/dl.
Pipe flow with incompressible fluid has two regions. A thin layer {boundary layer} touches tube or obstruction and has viscous effects, because surface interacts thermally and mechanically with fluid. Center has flow with no turbulence.
Temperature, viscosity, and fluid depth relate {Rayleigh number}. Reynolds number and Rayleigh number together account for flow effects, viscosity, thermal conductivity, linear-expansion or volume-expansion coefficient, fluid depth, and temperature gradient.
Fluids have ratios {Reynolds number} of internal force to viscous force. Reynolds number measures fluid momentum change. If Reynolds number is small, smooth pipe decreases drag, because flow is laminar. If Reynolds number is high, vortices in smooth pipe increase drag.
Fluids have attractive electric forces among molecules {cohesion}|. Surface tension has cohesion.
Inside fluid, cohesive forces are symmetric and cancel each other. At fluid surfaces, cohesion pulls molecules closer together {surface tension}|, to make surface density more than inside density. Increased-surface-density layer width is 20 molecules. Air above fluid has small density and has little attraction for fluid. Surface chemical potential is greater than inside fluid, because net force is more, and fluid is denser and so more organized.
floating
Surface tension can make density great enough to float objects, such as steel pins.
examples
Glue, waterproofing, detergents, wicks in candles, blotters, towels, bubbles, milk drops, camphor dance, soap film, salts, and needles on water illustrate surface tension.
drops
Droplets have more surface area and high surface tension.
factors
Solute can lower solvent-molecule cohesion by disrupting cohesive forces and lowering chemical potential. Soaps and detergents lessen water surface tension by blocking water-molecule attractions. However, ions in water increase surface tension by increasing electric forces.
Gravity causes fluid molecules to press on molecules below {pressure, fluid} {fluid pressure}|. Deeper molecules have more pressure, because more molecules are above them. Pressure P, force F per area A, at point below fluid surface is density d times depth h times gravity acceleration g: P = F / A = (m * g) * (h / V) = (m / V) * g * h = d * g * h. Pressure is directly proportional to gravity acceleration, because acceleration times mass is force. Pressure is directly proportional to density, because density relates to molecule mass. Pressure is directly proportional to depth, because depth relates to molecule number. Pressure does not depend on total surface area, because pressure is force per unit area.
fluid level
Liquids rise to equal heights at all openings to atmosphere, because pressures and potential energies are equal at liquid surfaces.
cause
Random molecule motions cause fluid pressure. Random motion exerts force and pressure equally in all directions, even upward or at angle. Container wall slope has no effect. Pressure is the same at all points at same depth. The net effect of random motions is that pressure is perpendicular to fluid surface, because random motions are symmetric around perpendicular.
temperature
Temperature increase increases fluid pressure, because molecules move faster.
density
More and/or heavier molecules have higher density and exert more pressure.
gas
Gas has random translational kinetic energy per unit volume, making force per unit area on container walls. Random translational kinetic energy depends on mass and molecule average velocity. If volume decreases, pressure goes up. If temperature decreases, pressure goes down. If pressure decreases, volume goes up. If pressure increases, temperature goes up.
Because pressure transmits equally throughout fluids, pressure sum, and energy sum, is constant {Bernoulli's theorem} {Bernoulli theorem} {Bernoulli's principle}. At pipe points, energy conserves. Pressure stays constant throughout fluid.
If fluid goes through narrower pipe area, fluid speeds up, because mass cannot build up. Higher speed makes more kinetic energy and directed pressure. Sideways pressure decreases, to conserve energy. If fluid slows, kinetic energy goes down, and potential energy and sideways pressure increase.
examples
Fountains, tanks with holes, aspirators, sprayers, Venturi meters, wings, pipes, rubber tubes, and curve balls demonstrate Bernoulli's principle.
In confined fluids, force in one direction can transmit pressure to all directions {hydraulics}|. Increased pressure changes into increased random molecule motions. For example, pushing a piston in a long thin cylinder can do work on fluid and increase molecule random kinetic energy, which can do work on a piston in a short wide cylinder.
area
At depth, force per area pressure is the same throughout confined fluid. Small force acting over long distance on small area can make big force acting over short distance on large area, because energy in and energy out are equal. Large area can apply large total force.
examples
Dams, breathing using lung double cavity, manometer, Bourdon gauge, barometer, hydraulic brakes, hydraulic lifts, syringe, and fluid tank show hydraulic effects.
Simple-fluid hydrodynamics {hydrodynamics}| has no viscosity or heat exchange and follows Euler equation: mass m per volume V times acceleration g equals negative of pressure P gradient perpendicular to velocity vector: (m / V) * g = - dP / ds.
Equal areas have same pressure in confined fluid {Pascal's principle} {Pascal principle}.
Fluid discharge velocity from small hole at depth below open surface is square root of two times gravity acceleration g times depth h {Torricelli's theorem} {Torricelli theorem}: (2*g*h)^0.5.
Removing gas molecules {vacuum, gas}| reduces pressure. Vacuum pumps remove gas molecules.
Oil with iron filings {magnetorheological fluid} can turn solid in magnetic field.
Fluids {electrorheological fluid} can become solid, or have lower viscosity, in high electric fields.
Fluid mildly heated from bottom at first has temperature gradient with no net flow. More heat creates alternating hexagonal cells that allow hot fluid to rise and cold fluid to fall. Even more heat makes turbulent motion, with no net flow {Bénard problem}.
Heated fluids can have convection {Rayleigh-Bénard convection} with circular motions.
Two surfaces can stick to each other {adhesion, surface}|.
Fluids have electric forces between molecules and container surfaces. Fluids that physically adhere to surface can rise in small-diameter tubes {capillary rise}|. Clinging force pulls fluid up tube sides. Fluid rises until potential-energy increase balances air pressure.
When fluids leave holes, fluid tends to flow around hole edge {Coanda effect}.
If fluid is adhesive, fluid curves up container walls {meniscus}|. If fluid is not adhesive, fluid curves down container walls.
Van der Waals forces can cause molecules to bind to surfaces {physisorption}|. Vibrations then cause molecules to leave surface, within 10^-8 seconds, heating surface. Surface chemical bonds do not form or break.
Chemical bonds between surface molecules and fluid molecules can bind molecules to surfaces chemically {chemisorption}|. Molecule stays at surface from 1 to 1000 seconds and then has desorption. Chemisorption has activation energy. At low pressure and low absorption, chemisorption fraction depends on pressure. At high pressure or for strong electrical forces, chemisorption fraction depends on pressure inverse.
Chemisorbed molecules stay at surface from 1 second to 1000 seconds and then leave {desorption, surface}|, heating surface. Surface chemical bonds break. Desorption has activation energy.
Objects in fluids have more pressure on bottom surface than on top surface {buoyancy}|, because bottom surface is deeper in fluid. The greater force on bottom pushes object up. Buoyancy equals difference between object-bottom pressure and object-top pressure. Buoyant force is in opposite direction from gravity. Objects that sink have more force than fluid weight pushed up. Objects that sink are denser than fluid.
If objects float in fluid, fluid weight pushed up around object equals upward buoyant force on object {Archimedes principle, buoyancy}|.
Object in fluid pushes fluid out {displacement, fluid}|. Fluid tries to return to original position by gravity. Displaced fluid and object both want to occupy same place. Equilibrium happens when both forces push down equally.
Objects can sink until buoyant force balances gravity {floating}|. Displaced fluid and object both want to occupy same place, so object is at equilibrium when fluid force pushing down equals object force pushing down. Object that floats is less dense, including air spaces, than fluid. Submarines, fish, boats, balloons, and ice cubes demonstrate buoyancy. If floating-object mass center is not along buoyant-force line, object rotates around mass center.
Fluid mass can go past point or through area over time {fluid flow}|.
pipe
Fluid velocity at different pipe radii differs. Highest velocity is in center. Velocity is zero at pipe walls.
conservation
Same fluid amount at one point must be at another point. Otherwise, fluid builds. Same fluid volume passes any point, during time. At pipe points, inflow equals outflow.
pressure
Around pipe loops, pressures add to zero.
rate
Flow rate increases with increase in molecule velocity, temperature, pressure, and/or mean free path. Flow rate decreases with increase in cross-sectional area, molecule mass, and/or molecule collision frequency.
Flow in pipes can have constant velocity at each radius, with no sideways motion {streamline flow}|.
Flow in pipes can have sideways motion or different velocities at same pipe radius {turbulent flow}|. Trapped gases in fluid can cause turbulence.
High-speed flow and/or pipe edges can pull fluid apart, making vacuum spaces {cavitation, fluid}|.
Fluids have flow rate through area {flux, fluid}|. Flux is energy, mass, momentum, or charge change D divided by cross-sectional area A times time t: D / (A * t).
pipe
In pipes, masses entering and leaving cross-sectional areas are equal. Otherwise, fluid builds, or vacuum happens. Mass m flowing through pipe equals fluid density d times fluid velocity v times cross-sectional area A: m = d*v*A. For liquid, fluid density is constant, and fluid velocity going in vi times cross-sectional area at entrance Ai equals fluid velocity going out vo times cross-sectional area at exit Ao: vi * Ai = vo * Ao. For gas, fluid density varies, and fluid density at entrance di times fluid velocity going in vi times cross-sectional area at entrance Ai equals fluid density at exit do times fluid velocity going out vo times cross-sectional area at exit Ao: di * vi * Ai = do * vo * Ao.
Flux equals constant times gradient {Fick's first law of diffusion} {Fick first law of diffusion}: dm / (A * dt) = dC / ds, where m is mass, A is cross-sectional area, t is time, C is concentration difference, and s is distance.
Pressure, temperature, concentration, or force change over time relates to quantity change over distance {Fick's second law of diffusion} {Fick second law of diffusion}: dP / dt = dm / ds, where P is pressure, t is time, m is mass, and s is distance.
Other objects can cause object to tend to change motion {force, physics}|. Force requires interaction between two objects. All forces are pushes or pulls, such as when two objects collide. Gravity and electric forces are interactions of objects with second-object fields. Force F that object receives equals object mass m times object acceleration a: F = m*a. Force has direction and amount and so is vector. Mechanical force can be interaction between two colliding masses, but repulsions between electrons around molecules mediate contact between objects.
Gravity and electric force can act between two objects over distance {force field}.
exchange
All forces transmit bosons. In all field interactions, two objects exchange bosons. Gravity exchanges graviton bosons. Electromagnetism exchanges photon bosons. Strong nuclear force exchanges gluon bosons. Weak nuclear force exchanges W-particle and Z-particle bosons.
field
All forces are metric gauge fields. Bosons form field around object. Field changes space curvature, from flat to curved. When two objects interact, they go through curved space and change motion accordingly, just as cars turn on banked curves. Acceleration involves boson exchange. Boson exchange and curved spaced have identical effects, because bosons curve space, and space-curvature acceleration releases bosons to interact.
mass and distance
Bosons with no mass, such as gravitons and photons, can exchange to infinite distances. Bosons with mass, such as gluons, W particles, and Z particles, have short action distances.
time
Particles exchanged by forces take time to act, based on force strength. Strong force takes 10^-23 seconds. Electric force takes 10^-21 seconds. Weak force takes 10^-18 seconds to 1000 seconds. Gravitational force takes much longer.
electromagnetism
Electrical force is interaction between two charges. Magnetic force is interaction between two relativistic charges.
weak force
Electric force is 1000 times stronger than weak force.
gravity
Gravitation and electromagnetism are similar, because interactions cause both forces and both forces radiate in all directions. Because masses are positive, gravitational force is attractive. Because charges are positive and negative, electromagnetic force is attractive or repulsive. Electric-force to gravitational-force ratio is 10^36. If gravity is stronger, universe is smaller, and stars are smaller and exist shorter. If gravity is weaker, universe is larger, and stars are larger and exist longer.
Electromagnetic and weak forces unify {electroweak force}| {electric-weak unification theory} under SU(2) x U(1) Lie symmetry group, with gauge symmetry. An adjoint representation uses U(1) subgroup of SU(2), making electroweak Higgs field. W+, W-, and Z bosons and photons are equivalent at proton-diameter distances and high energies. Spontaneous symmetry breaking in current universe causes W+, W-, and Z bosons to have mass and photons to have zero rest mass.
All forces are metric fields {gauge invariant field} {gauge field}| that transmit bosons.
Gravitational force {gravity, mass} is interaction between two masses. Gravitons mediate gravity, by exchange at light speed. Gravity curves space. If only gravity shapes space-time, why does light speed, which depends on electromagnetic force, determine space-time boundaries?
Mass is always positive, and gravity is always attractive. Negative energy is repulsive {antigravity} {anti-gravity}. Negative internal pressure is also repulsive.
Atomic particles that have quarks and gluons interact {strong nuclear force}|. Strong nuclear force is positive and constant over distances more than 10^-14 centimeters, is repulsive over shorter distances, depends on quark number, and is 100 to 1000 times stronger than electric force.
Perhaps, nuclear forces {superweak force}| can mediate between strong and weak nuclear forces.
Leptons and quarks interact {weak nuclear force}|. Quarks and leptons have left-handed and right-handed spin states. Weak forces act on one or the other. For example, left-handed down quark can become up quark, making neutron into proton, electron, and neutrino {radioactive beta decay}. Right-handed down quarks have no change.
antiparticles
Particles that are right-handed or left-handed have weak nuclear force, but their antiparticles do not have weak force.
bosons
Weak-force W-particle and Z-particle bosons can come from vacuum with no conservation, except at very high energy.
distance
Weak nuclear force acts over less than 10^-16 centimeters.
strength
Weak nuclear force is 10^34 times stronger than gravity.
Relation between weak and electromagnetic forces became constant at 10^-12 seconds after universe origin, when U(2) symmetry broke at a rotation angle {electroweak mixing angle} {Weinberg angle}.
Motions {simple harmonic motion}| can oscillate along lines, with acceleration proportional to distance from center point. Molecule-bond vibrations, springs, pendulums, rigid-bar vibrations, rotations, guitar-string vibrations, bridge vibrations, and tall-building sway have simple harmonic motion.
force
Restoring-force strength depends on material type and distance from center. For molecule bonds, spring constant depends on electrical forces between atoms. Restoring force F equals negative of spring constant k expressing restoring force strength times displacement x: F = -k*x. Restoring force is negative because it opposes displacement.
amplitude
Amplitude depends on input energy, which causes more or less displacement.
period
One oscillation takes one time period. Period depends on material restoring force. Period and amplitude are independent. Spring period T is 360 degrees expressed in radians 2*pi times square root of mass m divided by spring constant k: T = 2 * pi * (m/k)^0.5. Higher mass makes longer period. Higher spring constant makes shorter period.
energy
Potential energy PE equals half spring constant k times displacement x squared, which is average force, k*x/2, times distance x: PE = 0.5 * k * x^2. At center, force equals zero, and potential energy equals zero. At maximum displacement amplitude, force and potential energy are highest. At maximum displacement, kinetic energy equals zero, because motion stops as direction reverses. At center, velocity and kinetic energy maximize, because potential energy is zero.
velocity
Maximum velocity v is maximum displacement A times square root of spring constant k divided by object mass m: v = A * (k/m)^0.5. Average velocity is 4*A/T, where A is amplitude and T is period. Average velocity is 2 * v / pi, where v is maximum velocity.
friction
If friction damps simple harmonic motion, amplitude decreases, but frequency stays the same, because material is the same.
When pulled sideways and released, weight {pendulum} hanging by string or wire from point starts oscillating motion.
force
Pendulum restoring force is gravity. Gravity g pulls pendulum-bob mass m back toward center with force F from distance x, depending on displacement angle A: F = m * g * sin(A) = m * k * x.
distance
If pendulum displacement is small, displacement-angle sine equals displacement angle: sin(A) = A. For small displacement, displacement x is displacement angle, expressed in radians, times pendulum length L: x = A*L. For small displacement, constant k is gravity acceleration g divided by pendulum length L: k = g/L.
period
Pendulum period T is 360 degrees, expressed in radians 2*pi, times square root of gravitational-constant reciprocal 1/g: T = 2 * pi * (1/g)^0.5. Longer pendulums have longer periods. Weaker gravity makes longer period. Pendulum mass does not affect period.
Spring oscillation time T {period, oscillation}| is 360 degrees, expressed in radians 2*pi, times square root of mass m divided by spring constant k: T = 2 * pi * (m/k)^0.5. Higher mass makes longer period. Higher spring constant makes shorter period.
Springiness {spring constant, force} depends on length, cross-sectional area, and force strength between molecules. Stiff springs {spring, metal}| have high spring constant.
Motion can cause force in opposite direction {friction}|, from surface-bump collisions as surfaces slide over each other or from electrical forces between close surfaces.
process: sliding
As one surface slides over another surface, surface molecules collide, forcing object backward and upward {kinetic friction}.
process: rolling
If one surface rolls over another surface, collisions do not push surfaces upwards, because surfaces have no sliding. A continuously changing rolling-surface part contacts stationary surface. Bumps and grooves in the surfaces mesh deeper, so contact is greater {static friction}. Static friction is greater than kinetic friction.
factors: force direction
Friction force is directly proportional to force perpendicular to surface. If surface is horizontal, perpendicular force is gravity. If surface is not horizontal, perpendicular friction force F is gravity g times sine of incline-to-horizontal angle A: F = g * sin(A).
factors: area
Contact area between two surfaces affects friction force only slightly, because more area makes pressure and force less, and less area makes pressure and force more.
factors: speed
Faster speed makes less sliding friction, because moving surface rides higher over stationary surface.
factors: electric force
Friction depends on hydrogen bonds and van der Waals electric forces between surface molecules. Smooth surfaces can be in close contact, and so have high electrical attractions and high friction. For example, two smooth glass plates or plastic pieces stick together tightly. Rough-surface molecules are farther apart on average, have smaller electrical attractions, and have less friction.
factors: lubricant
Oil, grease, and graphite can fill surface depressions and keep two surfaces separated, so surfaces have little hydrogen bonding or van der Waals forces, and friction is small. Heavier weight oil flows less easily and sticks to surfaces more, but stickiness causes the objects to have more friction. Lighter weight oil flows more easily and sticks to surfaces less but has less friction. Best-weight oil balances ability to stay in bumps with ability to flow easily. Multi-weight oil flows easily when cold and gets thicker as it gets warmer.
tires
When cornering, radial tires keep more tread on road, so force goes more into road, instead of going into tire side. Radial tires act the same as bias-ply tires while going straight.
Tires with greater radius keep more tread area on road, but this has little effect. Tires with greater radius have larger sidewalls, which can flex more and so become hotter and weaker. Tires with stiff and narrow sides stay cooler and stronger but have harsher ride.
Wider tires have increased area, but more area does not affect friction much. Wider tires can find dry or debris-free road parts, while smaller tires contact only sand or water. Wider tires can aquaplane more on water or snow, because fluid cannot leave treads fast enough.
Friction force knocks molecules from surface through collisions and wears away surface {abrasion, surface}|.
Simple machines {simple machine} use small force exerted over long distance or large force exerted over short distance.
lever
Rods, such as crowbars, can have balance point near one end. Short radius moves short distance and exerts large force, as long radius moves long distance and needs only small force.
inclined plane
Ramps can lift heavy objects short vertical distances, by using small forces over long horizontal distances.
screw
Inclined planes can wind around axis, to allow lifting water short vertical distances using small forces over long horizontal distances.
Wheels on axles {wheel and axle} {axle and wheel} allow objects to move horizontally with no sliding friction. Pulley wheels on axles can hold rope, which can lift weight short vertical distances by pulling rope using small forces over long horizontal distances. Pulleys can alter force direction. One rope can wrap around two pulleys to make a block and tackle, to lift heavy objects short distances with small forces applied over long distances.
Construction levers {crane, machine} {derrick} lift and lower objects using counterbalancing weight.
Forces can redirect using movable connections {joint, connector}|.
Force on ladder steps transmits to both sides and then to ground. Ladders {ladder} have angle to ground. If angle is too large, frictional torque is too small to balance weight torque, and ladder feet slip.
Forces can redirect using connections {strut}| that exert force inward or outward.
Double inclined planes {wedge, machine}|, such as axes and knives, use small force over long distance to exert large force over short distance, to split things or push grooves into softer material.
Actual mechanical-advantage to ideal mechanical-advantage ratio {mechanical advantage}| {efficiency, dynamics} inversely relates to friction. Actual mechanical advantage is lower than ideal mechanical advantage. Actual force Fi exerted on machine divides into force Fo exerted on object-to-move {actual mechanical advantage}: Fo / Fi. Distance moved by output force divides into distance si moved by input force {ideal mechanical advantage}: si / so.
A dynamic quantity {momentum}| depends on mass and velocity. Momentum p equals mass m times velocity v: p = m*v. Constant force F times time change (tf - ti) makes momentum p: p = F * (tf - ti) = m * a * (tf - ti) = m*v. Force F is momentum change dp per time change dt: F = dp / dt.
Momentum {angular momentum}| can be through angles around rotation axes or points. Momentum p equals mass m times velocity v: p = m*v. Tangential momentum pt equals mass m times tangential velocity vt, which equals angular velocity w times circle curvature radius r: pt = m * vt = m*w*r. Angular momentum L equals tangential momentum pt times radius r: L = pt * r.
Moment of inertia I equals mass m times radius r squared: I = m * r^2. Angular momentum L equals moment of inertia I times angular velocity w: L = pt * r = (m*w*r) * r = (m * r^2) * w = I*w.
torque
Force F equals mass m times acceleration a: F = m * a. Tangential force Ft equals mass m times tangential acceleration at: Ft = m * at = m * dv / dt = m * r * dw / dt, where dv is velocity change, dt is time change, dw is angular-velocity change, and r is curvature radius. Torque T equals tangential force Ft times radius r: T = Ft * r = (m * at) * r = m * r * (dw / dt) * r = (m * r^2) * dw / dt = I * (dw / dt) = I * aa, where m is mass, at is tangential acceleration, r is curvature radius, dw is angular-velocity change, dt is time change, I is moment of inertia I, and aa is angular acceleration.
Force F equals momentum change dp over time change dt: F = dp / dt. Tangential force Ft equals tangential momentum change dp over time change dt: Ft = dp / dt = m * r * dw / dt, where m is mass, r is curvature radius, dw is angular-velocity change, and dt is time change. Torque T equals angular momentum change dL over time change dt: T = dL / dt = (m * r^2) * dw / dt, where m is mass, r is curvature radius, and dw is angular-velocity change.
Force acts over time {impulse}|, to change object motion.
Force tends to alter object structure. Force can apply over area {pressure, physics}|. Pressure P equals force F per unit area A: P = F/A. Pressure P is energy E divided by volume V: P = F/A = (F*s) / (A*s) = E/V. Pressure is energy density: P = F/A = F / s^2 = F*s / s^3 = E/V.
Force causes structure change {strain}| in force direction. Pressure changes length. For example, gravity pulls feet into floor and tends to flatten feet, shoes, and floor. Strain is proportional to stress.
Force and pressure {stress, physics}| tend to alter object structure. I-beam, twisting torsion bars, and standing up use material stresses, strains, and strengths.
Mechanical forces {compression}| can be pushes on materials.
Mechanical forces {tension, force}| can be pulls on materials.
Mechanical forces {torsion, force}| can be twists on materials. Torsion is simultaneous push in one direction and pull in another direction.
Stress S is proportional to length change dL divided by length L {Hooke's law} {Hooke law}: S = k * (dL / L). Pressure compresses material against restoring force. Restoring force F varies with material type k and distance compressed x: F = -k * x. Pressure is compression force F divided by area A: P = F/A. Pressure equals restoring force -k*x divided by area A: P = F/A = -k * x / A. Constant k depends on Young's or other elasticity modulus u, cross-sectional area A, and object length L: k = A * u / L.
If pressure exceeds limiting value {elastic limit}|, material inelastically changes shape permanently. Higher temperature lowers elastic limit.
If pressure is less than elastic limit, material returns to original shape after removing pressure {elasticity, physics}|. Elasticity can be bulk strength, shear strength, or tensile strength. Elasticity varies for different material axes.
If pressure exceeds elastic limit, material changes shape permanently {inelasticity}|.
Elastic modulus reciprocal {compliance}| can indicate shape-changing ease.
Elasticity {Young's modulus, elasticity} {Young modulus, elasticity} varies for different material types. Steel is hard and bouncy and has high Young's modulus. Wood is soft and has low Young's modulus.
Elasticity varies for compression {bulk strength}|.
Elasticity varies for torsion {shear strength}|.
Elasticity varies for tension {tensile strength}|.
All forces on objects result in net force {resultant, forces}|. Force-vector sum equals resultant-force vector. Vector resultants explain motions in sailing, billiard balls colliding, planes flying, crowbar leveraging, wrenches twisting, gears turning, wrecking balls breaking, roller coaster riding, and car rocking to get out of mud or snow.
Parallel forces {couple, physics}| can act in opposite directions on axis ends, to tend to cause rotation.
All system forces can add to zero {equilibrium, physics}|. All system velocities can add to zero {static equilibrium}. Objects have gravity centers. When objects move, gravity centers can rise {stable equilibrium}, fall {unstable equilibrium}, or stay the same {neutral equilibrium}.
Pressure, temperature, concentration, or force can change over distance {gradient, physics}|. Gradients are like forces and cause flow in resistant medium. Flow increases until reaching terminal velocity. Terminal velocity determines flux.
Force pushes or pulls objects along line between interacting-object centers. When one object exerts force in a direction on another object, second object exerts same force in opposite direction on first object {reaction, forces}. Relation is symmetric.
examples
Rockets burn fuel. Hot gas pushes rocket forward, while rocket pushes gas backward in opposite direction.
For objects on coil springs, coil spring pushes object up, while object pushes coil spring down.
When walking, foot pushes back against ground, and ground pushes back on person, to send body forward.
An hourglass on a balance has dropping sand, but balance stays still, because weight is constant. Downward collisions balance upward force.
Objects can spin or orbit or rotate around balance point {rotation, force}. Rotations rotate around point {center of rotation} {rotation center} {balance point}. Forces or weights are at distances {radius, rotation} from balance point or rotation center. On levers, forces can act perpendicular, or at an angle, to radius. Weighing balances and seesaws are levers. Weights on strings can orbit, so string provides centripetal force, and spin provides centrifugal force.
Spinning objects, including Earth, bulge at equator {equatorial bulge} and flatten at poles, by centrifugal force.
Just as mass traveling in straight line has inertia that tends to keep velocity constant, mass rotating around axis has inertia {moment of inertia}| {inertia moment} that tends to keep angular velocity constant.
mass
Inertia depends directly on mass. Moment of inertia substitutes for mass when quantities use angular velocity instead of velocity. When mass rotates around rotation center at radius r, tangential momentum pt is mass m times tangential velocity vt: pt = m * vt. Angular momentum L is moment of inertia I times angular velocity w: L = I * w. Tangential velocity vt varies directly with angular velocity w: vt = w * r. Tangential momentum pt varies directly with angular momentum L: L = pt * r. Moment of inertia I depends on mass m and radius r: L = I * w = pt * r = m * w * r * r = (m * r^2) * w. Moment of inertia I is mass m times square of distance r from axis or point: I = m * r^2.
summation
Masses have volume, so object points have different radii from rotation center. Total moment of inertia is sum of moments of inertia at each radius.
Thin-ring moment of inertia equals total mass m times square of distance r from ring center to ring middle: m * r^2.
Disk or cylinder moment of inertia equals half total mass m times square of distance r from disk or cylinder center to outer edge: 0.5 * m * r^2.
Pipe or doughnut moment of inertia equals half total mass m times sum of squares of distances from pipe or doughnut center to inner edge ri and outer edge ro: 0.5 * m * (ri^2 + ro^2).
radius
Objects with moment of inertia around rotation center have moment of inertia around any axis parallel to rotation axis. New moment of inertia Inew is old moment of inertia Iold plus total object mass m times square of distance d between axes: Mnew = Mold + m * d^2.
Forces {torque}| can tend to cause motions around rotation centers.
acceleration
Torque causes angular acceleration. For example, force can act on a rigid rod that can turn around a balance point. Force can act perpendicular to rod or at another angle. Torque T is force F times radius r from balance point times sine of force-to-radius angle A: T = F * r * sin(A), which is cross product of force and radius vectors: T = F X r. Torque-vector direction is perpendicular to both force vector and radius vector and parallel to axis.
moment of inertia
Tangential force Ft equals mass m times tangential acceleration at, which equals angular acceleration aa times radius r: Ft = m * at = aa * r. If torque acts perpendicular to radius, torque T equals moment of inertia (I = m*r^2) times angular acceleration aa: T = Ft * r = m * at * r = m * aa * r * r = m * (r^2) * aa = I * aa.
examples
Frisbees and yo-yos have torques. Torque causes car front to fall when car stops. Torque causes car front to rise when car accelerates. To open door, push farthest from hinge to apply least force, because radius is greatest. Spins in ice-skating begin with torque. Gymnasts and divers apply torque. Torque causes spin on footballs, bullets, bicycle wheels, helicopter blades, and propellers. Scales use opposing torques to weigh objects.
equilibrium
When lever is not moving around balance point {equilibrium}, right Tr and left Tl side torques, F * r * sin(A), are equal: Tl = F1l * r1l * sin(A1l) + F2l * r2l * sin(A2l) + ... = F1r * r1r * sin(A1r) + F2r * r2r * sin(A2r) + ... = Tr.
In curved motion, force can go away from curvature center {centrifugal force}|, along radius direction. For example, Moon is in orbit around Earth.
In curved motion, force can go toward curvature center {centripetal force}|, along radius direction. For example, Moon is in orbit around Earth.
Moon falls toward Earth center by Earth gravity {free fall}|. Gravity is centripetal force. Orbital speed moves Moon tangentially in orbit. Tangential movement accelerates Moon away from Earth along radius. This acceleration is centrifugal force. Centrifugal force and centripetal force are equal, and motion rates away from and toward center are equal, so Moon maintains approximately same distance from Earth.
weightless
Moon in orbit has no weight, because centrifugal force equals centripetal force, just as astronauts in orbit are weightless. If jumping from height, in free fall, one feels weightless, because no force is opposing fall.
Kepler formulated three planetary-motion laws {Kepler's laws} {Kepler laws}.
first law
Radius from orbiting body to Sun sweeps out equal areas in equal times, because velocity is slow at large radius and fast at small radius.
second law
If object in orbit moves closer, speed increases as potential energy changes to kinetic energy and moves object back outward. If object in orbit moves farther away, speed decreases as kinetic energy changes to potential energy and moves object inward.
If spinning object becomes more compact, radius decreases and speed increases as potential energy changes to kinetic energy. If spinning object expands, radius increases and speed decreases as kinetic energy changes to potential energy.
third law
Acceleration cubed is directly proportional to time squared, because acceleration is highest at greatest curvature point, where velocity is highest.
Two objects in different orbits interact by gravity or electromagnetism to make torque that changes object spin axis {nutation}|.
Earth spins on an axis that is at an angle to axis of Earth orbit around Sun. Sun gravity causes torque on Earth axis and causes it to rotate {precession}| {precession of the equinoxes, Earth}, as angular velocity around axis interacts with angular velocity around orbit. Object spin and orbital motion interact to cause spin-axis precession.
Energy {heat, energy}| {thermal energy} can be total random translational kinetic energy and total potential energy, in all directions, that holds atoms apart. Heat as kinetic energy raises temperature. Heat as potential energy makes solid into liquid and liquid into gas.
work
Heat energy has no direction. Heat energy in solids or liquids can do no work.
kinetic energy
Heat translational kinetic energy equals number of molecules times temperature.
potential energy
Solids have little heat potential energy. Liquids have some heat potential energy. Gases have much heat potential energy.
Drying, oxidizing, or reducing can use temperature just below melting {calcine}|.
Heat tends to push molecules apart {expansion, matter}| {thermal expansion} {heat expansion}. Temperature increase adds random translational kinetic energy to material and makes molecules collide faster, so molecules spread more. Material molecules have attractive forces, which resist random motion.
coefficient
Higher temperature makes material volume bigger by a ratio {coefficient of volume expansion} {volume-expansion coefficient}. Higher temperature makes material length longer by a ratio {coefficient of linear expansion} {linear-expansion coefficient}. Length change dL equals length L times temperature change dT times linear-expansion coefficient c: dL = c * L * dT. Volume change dV equals volume V times temperature change dT times volume-expansion coefficient c: dV = c * V * dT.
coefficient: gas
All gases have same volume-expansion coefficient, because gases approximate ideal gas.
examples
Road cracks, erosion, and thermostats with bimetallic strips illustrate thermal expansion.
Higher pressure forces most-substance molecules together and tends to make molecules go to lower potential energy {Joule-Thompson effect}. Expanding gas cools gas, as random translational kinetic energy changes into random potential energy.
state
If material is under pressure, state change happens at higher temperature. Higher temperature makes more kinetic energy to overcome higher pressure that keeps molecules together.
ice
Ice is less dense than water, so ice tends to melt under higher pressure. For example, pressure of ice-skate blade melts ice under blade to allow skating. If pressure on melted ice decreases, ice freezes again {regelation, pressure}.
altitude
Making cake at high altitude requires higher temperature, because lower air pressure makes air hold less heat.
Molecule motions and collisions make average random translational kinetic energy {temperature, heat}|. Average gas-molecule velocity at room temperature is 500 meters per second.
Materials can have no kinetic energy and no heat potential energy {absolute zero temperature}|, at -273.16 degrees Celsius.
Temperature scales {Celsius, temperature scale}| {centigrade} can set water freezing point to 0 C and boiling point to 100 C, at sea level.
Temperature scales {Kelvin, temperature}| can set water freezing point to 273 K and boiling point to 373 K, at sea level.
Surface temperature is proportional to energy emitted per unit area {Stefan's law} {Stefan law}.
Chemical reactions, engines, and collisions have force, energy, and heat transfers {thermodynamics}|.
heat
Energy transfers use work, through directed kinetic energy, or heat, through temperature change or state change. Friction changes some directed energy into random energy and increases temperature. Systems can minimize friction by slowing and by using lubricants.
comparison
Thermodynamics is about extensive quantities. Statistical mechanics is about intensive quantities. Thermodynamic quantities are number of moles times Avogadro's number times corresponding statistical-mechanics quantity. Molecular-property time averages give observable thermodynamic properties.
potentials
The six thermodynamic potentials are baryon-number density, total mass-energy density, isotropic pressure, temperature, entropy per baryon, and baryon chemical potential. Rest frame is stationary or moving fluid. Baryon number density and entropy per baryon determine composition. Baryon number is constant in fluid, because density is constant, so gradient equals zero. Systems can only create entropy, not destroy it. Shock waves increase entropy. Heat flows increase or decrease entropy.
Material transport {heat transport} properties, such as electric conductivity, thermal conductivity, viscosity, diffusion, effusion, and dissolution, depend on molecular properties such as temperature, pressure, collision frequency, and kinetic-energy range.
Heat flows have laws {thermodynamics laws}. When heat becomes another energy type or another energy type becomes heat, total energy does not change {energy conservation, first law} {first law of thermodynamics}. Heat flows from objects with higher temperature to objects with lower temperature, and energy must make heat flow from cold object to hot object {second law of thermodynamics}. Entropy is zero at absolute zero temperature {third law of thermodynamics}, because random motion is zero and system has complete order. Two systems in thermal equilibrium with third system have same temperature {zeroth law of thermodynamics}.
Systems react to change, such as energy change, to oppose further change {Le Chatelier's principle} {Le Chatelier principle}. As system resists change, directed work energy becomes random translational kinetic energy, through temperature and pressure change.
Systems with energy flows can have steady or periodic flow {steady state, thermodynamics}, rather than reach equilibrium. Movement rate or flux depends on gradient or force, so flow rate equals force or gradient sum. Steady states are irreversible thermodynamically. Entropy minimizes, because systems with forces or gradients can reduce entropy.
Perhaps, motion never slows {perpetual motion}|. Perpetual motion of first kind violates extended Le Chatelier's principle. Perpetual motion of second kind violates extended Le Chatelier's principle. Perpetual motion of third kind violates the principle that there must always be friction.
In heat exchange, heat lost by object equals heat gained by other object {conservation of heat energy} {heat energy conservation}| {law of heat exchange} {heat exchange law}.
Energy exchange can change potential energy, translational kinetic energy, and heat energy and change pressure and volume {enthalpy}|. Enthalpy equals total system energy E plus product of pressure P times volume V: H = E + P*V. Pressure times volume is work. Under constant pressure or volume, enthalpy is heat that system makes. For solids or liquids, enthalpy equals energy, because volume does not change.
Systems have energy {free energy}| available to do work. Free energy is energy from order loss plus potential energy converted to kinetic energy.
purpose
Free energy can show if process is spontaneous.
heat energy
Temperature times entropy is heat energy taken from surroundings.
work
Pressure times volume is work on system.
Helmholtz free energy
For constant temperature, free energy {Helmholtz free energy} is system energy minus heat energy: E - S*T.
Gibbs free energy
For constant pressure and temperature and changed volume, free energy {Gibbs free energy} is Helmholtz free energy plus work energy: E - S*T + P*V. Gibbs free energy G is enthalpy H minus temperature T times entropy S: G = H - T*S. Gibbs free energy is net work that system can do.
Arrhenius free energy
For changed temperature, free energy {Arrhenius free energy} is net work that system can do.
chemical potential
Gibbs free energy per mole u, the chemical potential, changes with absolute temperature T and mole fraction x: u = u0 + R * T * ln(x), where R is gas constant. Gibbs free energy per mole u changes with absolute temperature T and partial pressure P: u = u0 + R * T * ln(P).
free energy change
If system is not in equilibrium, something flows from higher to lower chemical potential. Free-energy change is negative. System changes spontaneously. However, spontaneous change does not happen if no pathway exists for energy change. To minimize free energy, system can lower potential energy, by reducing pressure, or increase entropy, by increasing temperature.
Isolated systems can have no work from outside. No energy transfers in or out of closed systems. Only entropy changes affect free energy.
Isothermal systems have only work and have no entropy change, because temperature is constant.
If temperature is low, entropy is small, so reaction makes heat to lower potential energy. If temperature is high, entropy is more important, and reaction heat can be small or large. At low pressure, more gas can evolve.
free energy change: equilibrium constant
In chemical reactions, free-energy change depends on equilibrium constant. Free-energy change equals gas constant times absolute temperature times natural logarithm of equilibrium constant.
free energy change: substances
For reactants, substance chemical potential times substance moles subtracts from reactant free energy. For products, substance chemical potential times substance moles adds to product free energy. Free-energy change in systems with one substance equals chemical potential a times change in number n of moles: a * n.
Chemical-reaction product and reactant concentrations depend on free-energy changes. Free-energy change equals -R * T * ln((ap1^np1 * ap2^np2 * ... ) / (ar1^nr1 * ar2^nr2 * ... )), where R is gas constant, T is temperature, api is product chemical potential, npi is chemical-equation product number of moles, ari is reactant chemical potential, and nri is chemical-equation reactant number of moles. Chemical reactions, and all physical changes, are spontaneous if they release free energy.
reaction
To reverse reactions, second reaction, with more free energy change, must couple to reaction. Total free energy change then favors reverse reaction. Diffusion, evaporation, and solvation take energy from surroundings, or use their thermal energy, to drive other reactions.
Heat can exert force in direction and so do work {work, heat}|. Possible work energy is difference in heat energy between hotter region and colder region, which is available heat energy. Machines have ratio {efficiency, work} between work actually done and heat available or input work. Efficiency is high temperature Th minus low temperature Tc divided by high temperature: (Th - Tc) / Th. Engines have efficiency of 30%.
Ideal engines have four stages {Carnot cycle}|: isothermal heat gain, adiabatic gas expansion, isothermal heat loss, and adiabatic gas contraction.
Temperature increase causes material to increase random translational kinetic energy and so absorb heat {heat capacity}|. Material can absorb heat and gain random translational kinetic energy, so temperature rises. Heat capacity is heat needed to raise one gram of material one degree Celsius. Heat H equals mass m times heat capacity c times temperature change T: H = m * c * T.
factors
Heat capacity depends on material type. Chemicals can hold more or less heat depending on possible electric dipole states. Metal atoms have no vibrations and rotations. Metals have low heat capacity, because all heat goes into random translational motion, rather than into vibrations or rotations. Diatomic molecules are linear molecules. Diatomic molecules have medium heat capacity, because they have few vibrations and rotations. Water is triatomic, is asymmetric, and has hydrogen bonds between molecules. Water has high heat capacity. Large complex molecules in gasoline, clays, and ceramics have high heat capacity. Crystal structure can have chemical bonds, hydrogen bonds, van der Waals forces, or ionic bonds, allowing many vibration modes and high heat capacity.
material heat capacity divided by water heat capacity {specific heat}|.
In reactions {explosion}|, temperature increase can increase reaction rate, which then increases temperature, which then increases rate, and so on. Gas production increases rapidly. Gas propels outward from center if reaction makes heat more rapidly than heat can dissipate by thermal radiation or gas loss. Randomly moving molecules tend to bounce outward, because surface area is greater toward perimeter and smaller toward center. Explosions require heat to stay high enough to burn gas before gas can move far.
burning
Burning does not explode, because gas has unconfined gas expansion or has much thermal radiation, so heat spreads out by thermal radiation or gas loss faster than reaction makes heat.
Ignition can spread flame through flammable gas at subsonic speeds {deflagration}|, as heat diffuses through gas. Gas expands evenly.
Ignition can spread flame through flammable gas at supersonic speeds {detonation}|, because shock waves compress gas. Detonation causes engine knocking, because gas expands unevenly.
In explosion-like reactions {implosion reaction}|, gas amount can decrease as temperature increase increases reaction rate, which then increases temperature, which then increases rate, and so on, because gas is reactant and products are not gases.
Heat change can happen over time {heat flow}. Heat flow is from high-temperature region to low-temperature region. Heat flow converted to translational kinetic energy exerted in direction can do work. Engines use adiabatic and isothermal heat-flow stages to perform work.
Heat flow can have constant heat {adiabatic}|. Temperature goes up in one location and down in another location.
Heat flow can have constant pressure {isobaric}|, typically in systems open to atmosphere.
Heat flow can have constant temperature {isothermal}|. Heat flows into or out of heat sinks or sources.
In heat flow {conduction, heat}|, collisions among molecules can transfer translational random kinetic energy.
materials
Fluids are good heat conductors, because molecules move freely. Metals are good heat conductors, because electrons move freely. Diamonds have high thermal conductivity, because crystal vibrations transfer heat.
area
Conductive-heat flow rate increases as contact area increases. Adding fins to surfaces or roughing up surfaces increases surface area and conducts heat better.
temperature
Conductive-heat flow rate increases as temperature difference increases. At room temperature, good conductor, such as metal, feels cool to touch, because heat moves quickly away from warmer human body. Poor conductor, such as plastic or wool, feels neither cool nor warm. Steering wheel covers and seat covers reduce heat conductivity.
In heat flow {convection, heat}|, mass can move in another mass, as hotter fluid at lower density rises and cooler fluid at higher density falls or as masses mix, flow, or blow. Convection is non-random motion that transfers heat by mass movement. Convective-heat flow rate increases as temperature difference increases. Blowing on something to cool it uses convection. Radiators use convection.
Electromagnetic-radiation emission or absorption {radiation, heat}| can transfer heat.
fire
Fire is electromagnetic radiation, emitting infrared and visible light from excited atoms in hot gas. Other objects can absorb radiation energy from fire and become hotter.
infrared
Heat radiation is typically infrared radiation. Infrared radiation is high in materials above 100 degrees Celsius.
color
White or shiny surfaces do not absorb radiation well, reflect radiation back into themselves at surfaces, and do not radiate at all frequencies well. Black surfaces absorb radiation well, do not reflect radiation back into themselves at surfaces, and radiate at all frequencies well.
Ideal objects {black body} can emit maximum heat radiation and have Planck distribution of radiation wavelengths and energies {black-body radiation}|.
Gas molecules move randomly, have elastic collisions, are point-like, and have no interactions {kinetic theory}|. Ideal gases follow kinetic theory. Gas molecules have cross-sectional area, and hydrogen bonds and van der Waals forces make molecules slightly attract, so real gas molecules do not move completely randomly and have somewhat inelastic collisions.
molecular collisions
In gases, one cubic centimeter has 10^28 molecular collisions per second. Collision frequency increases as mass decreases, temperature increases, cross-sectional area increases, and density increases.
molecular velocity
Gas-molecule collisions distribute speeds and directions. Molecular-velocity distributions are Boltzmann distributions. Some molecules have low velocity. Most molecules are near average velocity. Few molecules have very high velocities. Average gas-molecule velocity at room temperature is 500 meters per second. Molecular velocity increases as mass decreases or temperature increases.
Maxwell envisioned a demon {Maxwell's demon} {Maxwell demon} that can see particle motions and act on particles individually, so perpetual motion of second kind can happen. However, demon, light, and energy are all system parts, so perpetual motion cannot happen.
On average, particles travel short distances {mean free path}| between collisions. Mean free path is collision-frequency inverse and measures average distance between gas molecules. Mean free path decreases as mass decreases, temperature increases, cross-sectional area increases, and density increases.
Systems have different motions and kinetic energies {degrees of freedom, partition}, such as translations, rotations, and vibrations.
translation
All particles can have translations. Average random translational kinetic energy determines temperature.
rotations
Spherically symmetric molecules cannot have net rotational motion. Linear molecules can have one rotational motion state. Two-dimensional molecules can have two rotational motion states. Three-dimensional molecules can have three rotational motion states.
vibrations
Molecules with chemical bonds can have vibration states. Vibrations can involve one bond and be along bond axis. Vibrations can involve two bonds and be across bond axes. Molecule symmetries can cancel vibration states.
partition
Heat can go equally into all available energy states {partition of energy, heat}|. If molecule has more rotation and/or vibration states, raising temperature requires more energy, because some heat does not become average random translation kinetic energy.
partition: heat capacity
Material heat capacity depends on molecular-motion degrees of freedom. Molecules with more rotation and/or vibration states have higher heat capacity.
partition: equipartition
Motion-type average kinetic energies must be the same {equipartition, energy} {energy equipartition} {principle of equipartition of energy}, because energy transfers freely among states by collisions.
amount
Partition average kinetic energy KE is half Boltzmann constant k times temperature T: KE = 0.5 * k * T.
As substance state changes, one mole loses or gains heat {latent heat}|. Latent heat changes molecular conformation and depends on substance type. Total heat Q needed to change state depends on substance mass m and latent heat L: Q = m*L. For liquid-to-gas state change, one mole of liquid gains heat {heat of vaporization} {vaporization heat}. For solid-to-liquid state change, one mole of solid gains heat {heat of fusion} {fusion heat}.
Heating or cooling material can change phase {state change}| {change of state}, by changing chemical arrangement. Molecules spread farther apart, or pack closer together, and change potential energy. Phase changes can happen when increased heat translational kinetic energy causes increased volume, which favors new electrical-attraction structures.
types
State change can be condensation, vaporization, solidification, sublimation, or fusion.
time
Phase change is usually rapid.
temperature
Temperature is constant during state change, because added or removed kinetic energy goes into potential energy change, so average random translational kinetic energy is constant.
pressure
More pressure tends to lower state from gas to liquid to solid, because it compresses molecules and so lowers potential energy.
Adding heat to liquid can increase liquid vaporization until vapor pressure equals air pressure {boiling}|. Heating fluid makes bubbles. Bubbles are liquid vapor, not air bubbles. Boiling is only on pot bottom, because bottom is hottest.
Liquid-to-gas state change is at a temperature {boiling point}| and pressure.
Surface-molecule collisions make some molecules have enough energy to leave surface and make vapor, which has pressure {vapor pressure}|. Molecules that left liquid before can later fall back into liquid from vapor, so vapor pressure depends on outside pressure and temperature. Substances in liquid mixtures contribute partial pressure to total vapor pressure. Total vapor pressure equals sum of partial pressures. Mixed-liquid vapor pressure is less than pure-liquid vapor pressure. Vapor pressure equals mole fraction times pure-vapor pressure: P = f * P0.
State change from liquid to gas is easier if material has weaker bonds between molecules {volatility}|. Materials with small non-polar molecules, globular shape rather than linear shape, and small forces between molecules are volatile. Volatility is high if chemical potential is high. Solute amount that can vaporize depends on boiling point and vaporization enthalpy. If both are low, solute disrupts easily and leaves.
Vapor and liquid {azeotrope}| can have same composition, if they form third material or help each other dissociate.
Partial pressure equals substance mole fraction in liquid times total pressure {Dalton's law} {Dalton law}. Total vapor pressure equals sum of partial pressures.
Liquid can change to gas {vaporization}|. Vaporization causes drying. Liquid-to-gas state change is at a boiling point temperature and pressure. As liquid becomes gas, gas absorbs heat and cools surroundings, as in refrigeration and air conditioning.
Gas can change to liquid {condensation, gas}|. Gas-to-liquid state change is at a temperature {condensation point} and pressure. Cold surfaces cool nearby air and cause air to lose water, which forms surface droplets.
Solid can change to gas {sublimation, heating}|. Solid-gas state change is at a temperature {sublimation point} and pressure.
Solid can change to liquid {fusion, melting}| {melting}. Solid-to-liquid state change is at a temperature {melting point} and pressure.
If pressure on melted ice decreases, ice freezes again {regelation, ice}|.
If liquid has no dirt, bubbles, or other crystallization initiators, it can cool below freezing point {supercooling}| without solidifying.
Liquid or gas can turn into solid {solidification}| {freezing} upon heat loss or removal. Solid-liquid state change is at freezing-point temperature and pressure.
about mixture-solidification temperature {eutectic}.
Physical systems have particles with properties, locations, times, motions, energies, momenta, and relations. Particles can be independent or depend on each other.
degrees of freedom
Related particles have motion restrictions. Particles with no relations are free to move in all directions by translations, vibrations, and rotations. Systems have interchangeable states {degrees of freedom, entropy}. More particles and more particle independence increase degrees of freedom.
order
Order depends on direction constraints. Ordered systems have few possible states. Disordered systems have many possible states. Systems with high heat have more disorder because kinetic energy goes in random directions and potential energy decreases. Systems with work have less disorder because kinetic energy goes in one direction and potential energy increases in direction against field. Systems have disorder amount {entropy, heat}|.
information
Systems with no relations have no information, because particles move freely and randomly, with no dependencies. Systems with relations have information about relations and dependencies. Systems with more degrees of freedom, less order, and more entropy have less information. Systems with fewer degrees of freedom, more order, and less entropy have more information. Because entropy relates to disorder, entropy relates to negative information.
information: amount
The smallest information amount (bit) specifies binary choice: marked or unmarked, yes or no, 0 or 1, or on or off. The smallest system has two possible independent states and one binary choice: 2^1 = 2, where 2 is number of states and 1 is number of choices. Choices can always reduce to binary choices, so base can always be two. Systems have number of binary choices, which is bit number.
information: probability
The smallest system is in one state or the other, and both states are equally probable, so states have probability one-half: 1/2 = 1 / 2^1. State probability is independent-state-number inverse.
information: states
Systems have independent states and dependent states. Dependent states are part of independent states. Systems can only be in one independent state. Particles have free movement, so independent states can interchange freely and are equally probable. Particles have number {degrees of freedom, particle} of independent states available. Systems have number of states. Number is two raised to power. For example, systems can have 2^6 = 64 states. States have probability 1/64 = 1 / 2^6. 6 is number of system information bits. Systems with more states have more bits and lower-probability states.
information: degeneracy
Different degenerate states can have same properties. For example, systems with two particles can have particle energies 0 and 2, 2 and 0, or 1 and 1, all with same total energy.
information: reversibility
Particle physical processes are reversible in time and space. Physical system states can go to other system states, with enough time.
entropy: probability
Disorder depends on information, so entropy depends on information. Entropy is negative base-2 logarithm of probability. For example, for two states, S = -log(1 / 2^1) = +1. For 64 states, S = -log(1 / 2^6), so S = -log(1 / 2^6) = +6. More states make each state less likely, so disorder and entropy increase.
entropy: degeneracy probability
Degenerate-state groups have different probabilities, because groups have different numbers of degenerate states. Groups with more members have higher probability because independent states have equal probability. Entropy depends degeneracy pattern. Going to lower probability group increases system order and has less entropy. Going to higher probability group decreases system order and has more entropy.
Lowest-probability groups are reachable from only one other state. High-probability groups are reachable from most other states. Systems are likely to go to higher-probability groups. Systems move toward highest-probability group. In isolated closed systems, highest-probability group has probability much higher than other groups. If system goes to lower-probability group, it almost instantly goes to higher-probability group, before people can observe entropy decrease. Therefore, entropy tends to increase.
entropy: additive
Entropy and disorder are additive, because they depend on independent states, degrees of freedom. Systems with independent parts have entropy equal to sum of part entropies. If parts are dependent, entropy is less, because number of different states is less.
entropy: heat
Heat is total random translational kinetic energy. Temperature is average random translational kinetic energy. Entropy S is heat energy Q, unavailable to do work, divided by temperature T: S = number of independent particle states = (total random translational kinetic energy) / (average random translational kinetic energy) = Q/T. Kinetic energy is random, and potential energy holds molecules apart in all directions, so heat has no net direction. Average direction is zero.
entropy: energy
At constant pressure and temperature, entropy is enthalpy change divided by temperature, because heat is enthalpy change at constant pressure and temperature. At constant volume and temperature, entropy is energy change divided by temperature, because heat is energy change at constant volume and temperature.
entropy: gravity
If no gravity, entropy increases as particles spread, because particle occupied volume increases. If gravity, entropy increases as particles decrease separation, because potential energy becomes heat though particle occupied volume decreases. If antigravity, entropy increases as particles increase separation, because potential energy becomes heat and particle occupied volume increases.
entropy: mass
Entropy increases when particle number increases. Matter increase makes more entropy. Entropy increases when particles distribute more evenly, toward thermal equilibrium, and have fewer patterns, lines, edges, angles, shapes, and groupings.
entropy: volume
If there are no forces, volume increase makes more possible molecule distributions, less order, and more entropy. If there are forces, volume decrease makes more possible molecule distributions, less order, and more entropy. See Figure 1.
entropy: directions
Energy dispersal increases entropy, because disorder increases. Increasing number of directions or motion types increases entropy. Mixing makes more disorder and more entropy. More randomness makes more entropy. More asymmetry makes more disorder and more entropy.
entropy: heat
Work makes more friction and heat and more entropy. Making heat makes more randomness and more entropy.
entropy: fields
Field strength decrease disperses energy and makes more entropy.
entropy: force and pressure
Pressure decrease disperses energy and makes more entropy. Force decrease disperses energy and makes more entropy.
entropy: volume
At phase changes, pressure change dP divided by temperature change dT equals entropy change dS divided by volume change dV, because energy changes must be equal at equilibrium: dP / dT = dS / dV, so dP * dV = dT * dS. Volume increase greatly increases entropy.
entropy: increase
Systems increase entropy when disorder, degrees of freedom, and disinformation increase. Information decrease makes more interactions and more entropy. Order decrease, as in state change from liquid to gas, increases entropy.
entropy: decrease
Many factors increase order, regularity, or information, such as more regular space or time intervals, as in stripes and waves. Higher energy concentration, more mass, larger size, higher density, more interactions, more relations, smaller distances, closer interactions, more equilibrium, more steady state, more interaction templates, more directed energy, and more filtering increase order.
More reference point changes, more efficient coding or language, better categorization or classification, more repetition, more shape regularity, and more self-similarity at different distance or time scales increase order. More recursion, bigger algorithms, more processes, more geodesic paths, more simplicity, lower mixing, and more purity increase order. More reconfigurations, more object exchanges, and more combining systems increase order. Fewer functions, fewer behaviors, more resonance, fewer observations, more symmetry, more coordinated motion, and more process coupling increase order.
Larger increase in potential energy increases order, because energy concentrates. Higher increase in fields increases order, because energy increases. Fewer motion degrees of freedom, as in slow and large objects, increase order. More same-type, same-range, and same-size interactions increase order. More and equal influence spheres increase order. Higher space-time curvature increases order. More constant space-time curvature increases order.
Lower harmonics of Fourier series increase order. Fewer elements in Fourier series increase order.
entropy: closed system
In closed systems, entropy tends to increase, because energy becomes more random. Potential energy becomes random kinetic energy by friction or forced motion. Random kinetic energy cannot all go back to potential energy because potential energy has direction. Work kinetic energy becomes random kinetic energy by friction or forced motion. Random kinetic energy cannot all go back to work energy because work energy has direction. Heat energy is already random. Only part, in a direction, can become potential energy or work kinetic energy. Radiation becomes random kinetic energy by collision. Random kinetic energy cannot all go back to radiation energy, because radiation energy requires particles accelerated in direction.
universe entropy
Universe is isolated closed system. It started in low-entropy state and moves to higher entropy states.
Perhaps, at beginning {hot big bang} {primordial fireball}, universe had one particle at one point with smallest possible volume, and so no relations among parts. There were no space fields and no tidal effects. Universe had highly concentrated energy at high temperature and so large contracting forces and high pressure. Particle number remained the same or increased, as particles and radiation split.
Universe expansion increased space volume. Space points became farther from other points. Expansion was greatest at first. Then expansion slowed, because all particles had gravity.
As universe cooled, it created particles, in evenly distributed gas. Entropy increased but was still low.
As universe cooled, gas-particle gravity formed galaxies and stars. Condensed gas had higher entropy but was still low. Potential energy converted to heat as infalling particles collided. Heat and mass concentrated in stars.
Stars are hot compared to space, so stars can transfer energy to planets and organisms. Stars undergoing nuclear fusion make visible light. Visible light has higher energy than heat infrared radiation. On Earth, temperature stays approximately constant. Therefore, visible light energy that impinges on Earth is equal to energy that radiates away from Earth as heat. Because sunlight has higher energy per photon, fewer sunlight photons land on Earth than Earth emits as infrared heat photons. Entropy increases in space, and total universe entropy increases. On Earth, order increases and disorder decreases, mostly in organisms. From universe beginning until now, universe entropy increases, while small-region physical forces and particle motions can cause entropy decreases.
Now, universe has many photons, large volume, negligible forces, and even matter distribution, so universe entropy is now large. For example, cosmic microwave background radiation has many randomly moving photons, from soon after universe origin. Photons mostly evenly distribute. They fill whole universe and have little effect on each other. As universe expands, their entropy becomes more.
Now, universe has many galaxies with central black holes and has black holes formed after supernovae. Black holes are mass concentrations denser than atomic nuclei. Black holes have very high entropy, because particle number is high, volume is small, mass evenly distributes, gravitational force and fields are high, and density is high. Black holes make universe entropy large now.
In the future, universe entropy will increase. Universe will have more black holes and can evolve to have only black holes. Universe will have more local forces. Universe will have more volume. Universe will have more particles. Universe will have more-even particle distribution.
Systems have particles, which have position and momentum, and energy and time {information, physics}.
spin system
Particles can have spin up clockwise or spin down counterclockwise. Particle spin state encodes one binary choice or information bit: 0 or 1. For quanta, such as spin, state holds one quantum information bit. One kilogram of plasma in one liter of space can hold 10^31 bits.
Particles with known spins can carry input or output information. For interacting particles, quanta can entangle, and information can entangle. Particles can interact to represent computation to calculate. Bits can change electromagnetically every 10^-20 second. Signals travel 3 * 10^-12 seconds to next particle. System particles entangle and change in parallel. One kilogram of plasma in one liter of space can have 10^51 particle-spin changes each second.
system information
Information required to describe system state depends on degrees of freedom. Mole of chemical has 10^23 degrees of freedom because it has that many molecules. In regions that have no boundary and have uniform matter and energy, entropy is proportional to volume.
black hole
By Bekenstein-Hawking formula, one-kilogram black holes have radius 10^-27 meters. Event-horizon surface can hold 10^16 bits. Bits can change every 10^-35 second. Surface can have 10^51 particle-state changes each second. Light can cross black holes in 10^-35 seconds, so physical processes have time to work serially.
Observers cannot see past event horizon, so horizon surface must contain all information about region volume inside black hole. Gravity causes information bits to interact, so entanglement over surface area can have enough information states to hold information about inside volume. Black-hole entropy is directly proportional to event-horizon surface area. Planck area is 10^-66 cm^2. One Planck area has 0.25 information bit, so four Planck areas make one bit.
One-kilogram black holes emit gamma and higher-energy rays by Hawking radiation and disappear in 10^-21 second. Smaller black holes dissipate faster and emit higher energies.
Perhaps, rather than particle spins and states, branes or strings hold states.
cell size
Bit-change maximum rate is maximum clock frequency, and so time has quanta. Rate depends on gravitational constant, light speed, and Planck constant.
Space has quanta, with cell sizes. Black holes have maximum energy per volume and have definite ratio between mass and event-horizon radius, so maximum mass-energy is proportional to space-time radius. Cell size varies with radius cube root. For universe, cell diameter is 10^-15 meters.
Energy and time relate, so space cell size depends on space-time radius. The holographic principle can derive from uncertainty principle.
universe
Universe has age 1.3 * 10^10 years and so radius 1.3 * 10^10 light-years. Universe is close to maximum density, so it can hold 10^123 bits. It can have 10^106 bit changes each second, in parallel, from 10^72 joules. It has 10^92 bits of ordinary matter, which can change bits at 10^14 Hz. It has less than 10^123 bits of dark energy, which can change bits at 10^18 Hz.
Total entropy in universe, inside and outside black holes, cannot go down {generalized second law} (GSL).
Information that went into black hole can come out by entanglement of two surface virtual particles, one of which interacts with original matter and information at singularity {Horowitz-Maldacena model}.
Information bits can change no faster than quantum time. Heisenberg uncertainty principle relates energy and time. Quantum time t depends on quantum energy E {Margolus-Levitin theorem}: t >= h / (4*E), where h = Planck's constant.
Masses in volumes can hold limited entropy amounts {universal entropy bound}.
Energy and mass within volume can hold limited entropy amount {holographic bound}. Holographic bound is number of event-horizon-surface Plank areas divided by four. Maximum-entropy regions relate mass and volume, so area can define region information. Black-hole formation prevents volume entropy increases from overtaking surface entropy increases and so limits entropy.
Particle systems have total energy and distribute energy among particles {statistical mechanics}|.
energy: particle
Particles have energy levels. Particles have possible energy levels. Particle energy level cannot be zero, because particles must move and so have kinetic energy. Particles always have at least minimum ground-state energy, because energy has quanta.
energy: distribution
Some particles have lower energy, and some have higher energy {distribution, energy}. Particles exchange energy by collisions or electronic transitions. Systems have average particle energy, which is higher for higher temperature and/or work. Large systems typically have only one particle-energy distribution, which has highest probability.
energy: total
Sum of particle energies equals total energy. Total energy equals average particle energy times particle number.
energy: types
Particles can have translational energy, vibrational energy, rotational energy, and electronic-transition energy, with different ground states and different quanta. At normal temperatures, vibrational energy is at ground state, electronic-transition energy is at ground state, and rotational energy is above ground state. Total energy distributes equally among possible translation, rotation, vibration, and electronic-transition energy levels, if there are pathways. Systems with large energy quanta have few particles at high-level energies. Systems with small energy quanta have more particles at high-level energies. Energy change does not change particle distribution much.
entropy
Energy distributions have entropy. Entropy change changes particle distribution. Systems with few particles or low temperatures have quantum states, easy transitions among states, and minimal entropy. Systems with many particles or high temperature have thermal states. Black-hole event horizons have random kinetic energy and cause thermal states. Thermal states have random kinetic energy and have maximum entropy.
entropy: degeneracy
Different particle-energy distributions can have same number of particles at each energy level. For example, if two same-type particles exchange energies, system has different particle-energy distribution, same total energy, and same number of particles at each energy level. Different particle-energy distributions with the same energy and same number of particles at each energy level make system phase. System has largest phase, which has highest probability and most even energy distribution possible at total energy. Largest phase has highest entropy.
state: fluctuation
If system is in largest phase, particles have lowest probability of returning to smaller regions, because largest phase has highest probability. If particle collision results in smaller phase, in shortest possible time, system returns to largest phase, because largest phase has highest probability. Hawking radiation requires large phase fluctuation.
For systems with many molecules at equilibrium at temperature, frequency distributions {Boltzmann distribution, statistics}| can plot frequency against molecule energy. y(E) = e^(- E / (k * T)), where E is molecule energy level, y(E) is frequency for molecule energy level, e is natural-logarithm base, k is Boltzmann constant, and T is absolute temperature.
energy
Particle-energy probability is partition number and is relative frequency of that energy in Boltzmann distribution. Most-probable energies are near average energy. Total energy is integral of Boltzmann distribution.
comparison
At temperatures above 50 K, Boltzmann distributions look like Gaussian distributions.
equilibrium
Systems at equilibrium have Boltzmann distribution, because that distribution has much higher probability than other distributions with same total energy. Boltzmann distribution has the most combinations that can give total energy.
equilibrium: entropy
For that reason, Boltzmann distribution has lowest probability of molecule being in any one energy level, so Boltzmann distribution has the most entropy and least order. Entropy S equals Boltzmann constant k times combination-number C natural logarithm: S = k * ln(C).
Molecular properties {canonical properties} can be at constant temperature {canonical property}, at constant temperature and volume {grand canonical property}, or in isolated adiabatic systems {microcanonical property}.
If system molecules are indistinguishable, some particle-energy distributions have same numbers of particles at each energy level {degeneracy, system}|.
Particles have different possible motions and kinetic energies {degrees of freedom, energy}.
Physical systems can have different numbers and energy levels of particles {distribution of energies} {energy distribution}. Particles can be molecules, atoms, photons, or subatomic particles.
energy quanta
Particle energy cannot be zero, because particles are always moving and so have kinetic energy. Particle energy has quanta, by quantum mechanics, so particles have lowest energy level {ground-state energy}. Particle energies increase from ground-state energy by discrete energy quanta. Possible particle energies are ground-state energy, ground-state energy plus one quantum, ground-state energy plus two quanta, and so on. For total energy, possible energy levels have numbers of particles. Systems have particles at ground-state energy, particles at ground-state energy plus one quantum, particles at ground-state energy plus two quanta, and so on. Particle number at high energy levels is small compared to number at low energy levels, because elastic collisions distribute energy among energy levels. High particle energy has low probability. Infinite particle energy has zero probability.
system energy
Closed systems have constant total energy. Total energy is ground-state energy times particle number, plus any quanta. Sum of particle energies makes total energy. Product of particle number and ground-state energy is minimum system energy.
energy distribution
For example, two-particle system can have one particle with energy 3, one particle with energy 1, and total energy 4. For closed systems, particle collisions can change energy distribution, but total energy stays constant. For example, the two-particle system can have one particle with energy 2, one particle with energy 2, and total energy 4.
energy distribution: low-energy example
Two-particle system can have ground-state energy Q0, one particle at ground-state energy, E1 = Q0, and another particle at one quantum energy level Q above ground-state energy, E2 = Q0 + 1*Q. Total energy is E1 + E2 = Q0 + (Q0 + 1*Q) = 2*Q0 + 1*Q. See Figure 1.
energy distribution: equivalent distributions
For closed systems, different energy distributions can result in same total energy. For example, twelve-molecule systems can have energy distributions in which each particle has energy Q1a and total energy is 12*Q1a. By particle collision, system can have energy distribution with six molecules one quantum Q above Q1a and six molecules one quantum Q below Q1a. System still has total energy 12*Q1a.
For two-molecule system with total energy 2*Q0 + 2*Q, both molecules can have energy Q0 + 1*Q. After collisions, first molecule can have energy Q0, and second molecule can have energy Q0 + 2*Q, or first molecule can have energy Q0 + 2*Q, and second molecule can have energy Q0. See Figure 2. All three energy distributions have same total energy.
probability
In closed physical system, all energy distributions have same total energy, and all distributions are equally likely, because collisions transfer energy freely between particles.
probability: distinguishable particles
If particles are distinguishable, energy distributions are unique and have equal probability. For example, system can have total energy 6, ground-state energy 2, quantum 1, and 2 particles. If particles are distinguishable, such as E1 and E2, three energy distributions are possible. E1 = 2 and E2 = 4. E1 = 4 and E2 = 2. E1 = 3 and E2 = 3. Distribution E1 = 3 and E2 = 3 has one-third probability. Distribution E1 = 2 and E2 = 4 has one-third probability. Distribution E1 = 4 and E2 = 2 has one-third probability. Distributions are equally likely. See Figure 3.
In this system, particles cannot have energy 0 or 1, because ground-state energy is 2. Only these three cases make total energy 6.
probability: indistinguishable particles
Typically, some system particles are exactly the same and so indistinguishable. For example, all electrons are the same. If particles are indistinguishable, some energy distributions appear the same.
For example, system can have total energy 6, ground-state energy 2, quantum 1, and 2 indistinguishable particles. Two energy distributions are possible: energy 2 and energy 4 or energy 3 and energy 3. Cases E1 = 2 and E2 = 4, and E1 = 4 and E2 = 2, are now indistinguishable. Energy distribution 3 and 3 happens once and has one-third probability. Energy distribution 2 and 4 happens twice and has two-thirds probability.
probability: degeneracy
If particles are indistinguishable, some energy distributions have same numbers of particles at each energy level. In the example, two energy distributions have one particle at level 2 and one particle at level 4, so degeneracy is two.
Degenerate energy-distribution probability is degeneracy divided by number of distributions when particles are distinguishable. In the example, number of energy distributions with distinguishable particles is three. Degeneracy of "energy 2/energy 4" distribution is two, and probability is 2/3. Degeneracy of "energy 3/energy 3" distribution is one, and probability is 1/3.
For degenerate distributions, more degeneracy makes higher probability. Degeneracy is greater if most particles are near average energy and particles have Boltzmann energy distribution. Maximum degeneracy spreads particles maximally. For many-particle systems, highest probability is many orders of magnitude above second-most-likely distribution.
partition number
If system has constant total energy, distribution degeneracy {partition number, distribution} is (total number of particles)! / (number at ground-state energy)! * (number at ground-state energy plus one quantum)! * ... * (number at ground-state energy plus infinite number of quanta)!, where ! means factorial. Above 50 K, for thermal distributions, partition number maximizes according to Boltzmann distribution.
probability: particle
Particle has probability that it is in energy level. If system is in energy distribution with maximum degeneracy, particle has lowest average probability that it is in energy level, because particles spread most evenly. Other energy distributions increase average probability that particle is in energy level, because they concentrate particles more.
system state
With collisions, systems tend to go to most degenerate energy distribution, from which the most collisions make same degenerate distribution, because it repeats itself the most. This is why the most-degenerate energy distribution has highest probability. Isolated systems soon reach this single stable state and stay there.
Complex systems can have sets {ensemble, system}| of identical objects. Sets have statistical properties. Linear ensemble operators can calculate set-property average values, while varying initial conditions.
Collisions interchange energy, so average energies of system kinetic-energy sources are equal {equipartition, statistical mechanics}|.
motions
For particles, kinetic energy partitions equally into available motion states {degrees of freedom, motion}. Particles have different possible motions. Translations can independently be in three spatial dimensions. Vibrations depend on chemical-bond stretching-and-bending modes. Rotations can be in zero to three rotation dimensions, depending on molecule symmetry.
energy
If temperature is above 50 K, average partition energy E is half Boltzmann constant k times absolute temperature T: E = 0.5 * k * T. System partition function is product of particle partition functions.
In rare systems, all phases solve energy equation, and system reaches equilibrium {ergodic hypothesis}|. Though few systems are ergodic, real systems come arbitrarily close to ergodic {quasiergodic hypothesis}. Quasiergodic systems are fractal, because one trajectory cannot fill up space but can pass close to all points. If trajectory does not follow simple law, system uses statistical law.
Plane rectangles can have many square cells and some particles {gas in box model}. Connected cells make a region with percentage of total cell number. For example, box can have 10 cells, with one-cell region in one corner. Probability that one particle is in region is 1/10.
Statistical thermodynamics applies to systems {ideal fluid} in which the only interactions among particles are elastic collisions, with no forces between molecules. Particles can have cross-sections.
Boltzmann distribution gives number of molecules at each energy level {partition function}| {canonical partition function}, for a temperature.
Probability {partition number, energy} of particle energy level is relative frequency of that energy in Boltzmann distribution. Most-probable energies are near average energy.
Molecule collisions make fast and discrete molecule-energy changes {quantum energy change}.
temperature
Energy fluctuations Q depend on Boltzmann constant k times absolute temperature T: Q = k*T. At higher temperatures, energy change is more, and molecule energy levels are farther apart. At low temperature, quanta are almost equal, and molecules have ground-state energy Q0 plus multiple of energy quantum Q: Q0, Q0 + 1*Q, Q0 + 2*Q, Q0 + 3*Q, and so on.
factors
Quantum increases if volume decreases, system does work, mass decreases, temperature decreases, pressure decreases, fields decrease, or electrons transition to lower orbits. In those cases, overall energy decreases, so quanta are bigger.
entropy
If system has only entropy changes, quanta stay the same.
high energy
If quanta are large, high energies are hard to reach.
Spontaneous processes {spontaneous process}| lower free energy. Particles move along geodesics. Electrons move along zero-field lines. Particles orbit at lowest orbit.
Coldness can be very low {cryogenics}|. Lasers cool by slowing atoms to 50 microKelvin. Magnetic fields can trap and compress gas. Cooling can be by both lasers and magnetic traps {magneto-optical trap} (MOT).
Evaporation cools to below 50 microKelvin by removing hottest atoms. Time-averaged orbital potential (TOP) magnetic trap allows evaporation at point that moves in circle to build gas ellipsoid. Ioffe trap holds plasmas. Ioffe-Pritchard trap can use parallel magnetic fields or other arrangements to form various gas shapes.
properties
Quantum-mechanical effects can change low-temperature material properties, such as superconductivity.
Metal superconductors have bound-electron pairs, each with same spin, which make metal ions, streamline flow, and make bosons that can condense {Bose-Einstein condensation, cryogenics} (BEC). All bosons in same quantum state can condense from gas to make liquid. Repulsive bosons condense better.
At low temperature, fluids {superfluid}| can have no viscous resistance. Liquid helium is the only known superfluid. Vortexes but no overall rotation can appear in spun superfluids. Space-time can be like fluid, and black-hole event horizon can be like superfluid with quantum-phase transition. General relativity has same equations as sound waves in moving fluid.
Helium 4 can cool and compress to solid {supersolid}, with no viscous resistance.
High temperature can separate electrons from atoms and cause electrons to leave metal or metal-oxide surface {Edison effect} {thermionic emission, heat}|.
Thermionic emission leaves surface positive charge {space charge, thermionic}.
Electric force is attraction or repulsion between electric charges. Magnetism is moving-charge relativistic effects and so is apparent electric force {electromagnetism}|. If electric fields cancel, because positive and electric charges are equal, magnetic fields do not necessarily cancel, because both positive and electric charges can move relativistically.
Particle properties {charge, electricity}| can cause electric force. Electron charge {negative charge} is one negative unit. Proton charge {positive charge} is one positive unit. Total electric charge is sum of particle electric charges.
static electricity
Rubbing glass with cloth keeps protons on glass and puts electrons on cloth. Rubbing rubber with cloth puts electrons on rubber and keeps protons on cloth. Rubbing energy frees electrons from rubbed material surface. Quickly pulling the materials apart leaves net charge on both materials. Sliding on rugs rubs electrons off rug, and touching metal doorknobs makes electrons jump to metal.
static electricity: lightning
High winds, when hot air rises, rub higher cold air, separate electrons from air molecules, and take electrons away before they can recombine. Lightning carries electrons back to positive-charge regions or to ground.
strong nuclear force
Only strong nuclear force can change particle electric charge.
Surface voltages can move charges around semiconductors {charge coupling}|. Semiconductors have capacitance. Charge moves between capacitors at each clock pulse. Solid-state TV cameras and memory circuits use charge coupling. Semiconductors {charge coupled device} (CCD) can move free electric charges from one storage element to another, by externally changing voltage. Charge can vary by varying voltage and capacitance. Image sensors and computer memories use CCDs.
Electric forces on materials can pull electrons in one direction and protons in opposite direction {induction, charge} {charge induction}|.
dielectric
Conductors have free charges, so charges move to counter outside electric force, with no net charge. Dielectrics have no free charges, so induction pulls electrons and protons apart to make induced charge and dipoles.
factors
If electric field is more, electric force is more, and system has more dipoles. If atoms are small, smaller mass moves easier, and system has more dipoles.
factors: temperature
In polar materials, if temperature is lower, material has fewer random motions, and material has more dipoles. In non-polar materials, temperature has little effect.
factors: frequency
If electric-field frequency is more than 10^10 Hz, dipole moments cancel, because dipole moments change slower than field changes. If electric-field frequency is above 10^11 Hz, bending and stretching dipole moments cancel, because vibrations are slower than frequency, and only electrons affect polarization.
examples
Sifting sugar or streaming water through electric fields illustrates charge induction.
Outside electric force on dielectrics can pull electrons one way and protons opposite way, to separate charges {dipole}|. Negative charges are at one end, and positive charges are at other end, along outside-electric-field direction.
Instruments {electroscope}| can detect static electricity.
Friction can cause glow {St. Elmo's fire}| around objects in storms.
Objects can have stationary extra surface charges {static electricity}|. Electric charge is on material surface, because electrons repel each other to farthest points. More charges are at higher-curvature surface points, because repulsions are less where average distances are more. Sparks, van de Graaf generators, pith balls, cloths and rods, and electroscopes demonstrate static electricity.
Ions can have charge {valence, ion}, of -7 to +7.
Electric force depends on charge and distance {Coulomb's law} {Coulomb law}. Electric force F between two charges varies directly with charge q and varies inversely with square of distance r between charges: F = k * q1 * q2 / r^2 = (1 / (4 * pi * e)) * q1 * q2 / r^2.
permittivity
Electric-force constant k depends on medium electric permittivity e: k = 1 / (4 * pi * e).
distance
Force varies with distance squared, because space is isotropic in all directions, time has no effect, and field-line number stays constant as surface area increases. Sphere surface area = 4 * pi * r^2.
charge
Force depends on both charges, because force is interaction. Electric force depends on charge linearly, because charge directly causes force. Because charges can be positive or negative, electric force can be attractive positive or repulsive negative. If both charges are positive or negative, electric force is positive. If one charge is positive and one charge is negative, electric force is negative.
comparison
Electric force is very strong compared to gravity. Gravitational force and electric force equations are similar, because interactions cause both forces and both forces radiate in all directions.
voltage
dW = F * ds = q * dV. F = q * dV / ds = q * E.
In potential equations {d'Alembert equation, electromagnetism} for electric and magnetic fields, source-charge density and three current-density components make four potentials for each field.
For stationary magnet and moving wire in a circuit, electric force F makes electric current, and force varies directly with magnetic-flux (phi, depending on magnetic field B and surface area A) change over time {Faraday law of induction} {Faraday's law of induction}: F ~ d(phi)/dt, and phi = sum over A of B. Induced electric current makes magnetic field opposed to stationary magnetic field. For moving magnet and stationary wire, electric field E makes electric current in wire, and electromotive force on charges varies directly with magnetic-flux change over time. Faraday law of induction applies to both Maxwell-Faraday equation, for changing magnetic field and stationary charge, and Lorentz force law, for stationary magnet and moving wire in a circuit.
A constant {fine-structure constant} {coupling constant} measures electromagnetism force strength (Sommerfield) [1916]. It has no dimensions. It equals 7.297 * 10^-3 ~ 1/137. The fine-structure constant depends on electron charge, Planck constant, light speed, and permittivity or permeability or Coulomb constant. The coupling constant measures photon-electron force.
For changing magnetic field and stationary charge, changing magnetic field B makes electric field E {Maxwell-Faraday equation} {Faraday's law}: curl of E = partial derivative over time of B. This has an integral form {Kelvin-Stokes theorem}: line integral of E = integral over surface area of partial derivative of B with time.
For stationary magnet and moving wire in a circuit, Lorentz force F on charges makes electric current and electric force varies directly with electric charge q and with wire velocity v and magnetic field B cross product {Lorentz force law}: F ~ q * (v x B). Induced electric current makes magnetic field opposed to stationary magnetic field.
Flux equals integral of electric field E over area A, which equals sum of charges q divided by electric permittivity e {Gauss' law} {Gauss law}: integral of E * dA = (sum of q) / e. Gauss' law can find electric field and voltage.
Divergence of magnetic field B equals zero {Gauss law of magnetism} {Gauss's law of magnetism} {transversality requirement} {absence of free magnetic poles}: divergence of B = 0, or line integral over a surface of B = 0. Magnetic fields are solenoids. Magnetic "charges" are dipoles, and there are no magnetic monopoles.
Electric fields are like force lines {force lines} {lines of force}| radiating from center outward in all directions. Force lines per area equal electric field. Force lines have direction, from positive to negative, because test charges are positive. Force lines entering closed surfaces are negative. Force lines leaving closed surfaces are positive. For large charged objects, electric-field lines are perpendicular to surfaces, because force lines are symmetric around surface perpendiculars.
Electric-force lines pass through areas in directions {flux, electric} {electric flux}. Positive and negative fluxes from different sources add together. Does infinite flux exist? Perhaps, field lines cannot come closer than Planck length. Then flux has maximum density, and field has no infinities.
For electricity, energy W {electric energy}| is charge q times voltage V: W = q*V. Electric energy is in joules: W = F*s = (k * q * Q / s^2) * s = q * (k * Q / s) = q * (E / s) = q*V, where F is electric force, k is electric-force constant, Q is charge, E is electric field, and s is distance.
Electric charge causes potential energy {electric field}| that radiates in all directions. Electric fields can cancel each other, because charges can be positive or negative.
potential
Electric charges q Q make electric force F, which decreases with distance r squared: F = k * q * Q / r^2, where k is electric force constant. Electric-field strength intensity H changes with distance r from charge Q: H = F/q = k * Q / r^2, where k is electric-force constant. Electric field depends on material electric permittivity.
Different distances have different potential energies. Charge can move between two electric-field points, causing potential-energy-change potential difference. Electric-field energy change E is potential difference V times charge q: E = F*s = (k * q * Q / s^2) * s = q * (k * Q / s) = q*V. Potential-energy difference is work done by electric force as charge moves through distance.
examples: surface
Potential is equal all over large charged-object surfaces. Otherwise, electrons flow to lowest-potential location to equalize potential.
examples: plate
Electric field H above charged plates equals charge q divided by electric permittivity k: H = q/k.
examples: rod
Electric field H above long rods varies as reciprocal of distance d from rod: H = C * (1/d).
examples: point
Electric field H around point charges or spheres varies as reciprocal of square of distance r from center: H = k * Q / r^2.
examples: dipole
Electric field H around dipole varies as reciprocal of cube of distance d from dipole center: H = C * (1 / d^3).
For electricity, power P {electric power}| is current I times voltage V: P = dE / dt = V * dq / dt = V*I.
Pressure on crystals can cause voltage {piezoelectricity}|. Pressure polarizes crystals, such as quartz, mica, or lead zirconate titanate (PZT). Pressure changes polarized-material charge separation to make voltage. In reverse, applying electric field contracts crystals in field direction.
Tendency for charges to flow depends on electric energy per charge {voltage}| {potential difference}. Higher potential is positive and attracts negative charge. If two points have potential difference and path exists, charge flows from one point to the other.
energy
Because field is electric force F divided by charge q, voltage V is electric field H times distance ds moved in field direction: V = H * ds = (F/q) * ds = (F * ds) / q = W/q, where W is electric energy. Voltage is electric energy divided by charge.
field
Separating charges using work creates electric field, with voltage between charges. Batteries separate charges to create voltage. Electromagnetic induction creates voltage by separating charges. Voltage V equals area A times negative of field change dH divided by time change dt: V = A * -dH / dt. Voltage V equals negative of inductance I times current change di divided by time change dt: V = -I * di / dt.
Electric fields {wakefield} can pulse and so force electrons to accelerate.
Moving-electron and stationary-molecule interactions oppose electric current flow {resistance}| and turn electric energy into heat. Electrical resistance depends on path length, cross-sectional area, and material resistivity.
voltage
Resistance makes heat from electrical kinetic energy and, as potential energy decreases, drops potential across resistor.
current
For same voltage, more resistance makes less current, because flow slows.
factors
Resistance is more for poor conductors with few electrons that can move, for longer conductor length, or for less cross-sectional area. If cross-sectional area is more, conductor perimeter is more, fewer electron collisions happen, and resistance is less. If conductor length is more, distance is longer, and total resistance is more. If material resistivity is more, conductor has fewer free charges, and resistance is more.
factors: temperature
In conductors at higher temperature, resistance is more, because random motions are more. In insulators or semiconductors at higher temperature, resistance is less, because more electrons are free to move. Alloys have smaller resistance change when temperature changes, because alloys have fewer free electrons than pure metals.
resistor
Electrical devices {resistor} can have resistance. Conductor resistance R equals material resistivity r, which differs at different temperatures, times conductor length l divided by conductor cross-sectional area A: R = r * l / A.
resistivity
Resistivity is 10^-6 to 10^-1 ohm-cm for conductors, 10^-1 to 10^8 ohm-cm for semiconductors, and 10^8 to 10^21 ohm-cm for insulators.
examples
Resistance in incandescent light bulbs creates light. Resistance in electric heaters creates heat. Fuses, circuit breakers, voltmeters, ammeters, tube resistors, rod resistors, and coil resistors demonstrate electrical resistance. Lie detectors detect skin electrical resistance, which varies with sweat amount.
Resistance reciprocal {conductance}| measures current-flow ease.
At same temperature, electrical-conductivity to heat-conductivity ratio is the same for all metals {law of Wiedemann and Franz} {Wiedemann and Franz law}.
Materials {conductor}|, such as metals, can allow electrons to move almost freely.
dipoles
Because electrons are free to move, no dipoles form. Conductor dielectric strength is zero, and dielectric constant is infinite, because charges can move freely. Rubbing metal with cloth cannot rub off charges, because electrons move freely and quickly in conductors.
spark
If charge touches conductors, electrons flow to neutralize charge, typically making sparks.
spread
Potential difference between conductor points is zero, because all electrons already repel each other equally. No electrons flow.
compression
Compressing metal increases conductivity, because crystals have fewer imperfections.
Materials can allow electrons to move almost freely {conduction, electricity}|. Semiconductors allow electrons to move with high resistance. Insulators do not conduct electricity. Circuit loads are either conductors or semiconductors.
Electric force causes average drift velocity per unit force {mobility, charge} {charge mobility, conductor}|.
Metallic bonds are electron deficient and leave electrons free {free electron}. Metal has electrons that can move among atoms around metal surface. Outside electric force can pull electrons completely away from atoms.
In conductors, moving electrons have average time between collisions {mean free time}| and have average distance, mean free path, between collisions. Collisions tend to reduce charge velocities.
In conductors, voltage V equals resistance R times current I {Ohm's law} {Ohm law}: V = R*I.
Most materials {insulator}| {dielectric} allow no free electron movement. Air, vacuum, paper, and glass are insulators.
dipoles
Outside electric field separates electrons and protons, to make induced charge. Inducing charge can be easy or hard. Dielectric strength is ratio between material capacitance and vacuum capacitance. For vacuum, dielectric constant is 1. For insulators, dielectric constant is 1 to 8. For water, dielectric constant is 81, because water has high polarization and free dipole rotation. For conductors, dielectric constant is infinite. Lustrous metals have negative dielectric constant.
Materials have ease by which electric fields can go through {permittivity}|. Metals have free electrons and cannot have electric fields inside. Insulators have charges that move relative to electric field and oppose electric field. Empty space has no charges and allows electric field. Electric-force constant k inversely depends on permittivity.
Insulators have different abilities to make dipoles {polarizability}| {polarization, electricity}. If polarization is more, refraction index is more. Polarization K is refractive index n squared: K = n^2. Metal has free electrons and cannot make dipoles. Empty space has no charges and cannot make dipoles.
Materials {semiconductor}| can have electrons that can move from atom to atom in atomic-orbital conducting bands. Silicon and germanium are semiconductors. Semiconductor compounds include indium gallium arsenide and indium antimonide.
impurities
Semiconductors can be silicon with added gallium {P-type semiconductor} or arsenic {N-type semiconductor}. P-type semiconductors transfer electron vacancies. N-type semiconductors transfer electrons. Holes and electrons must move in opposite directions to complete circuit.
electric charge
If charge touches semiconductor, no change happens, because semiconductor electrons are not free to move.
Impurities {doping}| added to silicon or germanium supply more negative or positive charges, to make more conduction.
donor
Adding material with five electrons in highest orbital {donor impurity} adds extra electron. Antimony, arsenic, and phosphorus are donors {n-type semiconductor}.
acceptor
Adding material that has three electrons in highest orbital {acceptor impurity} results in extra proton {electron deficiency}. Gallium, indium, aluminum, and boron are acceptors {p-type semiconductor}.
junction
If p-type semiconductor touches n-type semiconductor, electrons in n-type semiconductor flow into holes in p-type semiconductor until reaching balance, with voltage across junction. p-type semiconductor has become slightly negative. n-type semiconductor has become slightly positive. No more free charges exist. Junction width is 50 atoms.
diode
If voltage across np junction makes p side positive, current flows greatly, because p side attracts electrons. If voltage across np junction makes p side negative, no current flows, because p side repels electrons. np junctions allow current in only one direction and allow current to be ON or zero OFF, like diodes.
Semiconductors can emit light {electroluminescence}| across pn junctions when current flows. Phosphors can glow if AC current passes through. Machine sprays glass panel with thin transparent metal layer, adds a phosphor layer, and adds thin metal foil. Electroluminescence is efficient and cool but allows only low light levels.
One electron and one hole {exciton} can bind electrostatically for 4 to 40 microseconds, 1150 nanometers apart. Electric forces cause free electrons and holes to drift in opposite directions, at same velocity. Electric force causes average drift velocity per unit force {charge mobility, exciton}. When electron meets hole, they merge. At constant force, ejections and recombinations are in equilibrium.
A thin layer of electrons is between two semiconductors. Near 0 K in high magnetic field [1982], pairs {quasiparticle, pair} of excited superposition of electron states have fractional charges {fractional quantum Hall effect}, with edge effects {edge state}. Fractional quantum Hall effect can extend to four dimensions, as on five-dimensional-sphere surfaces, which have three-dimensional edge states that emerge with relativity. Excitations can carry magnetic-flux units.
Adding electron-deficient materials, with three electrons in highest orbital, results in extra protons, because of electron vacancies {hole}|.
Two semiconductors can have insulator between them {Josephson junction}. Microwaves can supply energy to electron pairs. Voltage V is n = 1/2, 1, 3/3, or 2 times Planck's constant h times frequency f divided by electron charge e: V = (n*h*f) / (2*e). Third semiconductor can supply control current. Control current sets voltage at zero or one, using quantum-mechanical tunneling.
Two semiconductors, or semiconductor and conductor, can meet in region {junction}| 50 molecules thick. Contact point between metal and semiconductor has resistance that does not follow Ohm's law, because current depends on surface properties.
Semiconductors {metal oxide semiconductor} (MOS) can be metal oxides, which can be unipolar, rather than just bipolar.
Solid-state semiconductor circuit elements {transistor, electronic}| amplify current.
types
N-type semiconductor, P-type semiconductor, and N-type semiconductor can join in sandwich {NPN transistor}. P-type semiconductor, N-type semiconductor, and P-type semiconductor can join in sandwich {PNP transistor}. NPN or PNP transistors are bipolar transistors {junction transistor}, with two junctions. Weak signals control current flow. Junction transistors are current operated.
parts
Transistors have cathode emitter, anode collector, and base controller. Collector is between emitter and base. In PNP transistors, electrons flow from emitter to middle collector and from base to middle. In NPN transistors, electrons flow from middle collector to emitter and from middle to base.
process
Holes and electron diffusion across semiconductor np junction continues until electric force equilibrium, preventing further diffusing. Voltage is across emitter and base and across collector and emitter. Applying small positive charge to base attracts electrons and amplifies current 10^5 times. Electron flow from emitter to collector multiplies directly with voltage from base to emitter.
surface barrier
A depletion layer {space charge, transistor} between metal and semiconductor can control conductivity {point contact semiconductor} {surface barrier transistor}.
field effect
Electric field at right angles to silicon surface causes lateral conductance {field-effect transistor}. Insulated field plate can have field that induces conducting surface channel between two surface pn junctions {gate}, as in field-effect transistors, such as metal oxide semiconductors. Field-effect transistors have slow response and high impedance. They are voltage operated, rather than current operated.
electron tubes
Transistors can replace all electron tube types.
In PNP transistors, electrons flow from emitter to collector, and from other sandwich side {base, transistor} to middle.
In PNP transistors, electrons flow from emitter to middle {collector, transistor}, and from base to middle.
In PNP transistors, electrons flow from one sandwich side {emitter, transistor} to collector, and from base to middle.
At low temperatures, substances can be electrical conductors {superconductor} with no electrical resistance {superconductivity}|. Liquid oxygen, liquid nitrogen, and liquid hydrogen are superconductors. Organic crystals, metal oxides, and insulators can have superconductivity.
high temperature
Most high-temperature superconductors are copper oxide {cuprate} layers between other layers. Mercury-barium-calcium copper oxide superconducts at 164 K under 10,000 atm pressure and at 138 K at 1 atm.
Iron and arsenic layers between a lanthanum, cerium, samarium, neodymium, or praseodymium layer and an oxygen or fluoride layer can superconduct up to 52 K. Magnesium boride superconducts at 39 K. Bismuth, strontium, calcium, copper, and oxygen atoms can combine to make BSCCO high-temperature superconductor. Yttrium, barium, copper, and oxygen atoms can combine to make YBCO high-temperature superconductor.
cause
Large-scale quantum effects cause superconductivity, which happens when energies are small, such as at low temperature. Bosons in same quantum state can condense {Bose-Einstein condensation, superconductivity} (BEC) from gas to liquid. Repulsive bosons condense better. Materials can Bose-Einstein-condense at cold temperatures.
Fermions, such as electrons, form Cooper pairs at temperature lower than temperature at which material becomes degenerate Fermi gas. Both electrons have same spin. Making electron pairs makes positive metal ions. Cooper pairs have streamline flow through metal ions and travel with no resistance.
Fermions can pair more easily if attraction increases. Electrons can resonate {Feshbach resonance} in magnetic fields {magnetic resonance} and so pair better.
Magnetic flux has quanta in superconductors. Electric field has no quanta but quantizing it can make calculations easier.
magnetic field
Outside magnetic field can enter only short distance into superconductors with current, because photons acquire mass as electromagnetic gauge symmetry breaks spontaneously.
insulator
Forcing atoms {Mott insulator} in Bose-Einstein condensates to have definite positions changes quantum properties.
In superconducting materials, electrons distort positive-ion lattice to make phonons, which interact with other electrons, causing attraction and electron pairing {BCS theory}. Critical temperature is higher if more electrons can go to superconductive state, if lattice-vibration frequencies are higher, and if electrons and lattice interact stronger. Mercury-barium-calcium copper oxide does not follow BCS theory. Magnesium boride follows BCS theory. Liquid oxygen, liquid nitrogen, and liquid hydrogen follow BCS theory.
Fermions can pair {Cooper pair} at temperatures much lower than temperature at which system is degenerate Fermi gas.
Systems {degenerate Fermi gas} can have one fermion in each low-energy quantum state.
Devices {transition edge sensor} can detect photons at transition temperature to superconductivity.
Electric charges can flow past a point over time {current, electricity}| {electric current}. Flowing charge has one-tenth light speed.
Current I per area A {current density}| equals conductivity K times electric field E: I/A = K*E. Current I equals current density j times cross-sectional area A: I = (I/A) * A = j*A.
Current tends to stay on conducting-material surface {skin effect}, because electrons repel each other.
Charge flow can alternate directions over time {alternating current}| (AC). Alternating voltages cause electrons to oscillate. Commercial alternating current oscillates at 60 Hz in USA and 50 Hz in most other countries.
comparison
Direct current has less power loss and more average power than alternating current.
voltage
In alternating-current circuits, average or effective voltage equals maximum voltage divided by square root of two. In alternating-current circuits, average or effective current equals maximum current divided by square root of two.
transformer
Transformers change alternating-current voltage.
Charge flow can be in one direction only {direct current}| (DC). Direct current has less power loss and more average power than alternating current. Adding voltage sources alters direct-current voltage.
Charges must flow from source around loop back to source {circuit, electricity}| {electric circuit, flow}. At circuit points, current that goes in must equal current that comes out. Voltages around circuit loops must add to zero. Otherwise, electron density increases somewhere in circuit.
Earth {ground, electricity}|, or conductor leading to Earth, is an electron sink. Earth has many molecules and can absorb or give any number of charges without changing potential.
Circuit current and voltage follow laws {Kirchoff's laws} {Kirchoff laws}. Kirchoff's laws apply to circuit steady state. Transiently, Kirchoff's laws can break.
At circuit points, current coming in equals current going out. If current flowing in did not equal current flowing out, charge builds or falls at point and repels or attracts incoming charges, to make net charge return to zero.
Around circuit loops, voltages add to zero. If voltages around loop do not add to zero, extra voltage sends more charges to low-voltage point and so makes sum of voltages become zero again.
Wave-front amplitude or phase modifications {modulation, electricity}| can carry information on carrier waves. Information can vary amplitude {amplitude modulation, circuit} (AM) or frequency {frequency modulation, circuit} (FM). Frequency modulation carries temporal information along propagation line. Amplitude modulation carries spatial information perpendicular to propagation line.
Circuits {short circuit}| can have almost no resistance, allowing very high current.
RC, RL, and RLC circuits have reactance and resistance vector sum {impedance}|. Voltage equals impedance times current. Maximum power has equal source and circuit impedances.
Inductance and capacitance {reactance}| aid or impede current flow by storing and releasing energy, without heat loss.
comparison
Resistance opposes current and has heat loss.
phase
Reactance causes lag between voltage and current. In inductors, high frequency makes big current change and so large voltage. High inductance makes current changes make big voltage changes. Inductive reactance R equals two times pi times frequency f times inductance L: R = 2 * pi * f * L. In capacitors, high frequency makes voltage stay low, because little charge can build up. High capacitance requires large charge to make voltage, so voltage stays low. Capacitive reactance R equals reciprocal of two times pi times frequency f times capacitance C: R = 1 / (2 * pi * f * C).
Electrical devices {capacitor} {capacitance}| can store electrical energy. Electric-energy storage ability C is charge Q divided by voltage V: C = Q / V. Capacitance C equals material dielectric strength d times length l divided by cross-sectional area A: C = d * l / A.
field
In capacitors, electric field stores energy E: E = 0.5 * Q * V = 0.5 * C * V^2.
current
Electric-field energy builds as current flows. Electric-field energy tends to push current out. Current I is capacitance C times voltage change dV over time change dt: I = (1 / C) * dV / dt. Current and voltage are out of phase.
parallel plates
In parallel-plate capacitors, capacitance C equals electric permittivity e times dielectric constant k times cross-sectional area A divided by distance d between plates: C = e * k * A / d. Electric field between plates is constant and perpendicular to plates. If plates are farther apart, charge separation is more, voltage is more, and capacitance is less. If plate area is larger, charges spread out more, voltage is less, and capacitance is more. If dielectric constant is greater, material between plates has more polarization, field is less, voltage is less, and capacitance is more.
examples
Disk capacitors and rod capacitors work like parallel-plate capacitors. Two aluminum pie plates can make capacitor. Leyden jars can store charge as capacitance. Electrolytic capacitors allow only one-way current.
Circuits with current can store magnetic energy {inductance}|. Circuit devices {inductor} can store magnetic energy. Inductors are wire coils, so current makes strong magnetic field down coil middle. Soft iron bar can be in middle.
energy
Energy stored depends on current change compared to voltage. Inductance L is voltage V divided by current change dI with time change dt: L = V / (dI/dt). V = L * dI/dt. Magnetic-field energy builds as current flows. Magnetic-field energy tends to push current to stop. Current and voltage are out of phase. Magnetic-field energy E equals half inductance L times current I squared: E = 0.5 * L * I^2.
factors
Inductance increases as coil area increases, current-change frequency decreases, space magnetic-permeability increases, current decreases, voltage increases, coil-turn number decreases, and inductor length increases.
mutual inductance
Two coils with current have mutual inductance.
Circuit loads can be on separate wires {parallel circuit}|. Loads split currents. Voltages are equal. If circuit voltage sources are on separate wires, currents add, and voltages are equal. For resistances in parallel, resistance reciprocals add to equal total-resistance reciprocal, because cross-sectional area is more. Capacitances in parallel add, because area is more. For inductances in parallel, inductance reciprocals add to equal total-inductance reciprocal, because area is more.
Circuit loads can be in same wire {series circuit}|. Currents through loads are equal. Loads split voltages. If circuit voltage sources are in same wire, currents through loads are equal and voltages add. Resistances in series add, because length is longer. For capacitances in series, capacitance reciprocals add to equal total-capacitance reciprocal, because distance is more. Inductances in series add, because length is more.
Two circuits {coupled circuit} can share impedance, allowing energy transfer.
Circuits {filter circuit} can transmit frequency range, while blocking other frequencies. Filter circuits can remove frequency range, while allowing other frequencies, by differentiating or averaging. Circuits can choose different frequencies {selectivity, filter}. Frequency filtering sharpens edges, because edges have high frequency, and blurs have low frequency.
Circuits {LC circuit} can have inductor and capacitor. Energy flows from inductance magnetic field to capacitance electric field and then from capacitance electric field to inductance magnetic field, at resonance frequency. In resonating circuits, capacitance and inductance reactances are equal, so total reactance equals zero.
Circuits {RC circuit} can have resistor and capacitor. Voltage V depends on time t: V = Vo * e^(-t / (R*C)). Switching on circuit makes voltage build up in capacitor as field builds. Current lags behind voltage. At voltage alternation frequency, circuit resonates.
Circuits {RL circuit} can have inductance and resistor. Current I depends on time t: I = Io * e^(-t * R / L). At switching on, current changes fast, so voltage in coil is high. Then current becomes constant so voltage goes to zero, and current lags behind voltage. At current-alternation frequency, circuit resonates.
Thermionically emitted electrons can travel in beams, under side-electromagnet control, to fluorescent screens, where they excite phosphor crystals to make light {cathode ray tube}|. Cathode ray tubes are in TVs and oscilloscopes.
Circuit devices {detector} can select one signal from several.
Solid-state semiconductor circuit elements or vacuum tubes {diode}| can allow current to flow in only one direction. Diodes change alternating current to direct current. Tubes can have cathode emitter and anode plate. If plate is positive, emitted electrons flow toward plate. If plate is negative, no emitted electrons flow. For solid state, np junction allows charge to flow only in one direction, from P-type to N-type, with high resistance in other direction.
Circuit devices {mixer, signal} {electric circuit, mixer} can combine frequency signals.
Circuit devices {oscillator, circuit}| {electric oscillator} can change voltage waveform to other frequencies and amplitudes.
Devices {rectifier}| can change alternating to direct current.
Devices {rheostat}| can make variable resistance.
Circuit devices {wave shaper} can change voltage waveform.
Sunlight electric potential can make electric current in materials {photocell}|.
Shining light onto selenium {selenium cell} increases conductivity, because light increases electric field.
Instruments {ammeter}| can measure currents.
Instruments {galvanometer}| can measure small currents and current direction.
Instruments {ohmmeter}| can measure resistance.
Instruments {potentiometer}| can measure voltages at zero current, as ratio to exactly known voltage.
Instruments {voltmeter}| can measure electric potentials.
Devices {Wheatstone bridge}| can find resistance or capacitance in circuits using ratios. Wheatstone bridges eliminate voltage effects. AC current negates overall flow effects.
Current from a potential source P splits between two known resistances, R1 and R2, which have a galvanometer G across their endpoints to measure current and voltage (Figure 1). From one endpoint is an adjustable resistance Rv. From the other endpoint is an unknown resistance Rx. The resistances Rv and Rx meet at a point. If the ratio of the adjustable resistance Rv to the first known resistance R1 equals the ratio of the unknown resistance Rx to the second known resistance R2, the galvanometer has zero voltage and current.
For typical resistances, one can set R1 = R2, so Rv = Rx at galvanometer zero current and voltage.
If three resistances are known and one unknown, the measured voltage allows calculating the unknown resistance.
Wheatstone bridges can also find capacitance.
Relativistic electric-charge motion can caused electric force {magnetism, force}|. Magnetic fields have no net charge to stationary observers.
special relativity
Atoms and molecules have equal numbers of protons and electrons and so no net electric charge. Protons are in nuclei. Electrons orbit nuclei at 10% light speed. At that speed, motions have relativistic effects, and observers see length contraction. Stationary protons observe moving electrons, and electrons observe moving protons. Length contraction makes charges appear closer together along motion-direction line. Moving-charge density appears higher than stationary-charge density, making net electric force. Electric-charge number does not change, but relative distance decreases.
materials: iron
If electron orbits do not align, relativistic effects have all directions, and net force is zero. If electron orbits align, as in ferromagnetic materials, net force is not zero, and material has magnetism.
materials: conductors
Conductors have fixed protons and easily transferable electrons, with no net charge. Electric current moves electrons in wires at 10% light speed. Relativistic length contraction makes apparent increase in relative electric-charge density and apparent electric force. Current makes magnetism.
non-magnetic materials
People and non-magnetic materials have random molecule orientations and so no net magnetic effects.
no dipoles
Apparent electric charge in magnetism is not induced charge. Magnetism has no dipoles.
strength
At 10% light speed, relative electric-charge density increases by 1%, so magnetism is approximately one-hundredth electric-force strength. Larger currents make stronger magnetic forces. Electric generators and motors use many wires with high currents, to make strong magnetism.
direction
Electric longitudinal force between charges is along line between charge centers. Because it has no net charge, magnetic apparent-electric force cannot be along line between apparent charge centers. Magnetic transverse force is across line between apparent charges, along motion line, because apparent charge density increases only along motion line.
attraction and repulsion
Like electric force, magnetic force depends on interactions between charges. Like electric force, magnetic force can be attractive or repulsive. If apparent moving charges and stationary charges are both positive or both negative, magnetism is repulsive, because charges observe like charges. If apparent moving charges and stationary charges have opposite charge, magnetism is attractive, because charges observe unlike charges.
Wires at Rest with No Current
Charges are equal on both wires, and there is no movement and so no relativistic effects, so net force is zero. See Figure 1.
Wires at Rest and One Wire with Current
Stationary protons on wire with current see stationary protons and stationary electrons on other wire and so see no relativistic effects. Stationary protons on wire with no current see stationary protons and moving electrons on other wire and so see relativistic negative charge, making attractive force. Stationary electrons on wire with no current see stationary protons moving electrons on other wire and so see relativistic negative charge on other wire, making repulsive force. Moving electrons on wire with current see moving protons and moving electrons on other wire and so see relativistic effects, but they cancel. One force is attractive and one is negative, so net force is zero. See Figure 2.
Wires at Rest and Opposite Currents
Protons in both wires see stationary protons and moving electrons in other wire and so see relativistic negative charge on other wire, making attractive force. Electrons in both wires see moving protons and moving-twice-as-fast electrons and so see net relativistic negative charge on other wire, making large repulsive force. Net force is repulsion. See Figure 3.
Wires at Rest and Same Currents
Protons in both wires see stationary protons and moving electrons in other wire and so see relativistic negative charge on other wire, making attractive force. Electrons in both wires see stationary electrons and moving protons in other wire and so see relativistic positive charge on other wire, making attractive force. Net force is attraction. See Figure 4.
Stationary Conductor and Stationary Test Charge
See Figure 5. Stationary conductors, with equal numbers of fixed protons and easily movable electrons, have no net charge. Electric field from protons is equal and opposite to electric field from electrons, so there is no net electric field. Conductor is not moving relative to anything, so there are no relativistic effects. Stationary single negative test charge has electric field but feels no net force from conductor, because conductor has no net charge. Test charge is not moving relative to anything, so there are no relativistic effects. Net force is zero.
Stationary Conductor and Moving Test Charge
See Figure 6. Stationary conductors have no net electric field. Negative charge moves downward at constant velocity. Constantly moving charge has constant concentric magnetic field, which represents magnetic-force direction and strength that it exerts if it observes apparent charges. Test charge feels no net electric force from conductor, because conductor has no net charge. Test charge moves relative to both electrons and protons in conductor, so there is no net relativistic effect. Net force is zero.
Moving Conductor and Stationary Test Charge
See Figure 7. Conductor moves downward at constant velocity. Electric field from protons is equal and opposite to electric field from electrons, so there is no net electric field. Magnetic field from moving protons is equal and opposite to magnetic field from moving electrons, so there is no net magnetic field. Negative charge is stationary. Test charge feels no net electric force from conductor, because conductor has no net charge. Test charge moves relative to both electrons and protons in conductor, so there is no net relativistic effect. Net force is zero.
Moving Conductor and Moving Test Charge
See Figure 8. Conductor moves downward at constant velocity. Net electric and magnetic fields are zero. Negative charge moves downward at constant velocity. Test charge feels no net electric force from conductor, because conductor has no net charge. Test charge is not moving relative to either electrons or protons in conductor, so there are no relativistic effects. Net force is zero.
Moving Electrons in Stationary Conductor and Stationary Test Charge
See Figure 9. Conductor electrons move downward at constant velocity. Electric field from protons is equal and opposite to electric field from electrons, so there is no net electric field. Moving electrons make magnetic field. Negative charge is stationary. Test charge feels no net electric force from conductor, because conductor has no net charge. Test charge is not moving relative to protons in conductor, so there is no relativistic effect. Test charge moves relative to electrons in conductor and sees relativistic negative charge, making repulsive force.
Moving Electrons in Stationary Conductor and Moving Test Charge
See Figure 10. Conductor electrons move downward at constant velocity. Electric field from protons is equal and opposite to electric field from electrons, so there is no net electric field. Moving electrons make magnetic field. Negative charge moves downward at constant velocity. Test charge feels no net electric force from conductor, because conductor has no net charge. Test charge is not moving relative to electrons in conductor, so there is no relativistic effect. Test charge moves relative to protons in conductor and so sees relativistic positive charge, making attractive force.
Electric-charge relativistic motion causes weak electric force {magnetic force}| transverse to motion direction. Magnetic fields are electric fields caused by relativistic charge motions that make excess electrons or protons appear. Magnetic fields have no net charge to stationary observers.
examples
Wire in magnet field, tube and magnet, TV tube and magnet, two wires with current, and carpenter's bubble illustrate magnetic fields.
force
Magnetic force F equals moving charge q times velocity v times stationary-object magnetic field B times sine of angle A of approach to stationary object: F = q * v * B * sin(A). Magnetic force F equals wire current I times wire length L times stationary-object magnetic field B times sine of angle A between wire and stationary object: F = I * L * B * sin(A). Magnetic force F equals space magnetic permeability k' times wire current I1 times current I2 in other wire divided by distance r between wires: F = k' * I1 * I2 / r.
distance
Magnetic force depends on distance between wires, not distance squared, because relativistic effects are transverse to current motion.
Torques require moments {magnetic moment}. Magnetic moment M equals current i times coil area A: M = i * A. Magnetic moment equals pole strength p times path length l: M = p * l.
If positive current points in right-hand finger direction {right hand rule, magnetism}|, magnetic-field direction {north magnetic pole} points in thumb direction. The opposite direction is the other pole {south magnetic pole}.
Magnetic dipoles have magnetic force lines {magnetic field}| {flux density} {magnetic intensity} {magnetic induction}, from south pole to north pole. Magnetic field H is magnetic force F divided by pole strength p: H = F/p.
wire
Around wires, magnetic field H is space magnetic permeability k' times current I divided by two times pi times distance d from wire: H = (k' * I) / (2 * pi * d). Around solenoids, magnetic field H is space magnetic permeability k' times wire-turn number n times current I: H = k' * n * I. Around toroids, magnetic field H is space magnetic permeability k' times wire-turn number n times current I divided by two times pi times toroid radius r: H = (k' * n * I) / (2 * pi * r).
direction
Positive current in thumb direction makes magnetic field that circles conductor in right-hand finger direction {right hand rule, magnetic field}.
Numbers {magnetic flux}| of magnetic-field lines go through areas.
Magnetic field B times distance ds charge moves in field equals field magnetic permeability µ times current I {Ampere's circuital law} {Ampere circuital law} {Ampere's law}: integral of B * ds = µ * I. Current flows inside path of distance.
Because relativistic effects have small energies, atoms have quantized electric and magnetic fields. Magnetism quantum {Bohr magneton} is small magnetic pole.
Magnetic field relates to magnetic flux {Biot-Savart law}.
Energy conservation causes voltage from electromagnetic induction to make magnetic field opposed to original magnetic field {Lenz law}.
In dynamos or motors, electric and magnetic forces induce currents and voltages {electromagnetic induction}|.
outside force
If force moves conducting material through magnetic field or moves magnetic field near conducting material, protons and electrons in conductor move relative to protons and electrons that caused magnetic field. Moving protons and electrons make two electric currents that make two magnetic fields around conductor. Outside force provides energy to make magnetic fields.
However, no net charge moves, and test charges detect no electric current, because protons and electrons move together, so charges cancel.
induction
The original magnetic field interacts with both generated magnetic fields, setting up relativistic electric forces. Forces move electrons in conductor, but protons cannot move. Moving electrons make electric current opposite to movement and create magnetic field around current opposite in polarity to original magnetic field. Magnetic field created by moving electrons tends to resist relative movement between conductor and original magnetic field.
moving wire
For example, wire can moves through magnetic field. Moving wire moves wire protons and electrons, creating proton current and electron current, and currents make magnetic field around motion direction. Original magnetic field interacts with moving magnetic fields. Wire electrons are free and move down wire. Wire protons cannot move, though they feel magnetic force in opposite direction. Net current appears. Relativistic electric force separates electrons from protons, to make voltage that then makes current.
Energy for charge separation comes from outside mechanical energy used to move wire through magnetic field. Induced current makes net magnetic field that resists wire movement. Mechanical energy used to move wire makes electric field, induces current, and creates induced magnetic field.
energy transfers
In electromagnetic induction, potential energy in electric field causes voltage that makes current with kinetic energy, then current makes magnetic field with potential energy, then magnetic field slows current and builds voltage, which is potential energy in electric field. Cycle repeats.
Electric field and magnetic field, and voltage and current, are out of phase, because energy in one transfers to the other and then back again.
When electric-field change is zero and electric field maximizes, voltage maximizes and current is zero, and magnetic-field change maximizes and magnetic field is zero. As electric field decreases to zero, voltage decreases and current increases. As current increases, magnetic field increases and maximizes when current maximizes, electric-field change maximizes, and electric field is zero. As magnetic field decreases to zero, voltage increases and current decreases. As voltage increases, electric field increases and maximizes when voltage maximizes and electric field change is zero. Magnetic-field phase lags electric-field phase by 90 degrees.
examples
Electromagnetic induction happens in dollar bills in magnet, inductance coils, transformers, solenoids with iron bars, motors, and generators.
In conductors with current in magnetic fields, magnetic field pushes charges to conductor sides and makes electric field {Hall resistance, magnetism} opposed to magnetic field. Hall resistance varies with magnetic field and current.
semiconductor
In semiconductors, high magnetic field separates charges across width, not length, and so causes transverse current {Hall effect}.
quantum Hall effect
Quantum Hall resistance {quantum Hall effect} is inverse of small positive integer n times Planck's constant h divided by electron charge e squared: (1/n) * (h/e^2).
spin
In semiconductor ribbons with electric current, magnetic field from spin-orbit coupling causes excess electrons with one spin on one edge and excess electrons with opposite spin on other edge {spin Hall effect}.
In conductors with current in magnetic fields, magnetic field forces charge to conductor sides and makes electric field opposed to magnetic field {Hall resistance, current}, that varies with magnetic field and current.
Wire coil with current creates magnet with north and south poles {magnetic dipole}|.
field
Magnetic-field direction relates to current direction. By right hand rule, if positive current points in right-hand finger direction, magnetic-field direction points in thumb direction for north magnetic pole, and the opposite direction is south magnetic pole.
force
Like magnetic poles repel. Opposite magnetic poles attract. Force between magnetic poles equals space magnetic permeability k' times one magnetic-pole strength P times other magnetic-pole strength p, divided by distance r between poles: F = k' * P * p / r.
pole
Current i times path length L is pole strength p: p = i*L. Pole strength p equals charge q times velocity v: p = q*v.
infinitesimal
Infinitesimal wire loops can have unit current {elementary magnet}, to make idealized unit dipoles.
Materials have ease {permeability, magnetism}| {magnetic permeability} {mu, permeability} {µ, permeability} by which magnetic fields can go through. Permeability depends on ease with which magnetic dipoles form. Magnetic force constant k' directly depends on permeability.
types
Ferromagnetic materials have molecular magnetic fields that can align with outside magnetic field to enhance it. Non-magnetic materials and empty space have no magnetic fields and allow magnetic field. Diamagnetic materials have magnetic fields that oppose outside magnetic field. Paramagnetic materials have magnetic fields that slightly enhance outside magnetic field.
Crystals with impurities have greatly increased magnetization after crystal imperfections are overwhelmed by pressure {Barkhausen effect}.
Magnets cannot hold magnetism at high temperature {Curie temperature}, because random motions become great enough to cancel net magnetism.
In materials, all molecules in microscopic regions {domain, magnetism}| can have same magnetic-field alignment.
magnetization
After removing magnetization, domains return to original orientations {magnetic memory, domain}.
anistropy
Crystals magnetize differently on different axes {magnetocrystalline energy} {magnetocrystalline anisotropy}.
energy
Unaligned domains minimize magnetic-field potential energy {magnetostatic energy}. Boundaries between domains add potential energy {domain wall energy}. Domain-wall width increases by exchange energy but decreases by magnetocrystalline energy.
length
Crystals change length when magnetized, because domains shift {magnetostrictive energy}. Iron gets longer. Nickel gets shorter.
Electrical resistance can increase with increased magnetic field strength {extraordinary magnetoresistance} (EMR). Non-magnetic indium antimonide is a narrow gap semiconductor with high carrier mobility. Indium antimonide and gold lattice at room temperature has high EMR and so can be a magnetic-field sensor. Magnetic fields can change manganese oxide {manganite} from non-magnetic to ferromagnetic and metallic {colossal magnetoresistance} (CMR). Ferromagnetic layers with non-magnetic material between them {giant magnetoresistance} (GMR) are in disk-drive read heads.
External magnetic-field change changes material magnetization, after a time delay {hysteresis, magnetism}|. In motors and generators, external magnetic-field changes cycle, and material changes have time-delayed cycles {hysteresis loop}, with heat losses. Magnetic memory devices {twistor, memory} can use hysteresis loops.
Magnets can align all domains and have maximum magnetization {saturation, magnetism}|.
Magnetic materials {spin-glass} can have disordered magnetic domains that couple and make long-range effects.
Outside magnetic field causes weak, oppositely acting magnetism {diamagnetism}| in all materials. Outside magnetic field changes atom electron spins and electron orbits. Bismuth has the most diamagnetism. Two diamagnetic materials repel each other.
Solenoid coils can have large magnetic field that points down middle in one direction {electromagnet}|.
Outside magnetic field can induce weak enhancing magnetism {paramagnetism}| in materials, by affecting permanent magnetic dipole moment caused by unpaired-electron spin. Manganese, palladium, and metallic salts are paramagnetic. Paramagnetism is slightly stronger than diamagnetism. Higher temperature increases paramagnetism, by making longer dipoles. Two paramagnetic materials attract each other, because they have magnetic dipoles.
In materials, paramagnetism {ferrimagnetism}| can subtract from magnetic field. Manganese oxide is ferrimagnetic.
Materials can have asymmetric electron distributions in molecule outer orbits {ferromagnetism}|. Odd number of electrons allows materials to have permanent magnetism.
examples
Iron, nickel, cobalt, alnico alloy, liquid oxygen, lodestone, iron particles, magnetite, and ferrite have ferromagnetism.
alignment
Atom spins can align in same direction in microscopic domains. Electrostatic forces {exchange energy} align magnetic dipoles in domain. Magnets can align all domains in same orientation to make net magnetic field.
Hard ferromagnetic materials {permanent magnet}| holds magnetism even in another magnetic field. Soft-metal ferromagnets {soft magnet} lose or change magnetism in another magnetic field.
A metal disk {magnetic brake} rotating between two permanent magnets dissipates energy, because eddy currents make magnetic field opposed to permanent magnetic field and slow disk.
After removing magnetization, magnetic domains return to original orientations {magnetic memory, computer}.
Devices {solenoid}| can have wire coils. If current is in coils, magnetic field is sum of coil magnetic fields. Large magnetic field points down coil middle. Soft iron core in coil middle increases magnetic field by adding atom magnetic fields.
Devices {transformer}| can transfer voltage from circuit with alternating current to voltage from second circuit with alternating current. Transformers induce current in stationary-wire second coil using alternating current in first coil. Power in first coil equals power in second coil. Power is circuit voltage V times wire current I times wire-coil number n: V1 * I1 * n1 = V2 * I2 * n2.
Electronics can use electron charge and spin {spintronics} {magneto-electronics}. Flowing-electron spins {spin current} can align {spin-polarized}.
resistance
Electrical resistance {magnetoresistance} can change in different-polarization magnetic layers. Electrons take curved paths, slow in current direction, and decrease current. Computer hard drives can use magnetoresistant read heads [1998].
spin
Quantum spintronics can control single-electron spin. When nitrogen atoms replace carbon atoms in diamond, adjacent locations can be empty {nitrogen-vacancy center} (N-V center). Doped diamonds can semiconduct. N-V centers make single fluorescing electrons with two energy levels, with no ionization.
Mechanical energy can turn metal coil in magnetic field to generate electric current {generator, electricity} {electric generator}.
current
Electric current is in coil leading and trailing edges. Current changes direction with coil half turns, to make alternating current.
voltage
Voltage V equals magnetic field H times wire movement velocity v times wire-coil length l: V = H*v*l. Voltage V equals magnetic field H times area change dA divided by time change dt: V = H * dA / dt. Voltage V equals flux change dF divided by time change dt. V = dF / dt. Voltage V equals mutual inductance I times current change di divided by time change dt. V = I * di / dt.
example
Water from dams or steam from steam engines can turn wire coils around steel shafts {rotor, generator}, which are inside permanent magnets. Magnets and rotation cause electric current to flow in coils. Electric current changes direction as coil flips.
AC or DC
Rotor shaft {commutator, generator} can have separate conductors {brush, generator} on halves to allow current to leave rotor as direct current. Large-generator shafts {armature} collect alternating current directly.
Alternating current in coil has alternating magnetic field that can interact with outside magnetic field to make magnetic force on coil leading and trailing edges, and so turn coil {electric motor}|.
parts
Direct current or alternating current causes magnetic field in stationary wire coils {stator, motor} and in rotating wire coils {rotor, motor}. As rotor turns, current can go in forward or backward direction, changing magnetic field direction, because rotor shaft has separate conductors {brush, motor} on halves. Rotor magnetic field continually pulls into alignment with stator field, turning rotor by magnetic force. Rotation angular momentum starts cycle again.
torque
Magnetic force causes torque on coil and makes both magnetic fields tend to align. Coil torque T equals coil number n times magnetic field B times current i times coil area A: T = n * B * i * A. When magnetic fields align, force or torque is zero. Just before magnetic fields align, current reverses in coil. Current can reverse every half circle using commutators. Current can reverse using alternating current at needed frequency.
torque: direction
Right-hand palm points in magnetic-force direction, fingers point in magnetic-field direction, and thumb points in positive-current direction {right hand rule, torque}.
types
Series motors have low back emf, high field, and high current when starting and low current, high back emf, and low field when running. Shunt motors have constant field and lower current at high speed. Series and shunt motors can combine. Electric motors use direct current {induction motor}, alternating current {synchronous motor}, or either {universal motor}.
Current can reverse every half circle using devices {commutator, motor}|.
Voltage is between two different touching metals at different temperatures, because metals have different electronegativities {thermoelectric effect}|. If metal rod has different temperatures at ends, voltage is between ends.
If two different metals have different temperatures and contact at two different places, circuit forms {Seebeck effect}.
Thermoelectric-effect voltage can measure temperature {thermocouple}|.
Thermocouples can be in series {thermopile}|.
Mass acceleration or deceleration causes collisions with nearby particles, which collide with farther away masses, and so on, and the disturbance {wave, physics} continues outward at speed that depends on medium particle-connection strength.
mechanical waves
Water-table waves illustrate transverse mechanical waves. Long springs, such as slinkys, illustrate longitudinal mechanical waves. Tuning forks, guitar strings, bongs, and glasses with water at different levels illustrate mechanical longitudinal sound waves. Mechanical waves are in media, which determine wave velocity by electric forces between molecules.
longitudinal wave
Disturbances, such as collisions, can be along line between two masses. Imparting force requires acceleration. Molecules move toward nearby masses, hit them, and bounce backward. Hit molecules accelerate, move toward next masses, hit them, and bounce backward, and so on. Bounce-backs return masses to where they were before, and only heat remains, so no net mass moves. Only disturbance and energy move outward. Wave velocity depends on material elasticity.
transverse wave
Disturbances, such as plucking strings, can be perpendicular to line between two masses. Molecules accelerate transverse to line between two masses. Nearby molecules feel transverse pull, because molecules attract. Attractions eventually stop transverse motion and reverse it. Cycle repeats until only heat remains. No net mass moves along, or transverse to, line between masses. Only disturbance and energy move down line, in both directions. Wave velocity depends on material elasticity.
movement
Waves have to travel, because they must pass from mass to mass. Waves involve acceleration and decelerations.
properties
Mechanical waves displace mass from equilibrium position. Waves have maximum displacement amplitude before they return to equilibrium point. Wave trains have frequency of disturbances passing space point per second. Wave trains have period between disturbances. Waves have wavelength between first and second equilibrium points and have wavelength inverse or wave number. Waves have phase angle of displacement to amplitude. Waves have speed of disturbance travel.
electromagnetic waves
Charge acceleration or deceleration causes force-field change {half-wave, charge acceleration}, which travels outward at light speed. Charge-acceleration moments make photons, because photons have spin. After first acceleration or deceleration, reverse deceleration or acceleration can add half-wave disturbance in opposite direction, to make one complete wave. Repeated acceleration and deceleration can make wave train. Electromagnetic waves do not have position displacement, only field displacement.
Electromagnetic induction requires changing electric and magnetic fields. Electromagnetic-induction rate determines light speed and depends on electric-force strength. Changing electric and magnetic fields move induction point away from accelerating charge. Therefore, light cannot be at rest. Behind moving point, fields cancel. Photons are only at one point, so light has no motion relative to other reference points, and in vacuum, light has same speed for stationary and moving observers.
Electromagnetic induction does not need or have medium. Because light does not move in medium, light speed is not relative to medium. Light speed is absolute maximum speed.
Photons have no mass, so light has no inertia and moves as fast as anything can move. Light speed is maximum physical speed.
Light electric and magnetic fields from several sources add, because electromagnetic inductions add. In media, atoms and molecules absorb and emit light, and this slows light speed but does not change frequency or intensity.
Trigonometric functions {wave equation}| can describe waves. y = A * sin(2 * pi * f * t), where y is displacement, A is amplitude, f is frequency, and t is time. y = A * sin(2 * pi * x / l), where y is displacement, A is amplitude, x is position, and l is wavelength.
position and time
Wave equations are differential equations and include length and time. (D^2)H(x,t) / Dt^2 = (v^2) * (D^2)H(x,t) / Dx^2, where (D^2) indicates second partial derivative, H is function of displacement and time, v is wave velocity, x is position, and t is time. Solutions are waves. In springs, velocity depends on mass and material elasticity {spring constant, oscillation}. For strings, velocity depends on density, tension, and material. For solids, velocity depends on density and material elasticity {Young's modulus, oscillation}. For liquids, velocity depends on density and material elasticity {bulk modulus}. For gases, velocity depends on density, pressure, and molecule type: monatomic, diatomic, triatomic, and so on. For light, velocity depends on material magnetic permeability and electric permittivity.
Devices can reproduce input frequency with constant amplitude and/or phase (no distortion). Devices can reproduce input frequency with varying frequency, amplitude, and/or phase {distortion}. Devices can vary output with input frequency {linear distortion} or with voltage {nonlinear distortion} below or above linear-response range.
compression
Large voltages can have less relative gain than small voltages {compression, audio}. Compression creates lower harmonics.
clipping
Voltage can have limits {clipping}. Clipping creates higher harmonics.
overdriven harmonics
Non-linearly amplifying a tone and its fifth (ratio 3/2) can generate sum and difference frequencies of harmonic tones: higher and lower octaves, fifths, and fourths {overdriven harmonics}.
Sound changes frequency with source or observer movement {Doppler effect}|.
stationary case
When stationary sources emit sounds or light waves with one wavelength and frequency, stationary observers hear one pitch or see one color. See Figure 1. Only wave moves, at constant velocity, because medium does not change.
Source x emits maximum positive amplitude, a line in the diagram, once each cycle. In the diagram, wave travels left two spaces for each cycle line. From one cycle line to the next, observer encounters one peak. There is no Doppler effect.
moving-toward case
When sound-wave or light-wave source moves toward stationary observer, or observer moves toward stationary wave source, observer hears pitch increase or sees shift toward blue color. This is Doppler effect. When frequency increases, wavelength decreases, because only sound medium or electromagnetic-induction speed determines constant wave velocity. See Figure 2.
In the diagram, observer travels right one space for each line, at half wave speed. Observer movement brings it closer to next wave peak. From one line to the next, observer encounters one and one-half wave peaks. Frequency has increased.
See Figure 3. In the diagram, source travels left one space for each line, at half wave speed. Source movement brings it closer to previous wave peak. From one line to the next, observer encounters two wave peaks. Frequency has increased.
moving-away case
When sound-wave or light-wave source moves away from stationary observer, or observer moves away from stationary wave source, observer hears pitch decrease or sees shift toward red color. When frequency decreases, wavelength increases, because wave speed is constant. See Figure 4.
In the diagram, observer travels left one space for each line, at half wave speed. Observer movement brings it farther from next wave peak. From one line to the next, observer encounters one-half wave peaks. Frequency has decreased.
See Figure 5. In the diagram, source travels right one space for each line, at half wave speed. Source movement brings it farther from previous wave peak. From one line to the next, observer encounters two-thirds wave peaks. Frequency has decreased.
examples
As sound-emitting vehicles move closer, sound has higher pitch. As they move away, sound has lower pitch.
As light-emitting stars and galaxies move away from Earth as universe expands, Doppler effect makes emitted light have decreased frequencies, so light becomes redder {red-shift}.
Vibration can be along motion direction {longitudinal wave}|. Sound waves are longitudinal waves.
Mechanical-wave vibrations can be across motion direction {transverse wave}|. Guitar or violin strings vibrate transversely. Molecular interactions are at right angles to direction that wave travels, which is down string and back. Longer strings make lower frequency. Tighter strings make higher frequency. Larger diameter strings decrease frequency. Electromagnetic waves have oscillating transverse electric and magnetic fields.
Acceleration amount determines maximum displacement {amplitude, wave}. Mass displacement has distance oscillation. Zero-rest-mass displacement, as in electromagnetic waves, has field oscillation.
Sound and light have energy flow per second per area {intensity, wave}|, which is power per area.
Wavelength has an inverse {wave number}|.
Light rays travel in straight lines {straight-line motion}, because they follow least-action path.
At space points, wave trains can add {superposition, wave}|. Waves add without affecting each other. Waves are independent. Filtering other waves is subtracting and can leave one wave.
Wavelets add by superposition to make a wavefront {Huygen's principle} {Huygen principle}. See Figure 1.
When two different-frequency waves start from same source, waves superpose {heterodyning}| to make net wave with frequency {beat frequency} equal to difference between the original frequencies. Two frequencies can mix to make lower difference frequency. For example, if frequency-2 wave superposes with frequency-3 wave, frequency-1 wave results.
Light can have different wave-front shapes, such as plane, helix, or double helix {orbital angular momentum, light}. Diffraction gratings with fork or helical lens change plane-polarized light. After such transformation, light in phase makes circles with dark centers {cancellation by superposition}.
Spectrum low frequency can double in frequency {self-referencing}, to interfere with spectrum higher frequencies.
When light waves hit surfaces, surface points re-radiate light {wavelet}|.
If plate has one vertical slit {slit experiment, wave}, light diffracts around edge and makes horizontal diffraction pattern. The most-intense light goes straight through. Lesser light amounts are farther from center. If plate has two vertical slits {double-slit experiment} {Young's experiment} {Young experiment}, light diffracts through both slits and makes horizontal interference pattern, because the diffraction patterns add.
Double-slit experiments can have ring pattern with no interference or striped pattern with interference. Detectors that detect only half the particles cause half-striped and half-ring pattern.
Light can bounce off surfaces {reflection, light}|, as surface molecules absorb and re-emit light. Reflections are like elastic collisions. Plane mirrors and wave tanks show reflections.
wavefront
Wavefronts are moving space disturbances. Behind wavefronts, all wavelets cancel each other, because wavelets have random phases. Beyond wavefronts, nothing has reached yet. Wavefronts are moving edges. Wavefront oscillation and movement carry energy. At surfaces, wavefronts re-radiate.
angles
Reflection angle equals incidence angle. Because light travels straight, light has no sideways motion components, and light plane stays the same. Angles are the same, because light effects are symmetric.
images
Images from flat mirrors appear to be behind mirror and so are virtual images. Images appear at same distance from mirror as distance that objects are from mirror. Images have same size and orientation as objects. Reflections from flat surfaces only reverse right and left.
surfaces
Dielectrics can be mirrors.
polarization
At incidence angle 45 degrees, if reflection from plane mirror has 90-degree angle between reflected and refracted beams, light polarizes.
In reflection, incident light hits surface at angle {angle of incidence}| {incidence angle} to perpendicular.
In reflection, reflected light leaves surface at angle {reflection angle} {angle of reflection}| to perpendicular, as superposed wavelets add to make wavefront. Reflection angle equals incidence angle and is in same plane.
Curved mirrors {curved mirror} focus incoming parallel light rays onto point {focus, mirror}.
types
Curved mirrors {spherical mirror} can have constant radius. Spherical mirrors {convex mirror} can curve out. Curvature radius is positive if curve is convex. For convex mirrors, image is always virtual and erect. For convex mirrors, if object is inside focal point, image is bigger. For convex mirrors, if object is outside focal point, image is smaller.
Spherical mirrors {concave mirror} can curve in. Curvature radius is negative if curve is concave. For concave mirrors, if object is outside focal point, image is real and inverted. For concave mirrors, if object is inside focal point, image is virtual, erect, and bigger.
Curved mirrors {parabolic mirror} can have changing radius.
magnification
Ratio of image size I to object size O equals ratio of distance q of image from mirror to distance p of object from mirror: I/O = q/p.
focal length
Focal length F is spherical-mirror curvature radius R divided by two: F = R/2.
Image distance I and object distance O relate to focal point distance F {lens equation, mirror}: 1/F = 1/I + 1/O.
Find object image using incoming straight lines from object and outgoing straight lines to image {method of rays} {rays method}, which reflect from spherical mirror points.
Light can go from one medium into another medium {refraction}|.
reflection
Some light enters second medium, and some light reflects from surface. For greater refraction-index difference, reflection is greater, because electric fields interact more.
refraction
As wavefront hits surface between media, surface re-radiates light waves, and wavelets add, to make new wavefront in second material.
planar
Incident light and refracted light have same plane, because light travels straight and so has no transverse motion component.
speed
If second medium has different refractive index, incident light and refracted light have different speeds.
frequency
Light frequency stays the same in both materials, because electromagnetic induction does not use medium.
wavelength
Because velocity changes and frequency stays constant, wavelength changes, and incident light and refracted light have different angles to perpendicular. If second medium has higher refractive index, light bends toward perpendicular, because wavelength becomes shorter. If second medium has lower refractive index, light bends away from perpendicular, because wavelength becomes longer.
examples
Glass with different refractive indices appears warped. Refraction from air to water causes coins in fish tanks to appear in different positions than they actually are. Prisms, water glasses, and camera lenses use refraction.
Vacuums have no matter or electric or magnetic fields. Media have subatomic-particle electric and magnetic fields {refractive index}| {index of refraction}, which attract and repel light-wave electric and magnetic fields, decreasing light speed. Refractive index depends on electrical permittivity and magnetic permeability. Vacuum has refractive index 1. Glasses have refractive index near 1.5. Dense polar salts have refractive index 2.5. Teflon is transparent to microwaves but has high refractive index. Plasmas and metals have negative permittivity. No natural substances have negative permeability.
speed
In materials, velocity v equals light speed in vacuum c divided by refractive index n: v = c/n.
In crystals {anisotropic crystal}, refractive index can vary with light-propagation direction {birefringence}|. In birefringence, incident light divides into two light rays that polarize in planes at right angles. Isotropic crystals, glasses, liquids, and gases have the same physical properties in all directions. Most crystals are isotropic.
Different-frequency light does not focus at same point, because refractive index differs for different frequencies {chromatic aberration}|.
Higher frequencies refract more than lower frequencies {dispersion, refraction}. Higher frequencies travel slower than lower frequencies, because dielectric-dipole capacitance is higher, photon energy is higher, and electric forces are higher. Because wavelength is lower, percentage change is higher. Dispersion causes prism rainbows.
Incidence angle I and reflection angle R relate by media refractive indexes n {Snell's law} {Snell law}: nI * sin(I) = nR * sin(R).
If incidence angle is more than angle {critical angle}|, all light reflects, in total reflection, because reflection angle is 90 degrees or more. Critical angle depends on media refractive indexes.
If incidence angle is more than critical angle, all light reflects {total reflection}|, because refraction angle is 90 degrees or more.
Materials {opaque material}| that have free electrons absorb all light.
Materials {translucent material}| that have weakly bound electrons absorb some light and transmit some light, making blurry images.
Materials {transparent material}| that have tightly bound electrons have no absorption and transmit light with clear images.
Transparent curved surfaces {lens, physics}| can refract parallel light rays to point.
convex
For convex lenses, if object is inside focal point, image is virtual, erect, and smaller. For convex lenses, if object is outside focal point, image is real and inverted.
concave
For concave lenses, image is virtual and erect. For concave lenses, if object is inside focal point, image is bigger. For concave lenses, if object is outside focal point, image is smaller.
focus
Focal length F depends on lens refractive index n and radii R of sides: 1/F = (n - 1) * ((1 / Ri) - (1 / Ro)).
curvature radius
Curvature radius is positive if curve is convex. Curvature radius is negative if curve is concave.
size
Ratio of image size I to object size O equals ratio of distance q of image from lens to distance p of object from lens. I/O = q/p.
wavelets
Lenses perform spatial Fourier transforms.
Mirror or lens angular size {aperture}| is angle at focal point between two radii from ends of a spherical-mirror or spherical-lens diameter.
Spherical mirrors or lenses with large aperture deviate from parabolic reflection {spherical aberration}| at edges. Edges do not refract to focal point.
Units {diopter} can measure how much lenses converge or diverge light {dioptric power}. Zero diopters converges light from object at one meter to focus at one meter. Three diopters converges light from object at one meter to focus at one-third meter. Minus three diopters diverges light from object at one meter to focus at three meters.
Parallel light rays from one lens side go through lens to a point {focus, lens} {focal point}| on other lens side.
Images {real image} {image, object}| can form from actual light rays. Images {virtual image} can appear to be in locations where light rays cannot go. Images {erect image} can have same orientation as objects. Images {inverted image} can have opposite orientation as objects. Images can magnify or reduce objects.
Image distance I and object distance O relate to focal point distance F {lens equation, lens}: 1/F = 1/I + 1/O.
Lens surface can curve in {concave lens}.
Lens surface can curve out {convex lens}.
Lens combinations {achromatic lens} can eliminate chromatic aberration.
Lenses {aplanatic lens} can correct spherical aberration.
Microscopes {microscope}| have large lens that collects light to focal point, and second small, high-curvature lens that focuses small but near image. Microscopes {phase contrast microscope} can look for different light phases.
Two waves {standing wave} can travel in opposite directions from point and then reflect back from end barriers, so they reinforce each other {resonance, wave}| when they meet again, because they are in phase.
node
Resonating waves are stationary. In stationary waves, some points {node, wave} always have zero displacement.
wavelength
Fundamental standing-wave wavelength is two times distance between endpoints. Closed tubes have resonant wavelength one-quarter tube length. Open tubes have resonant wavelength one-half tube length. String resonant frequency is lower if string length is longer.
Systems can have standing waves {fundamental wave}| with lowest frequency.
Waves {harmonic wave, physics}| {overtone} can have frequencies that are fundamental-frequency multiples.
Waves can have frequency fundamental frequency times two {octave, wave}|, three {twelfth}, four {fifteenth}, five {seventeenth}, six {nineteenth}, and so on. Higher frequencies must have more energy to have significant amplitude.
Solitary, non-linear, stationary or moving waves {soliton}| can maintain size and shape. As wave components travel, solitons reinforce components by superposition. High-frequency components increase at same rate as they spread out, because they have different speeds. Solitons can be in plasma, crystal-lattice, elementary-particle, ocean, molecular-biology, and semiconductor boundary layers.
vacuum
Vacuum with periodic vacuum states can make soliton-antisoliton pairs.
quanta
Perhaps, massive elementary particles of 1000 GeV, or magnetic monopoles, are solitons. Solitons can allow bosons to make fermions and allow fermions to split.
One-dimensional soliton-antisoliton pairs can be in two or three dimensions and require vector fields {Sine-Gordon theory}.
Light appears to bend {diffraction}| around corners and edges. If light rays meet corners, corner re-radiates light in all directions, so some light goes to region behind edge. Wavelets add to form wavefront there. At most wavefront points, wavelets cancel each other, so light intensity is zero. At some wavefront points, sum is positive, and light appears behind edge at regular intervals. Shadows have diffraction patterns at edges.
sound
Diffraction is how people can hear sound around corners.
size
If obstacle or edge is smaller than wavelength, wave goes farther around obstacle or edge. If obstacle or edge is larger than wavelength, diffraction has smaller angle.
frequency
Higher-frequency light and sound have smaller diffraction, because wavelengths are smaller. Lower-frequency light and sound bend more.
Materials {diffraction grating}| can have regular repeating opening or ruling patterns, so surfaces are like many edges. Diffraction gratings can be for parallel rays {Fraunhofer grating} or spherical rays {Fresnel grating}. The many edges cause strong diffraction pattern, because more wavelets add together to make higher amplitude. If openings are small or rulings have close spacing, diffraction is more, because smaller edge can re-radiate more behind edge.
Transparent plates {phase plate} with varying thickness can delay light slightly, to change phase. Phase plates are diffraction gratings. If only parallel light rays reach phase plate, diffraction is regular. Phase differences cause intensity differences at various points, by interference effects.
Shadows {shadow}| have umbra and penumbra.
Shadows have a lighter part {penumbra}|, where diffracted light enters.
Shadows have a dark part {umbra}|, where no diffracted light enters.
If light wavelength is less than object diameter, light bounces off object {scattering, light}|. If light wavelength is more than object diameter, light goes around object.
example
Sky is blue, because blue light has small enough wavelength to scatter from air molecules, but other colors have longer wavelengths. Air molecules are large enough to block blue and some green light from Sun, but longer wavelengths go around air molecules. Scattered blue light goes all over sky to make it blue instead of clear. Sun is red at horizon, because light goes through more atmosphere to eye, and air scatters blue, green, and yellow light.
X-rays can have elastic scattering {Compton scattering} from stationary electrons in light elements. Scattered-radiation frequency decreases with increasing angle, so high frequencies are at narrow angles.
Two vibrators at similar frequency soon have same frequency and phase {entrainment}| {mode-locking}.
Two oscillators with similar frequencies soon have same frequency {virtual governor} {mutual entrainment}.
Molecular-vibration waves {sound, physics} can move through materials.
process
Molecules from outside material can collide with material, causing material molecules to move. Molecular movement causes collision with adjacent molecules. First molecules bounce backward, and second molecules move, causing collision with adjacent molecules, and so on. Collisions send longitudinal wave down motion line.
Sound compresses {compression, sound} material in front of it, leaving slight vacuum {rarefaction} behind compression. Compression pushes next material bit forward. Original bit bounces back to original position, so material does not move. Compression wave travels through material. Only wave and energy move.
speed
Medium determines sound-wave speed. Sound-wave speed increases with stronger interactions between molecules. Wave frequency and amplitude do not affect speed.
amplitude
Sound has kinetic energy {loudness, sound}. Kinetic-energy increase increases sound-wave amplitude, by moving molecules farther. Frequency, wavelength, and speed do not affect wave amplitude.
pitch
Sound has number {frequency, sound} of vibrations per second. People can hear sounds of 20 to 20,000 Hz.
Sound has frequencies at two, three, four, and so on, times fundamental frequency {harmonics, physics}. Higher harmonics have lower amplitude.
Outside-material vibration frequency determines sound-wave frequency. Materials can have resonance frequencies.
Sound waves travel in a medium, and the medium can be moving, making net sound-wave velocity faster or slower {Mach effect}.
Vibration quanta {phonon}| are sound-wave packets. Crystal phonon vibrations cause temperature gradient sideways to phonon direction, analogous to Hall effect for electromagnetism.
Surfaces can have acoustic waves {Rayleigh wave}. Earthquakes and radio waves can put Rayleigh waves in Earth or ionosphere. Ultrasonic surface acoustic waves can store, recognize, filter, and channel electronic signals in semiconductors, at 10^9 Hz.
Moving objects make sound {sonic boom} as they push air aside {shock wave}|. If object speed becomes the same as sound speed, waves of pushed-aside air travel as fast as sound. Waves are in phase and grow to make large wave. If plane travels faster than sound speed, sound is behind pushed-aside air, waves do not build up, and no shock wave builds.
Objects can go through air faster than air sound speed. Sound from object contact with air cannot travel away faster than sound waves build up. Wave constructive interference creates shock wave, which carries extra energy away when object breaks sound barrier, causing sonic boom. After passing sound speed, acoustic waves at sound speed are slower than object speed, with no more constructive interference.
Speech sounds {speech sound} have frequency range from 250 Hz to 2000 Hz and loudness range from 63 to 95 decibels.
Sounds {ultrasonic sound}| can have frequency greater than 20,000 Hz. Ultrasonic sound can visualize body insides and clean dishware.
Rooms {whispering gallery} can have focal points, where sound focuses {echo}|. Canyons and buildings can echo sound. Echoes work best with low amplitude and high frequency.
High-frequency sound can locate objects by echo pattern {echolocation}| {sonar, location}.
Electric charges have virtual photons streaming outward as straight lines in all directions, making electric field. Electric fields begin at electron edge, which emits virtual photons. Electric-field lines indicate electric-force direction. Each line is one photon stream, so electric-field lines are not about electric-force strength or electric-field strength. Electric-field-line area density, photons per area, is electric-field strength. Because area varies directly with squared dimension, electric force decreases as distance squared: 1/r^2. Electric field has virtual kinetic energy, which can transfer to other charges at field positions to become potential energy.
moving charges
Maximum charge velocity is typically one-tenth light speed. For constant-velocity charge, electric field moves at same speed and direction as charge. Virtual photons stream outward as straight lines in all directions.
Constant-velocity fields have no transverse or longitudinal field changes, and so no waves.
moving charges: magnetic force
According to special relativity, constant-velocity charge causes observer transverse to charge-motion-direction to see length contraction and so increased charge-motion-direction charge density. Length contraction makes flattened-spheroid charge shape, with short axis in motion direction and long axes in transverse-direction plane. Because total charge is same for moving and stationary charge, total field strength stays the same. Relativistically increased charge density along vertical direction causes increased electric force along horizontal direction. Therefore, relativistic length contraction makes electric field appear to observer stronger horizontally. According to special relativity, observer in front or back of constant-velocity charge does not see length contraction, only that charge approaching or receding. Total electric field strength is same as for stationary electron, because total charge is same. Because total charge is same as before, charge density must be less as observed from vertical direction, so electric field appears to observer weaker vertically. Vertical electric field is foreshortened in motion direction, because electron catches up to virtual photons. Vertical electric field is lengthened opposite to motion direction, because electron moves away from virtual photons.
Electric force due to relativistic length contraction and charge-density change, and not due to total charge, is magnetic force. (Stationary charges have no relativistic motion and so no relativistic electric force.) Adjacent magnetic force is a torus around moving charge. Just as electric forces act only on electric forces, magnetic forces act only on magnetic forces, because magnetic is perpendicular to electric and so does not affect electric.
Electric force has electric field. Electric force and electric field have same direction and relative strength. Because magnetic force is relativistic electric force, magnetic force has magnetic field. Magnetic force and magnetic field have same relative strength but perpendicular direction, because force is due to transverse relativistic length contraction and so is perpendicular to motion and field. Therefore, magnetic forces have magnetic fields perpendicular to electric force/magnetic force and perpendicular to charge-motion orientation. Moving-charge magnetic field is a torus adjacent to and around charge, transverse to motion direction. See Figure 1.
For positive charge moving in right-hand thumb direction, magnetic field is in curling index-finger direction, in a circle around moving proton, and magnetic force is outward from palm (right-hand rule). For proton moving vertically downward, magnetic field is in on left and out on right. Electron has negative charge, so magnetic field is out on left and in on right.
Magnetic field has virtual photons and so has virtual kinetic energy, which can transfer to other charges at field positions to become potential energy. Stationary charge has no magnetic field, because it has no relativistic length contraction.
accelerating charge
Charge acceleration pushes electric-field line transversely and stretches it sideways, causing tension and restoring force. Charge acceleration causes transverse electric field, while keeping radial field. Because virtual photons continually leave charge, transverse component moves outward along field line, so spatial transverse waves travel outward. See Figure 2.
Charge acceleration pushes magnetic-field line transversely and stretches it sideways, causing tension and restoring force. Charge acceleration causes magnetic field in charge-motion direction, transverse to magnetic field.
When stationary charge accelerates to constant velocity, electric-field lines curve toward motion direction, because charge and adjacent photon have higher velocity. When constant-velocity charge decelerates to zero velocity, electric-field lines curve away from motion direction, because charge and adjacent photon have lower velocity.
See Figure 3. Force causing electron deceleration also puts transverse upward pushing force on field lines and distorts electric-field lines. As electron slows down, electric-field-lines beginning at electron edge slow down, so horizontal electric-field lines begin to have transverse component upward.
See Figure 3. As electron slows down, electric-field upward transverse component increases over time. Changing electric-field flux (changing electric force) through an area causes relativistic length contraction transverse to area (in same plane) and magnetic-force change in toward or out from area (in same plane), and so causes induced magnetic field around area. Magnetic force has gradient in or out and so makes induced-magnetic-field gradient around. Faster change makes larger gradient.
Electric and magnetic fields interact, so they push/pull adjacent electric and magnetic fields. Interaction is strong and happens at light speed, so adjacency effect travels at light speed. Interaction is constant, so light speed is constant. All interactions are elastic, with no losses to heat or other energy, so induction has same effect later as at beginning.
Transverse effect travels inward and outward at light speed. Outward effect sees only undisturbed field line and so is the only effect and carries energy outward. Inward effect sees restoring force from stretched field line and so forces cancel and line returns to equilibrium, with no energy carried.
Electric-field increase (or decrease) causes magnetic-field increase (or decrease) that opposes electric-field increase (or decrease), by energy conservation.
See Figure 3. Induced magnetic field increases over time. Changing magnetic-field flux (perpendicularly changing magnetic force) through an area causes relativistic length contraction transverse to area (in same plane) and electric-force change around area (in same plane), and so causes induced electric field around area. Electric force has gradient in or out and so makes induced-electric-field gradient in or out. Faster change makes larger gradient.
Magnetic-field increase (or decrease) causes electric-field increase (or decrease) that opposes magnetic-field increase (or decrease), by energy conservation.
Changing electric field and magnetic field are in phase, because they both increase together and both gradients are in same direction.
Gradient and wave leading edge travels outward at constant light speed.
See Figure 3. Horizontal electric-field lines continue moving at constant velocity, because lines have same momentum, inertia, and kinetic energy as before.
See Figure 4. As electron slows down more, electric-field-line points at electron edge slow down more, so horizontal electric-field lines have greater transverse component. Electric-field upward transverse component increases more over time and so makes bigger induced-magnetic-field gradient. Induced magnetic field increases more over time and so makes bigger induced-electric-field gradient. Transverse fields have potential energy, so horizontal electric-field lines at transverse fields have less kinetic energy. Horizontal electric-field lines continue moving at constant velocity, because lines have same momentum, inertia, and kinetic energy as before.
See Figure 5. Metal plate stops electron within one electron width, so distance and time are small, and deceleration is high. Electron is at zero velocity, so current is zero. Electron has no kinetic energy and momentum. Original electric field is symmetric. Original electric field has same potential energy. Original magnetic field is zero. Original magnetic field has no potential energy.
See Figure 5. As electron stops, electric-field-line ends stop, so horizontal electric-field lines have maximum transverse component. As electron stops, electric-field upward transverse component has increased to maximum over time and so makes induced-magnetic-field gradient. Induced magnetic field has increased to maximum over time and so makes induced-electric-field gradient. Induced electric field is maximum.
See Figure 5. Horizontal electric-field lines continue moving at constant velocity, because lines have momentum, inertia, and kinetic energy.
See Figure 6. Deceleration has stopped, so electron and adjacent fields stop feeling upward force. Transverse electric-field stays constant at zero, and so makes no magnetic-field gradient and no magnetic field. Magnetic-field line feels no force, so transverse magnetic field stays constant at zero, and so makes no electric-field gradient and no electric field. Electron and adjacent electric-field line have no velocity, momentum, or kinetic energy. Gradient and wave leading edge travels outward at constant light speed. Adjacent virtual photon leaves electron and travels horizontally at light speed. Transverse electric-field-line component stretches farther downward. Transverse electric-field-line component moves outward at light speed. All interactions are elastic, with no losses to heat or other energy, so gradient has same effect later as at beginning. Original virtual photons of horizontal electric-field lines continue moving at constant velocity, because lines have momentum, inertia, and kinetic energy.
phase
When stationary charge accelerates to constant velocity, electric-field and magnetic-field transverse component increase in same direction and at same time (in phase). When constant-velocity charge decelerates to zero velocity, electric-field and magnetic-field transverse component decrease in same direction and at same time (in phase).
induction
Electric-field change over time (flux) through an area makes magnetic field around area, because of relativistic length contraction. Electric current makes magnetic-field torus around current. See Figure 1.
Magnetic-field change over time (flux) through an area makes electric field around area, because of relativistic length contraction. Magnetic-field flux change through torus cross-section makes electric field around torus cross-section. Current goes through torus hole, around, and back again to complete the circuit (displacement current). See Figure 1.
Stationary electric field has constant force, and so uses no energy to make magnetic field. Moving electric field changes over time and makes constant magnetic-field gradient, because electric-field-movement kinetic energy increases magnetic-field potential-energy over space. Accelerating electric field makes increasing magnetic-field gradient, because electric-field force increases magnetic-field acceleration over space. Accelerating magnetic or electric fields over space have force that causes increasing electric or magnetic fields over time. Fields over space have potential energy, and fields over time have kinetic energy, so energy alternates between kinetic and potential, making waves.
Constant stationary magnetic field has no affect, because it has no force, so magnetic-field energy remains potential energy. Moving magnetic field changes over time and makes constant electric-field gradient, because magnetic-field-movement kinetic energy increases electric-field potential-energy over space. Accelerating magnetic field makes increasing electric-field gradient, because magnetic-field force increases electric-field acceleration over space.
induction: energy conservation
Increasing (or decreasing) magnetic field increases (or decreases) electric field, which makes magnetic field that opposes original magnetic field, by energy conservation. Decreasing (or increasing) electric field decreases (or increases) magnetic field, which makes electric field that opposes original electric field, by energy conservation.
For downward current, acceleration increases magnetic field, and that makes upward electric field, which decreases magnetic field. For downward current, deceleration decreases magnetic field, and that makes downward electric field, which increases magnetic field.
Charge deceleration is against restoring force and builds potential energy. When deceleration stops, restoring force pulls back toward equilibrium, but potential energy transfers to kinetic energy and carries past equilibrium until restoring force pulls back to equilibrium.
Energy goes into adjacent electric-field transverse movement, as interchange between electric and magnetic fields makes wave travel outward. Therefore, energy dies down at past points.
Magnetic-field and electric-field changes have same displacement amount, but electric field has approximately one hundred times more energy. Most light-wave energy is in electric field, not magnetic field, because magnetism is relativistic effect.
To make electric field, virtual photons stream outward at light speed from electron in all directions. Electric-field lines are virtual photon streams. At electron constant velocity, photons also have same velocity as electron, so electric-field lines are straight.
The figure shows virtual photons streaming outward horizontally from electron transverse to electron motion direction. Electron and electric-field lines move downward at same velocity.
Because electric-field and magnetic field interact along line, line has tension, just as a taut string has tension, so line has restoring force if accelerated sideways, just like a taut string has restoring force. All interactions are elastic, with no heat losses, so forces and energies are the same all along electric-field lines from beginning to infinity.
Deceleration can knock field lines through space. Stronger deceleration makes farther and stronger fields.
gradient
Field induction around area circumference makes space gradients as tangents to circumferences. When electric-field flux change through area makes magnetic field around area, magnetic field has gradient around area. When magnetic-field flux change through area makes electric field around area, electric field has gradient around area.
speed
Because electric force is strong, electric and magnetic fields interact at light speed. Because magnetic field and electric field couple {electromagnetic wave induction}|, transverse field-line component moves outward along electric-field line at light speed. Electromagnetic interaction strength is constant, so light speed is constant.
wave
Waves are local effects that travel. Traveling field changes are independent of original charges.
For downward current, deceleration decreases magnetic field, and that makes downward electric field, which increases magnetic field. Electric and magnetic fields are in phase. Electric-field-line disturbance moves away from charge at light speed in a straight line. Transverse component makes traveling wave half {half-wave, wave}. All disturbances to electric-field lines travel outward at light speed. Previous points have no more disturbances, so only one half-wave exists at any time. No disturbances are left behind, because all energy has traveled away. Disturbance reaches farther positions in sequence out to infinity. See Figure 1 through Figure 7. Wave exists only at induction point and can only go straight-ahead. Wave has no physical effect except at moving single point.
elastic
Electric and magnetic interactions are elastic, with no losses to heat or other energies. Therefore, disturbances travel without losing energy. Inductions continue to infinity.
strength with distance
Inductions and other electric-field-line disturbances are transverse to electric-field lines. Because electric field oscillates in a plane, not area, intensity decreases directly with distance, not with distance squared. Transverse effects happen in one dimension, so wave strength decreases directly with distance: 1/r. At later times, transverse field component stretches more over space.
electric-field-line tension and restoring force
Guitar-string molecules attract each other by electric forces. Taut guitar strings have tension from these restraining forces. Pulling string sideways puts potential energy into the string, by stretching string electric forces, like springs. After releasing string, electric forces, like springs, pull string back by restoring force. Potential energy transfers to kinetic energy. Molecule kinetic energy carries molecules past equilibrium point, so they pull on string molecules in other direction.
Adjacent to pull and release point, molecule electric forces pull-and-restore adjacent string molecules, so transverse waves travel along string. Wave speed depends on molecule electric-force strength. Wave takes energy with it, so no energy is left at original disturbance point, and it no longer oscillates. Molecule electric forces bring displacement back to equilibrium at zero.
Electric-field lines are like strings. Like guitar strings, electric-field lines have tension, because electric fields couple to adjacent magnetic fields, and magnetic fields couple to adjacent electric fields. Electric-field and magnetic-field inductions cause adjacent electric-field line points to attract, like molecule electric forces. Pushing electric-field line sideways adds potential energy. Electric-field and magnetic-field inductions make restoring force that transfers potential to kinetic energy.
Electric-field-line-point transverse disturbance displaces adjacent points, which displace their adjacent points, so disturbance travels outward along electric-field line. Electromagnetic interactions are at light speed, so wave has light speed. Wave takes energy with it, so no energy is left at original disturbance point, and it no longer oscillates. After disturbance, electric-field and magnetic-field mutual-induction restoring force brings displacement back to equilibrium at zero.
metal plate
Plate-molecule electric force decelerates electron and so decelerates electric-field line and magnetic-field line at electron edge, transverse to motion direction.
electric and magnetic forces
When electric-field-line disturbance reaches test charges far away from original charge, test charges move along charge-motion direction, because transverse electric field is voltage and electromotive force. Electric-field change and magnetic-field change reach test charge at same time. Magnetic-force effect is one-hundredth electric force. For far-away test charges, radial electric force is smaller than disturbance force, because radial force decreases with distance squared but transverse force decreases with distance. Original-charge velocity and acceleration have only negligible effect on far test charges, because waves move at light speed but charges move much slower.
Test charges along accelerating-charge direction have no transverse effects, because push or pull is in same direction as accelerated charge.
not stationary
Stationary oscillating electromagnetic fields cannot exist, because electromagnetic induction requires field movement. Standing waves result from traveling-wave superposition.
medium
Light needs no medium, because electric/magnetic fields are their own medium.
situations: antenna
Alternating current accelerates many charges back and forth along one orientation (antenna), making transverse electric-field waves that expand in planes that go through acceleration direction. Electric-field lines transverse to oscillation direction have maximum transverse component. Electric-field line along oscillation direction has no transverse components. Electric-field lines between transverse and oscillation direction have decreasing transverse component.
Electric-field change causes magnetic-field change one quarter cycle later, by relativistic length contraction, and magnetic-field change causes electric-field change one quarter cycle later, by relativistic length contraction, so phases lag each other by 90 degrees. If magnetic-field gradient first increases to north, then electric-field gradient increases to east, then magnetic-field gradient increases to south, then electric-field gradient increases to west, and then magnetic-field gradient increases to north, and so on, because each drives the other along by transverse electric force. Inductions are at right angles, rotating around direction of motion by 90 degrees. 90-degree rotations result in linearly polarized waves.
Source charge accelerations affect electric and magnetic fields at same time, so changing electric field makes magnetic field and changing magnetic field makes electric field simultaneously, so electric field and magnetic field are always in phase.
As electric field increases, magnetic field increases, because magnetic fields are relativistic effects of electric fields. As electric field decreases, magnetic field decreases. When electric field maximizes, becomes zero, or maximizes in opposite direction, magnetic field maximizes, becomes zero, or maximizes in opposite direction. Magnetic-field and electric-field changes increase and decrease in synchrony (phase), because both fields couple. Transverse magnetic-field and electric-field accelerations are equal, in phase, and perpendicular.
When electric field and magnetic field are zero, and potential energy is zero, electric-field change and magnetic-field change maximize in space and time. When electric field and magnetic field maximize, and potential energy maximizes, electric-field change and magnetic-field change are zero in space and time. When electric-field change is zero and electric field maximizes, voltage maximizes and current is zero. When electric-field change maximizes and electric field is zero, voltage is zero and current is zero. When magnetic-field change is zero and magnetic field maximizes, voltage is zero and current maximizes. When magnetic-field change maximizes and magnetic field is zero, voltage maximizes and current is zero.
Fields elastically exchange potential and kinetic energy and make harmonic oscillations. Photons continue at same frequency.
Starting from stationary charge, voltage accelerates charge and adds kinetic energy. Increasing magnetic field increases electric field until increasing electric field has slowed increasing magnetic field and both are maximum, with potential energy maximum. The slower changing electric field decreases magnetic field, which decreases electric field, so both fall in phase, as potential energy becomes kinetic energy. As kinetic energy becomes potential energy in the opposite direction, and then potential energy becomes kinetic energy, the half-cycle repeats in the opposite direction, to complete one cycle. Oscillating current repeats the cycle, and the cycles move outward at light speed. Oscillating current induces electromagnetic waves of same frequency.
Light waves have electric-field and magnetic-field linear polarizations, at right angles. Electric field oscillates in plane that goes through charge-motion direction. Magnetic field oscillates in plane perpendicular to charge-motion direction.
Leading edge of wave rises transversely at angle determined by frequency, which depends on deceleration amount. Higher frequencies have steeper angles. Higher frequencies have greater curvatures at maximum displacement, because higher frequency means turnaround is faster.
situations: dipoles
For dipoles, charge acceleration increases as charge separation increases.
situations: atoms
Atom and molecule electrons can accelerate or decelerate and so change orbits, absorbing or making radiation. Molecule dipoles can rotate, vibrate, or translate, and so accelerate electrons, absorbing or making radiation.
situations: devices
Free charges in electric and magnetic fields accelerate free charges, as in vacuum tubes. When moving electrons hit metal plates, they decelerate and can make x-rays.
Electric-charge accelerations start electromagnetic waves {wave initiation} {initiation, wave}, because force makes radial electric field have transverse component adjacent to charge. Transverse component travels outward along electric-field line {wave propagation} {propagation, wave}, because electric-field (and magnetic-field) changes interact at light speed, because electromagnetic force is strong. Waves travel away from charges, because all energy travels outward, so no energy is left behind, and only wave leading edge (wave front) exists at any time. Wave has kinetic energy directly proportional to force that caused charge acceleration.
charge: stationary
Stationary charge makes constant electric field and no magnetic field. See Figure 1.
charge: moving
Charge moving at constant speed makes moving electric field and constant magnetic field. See Figure 2. Magnetic field is perpendicular to electric field, because magnetic field comes from relativistic length contraction that causes increased charge density along charge-motion direction, which observers see from side.
charge: acceleration
Accelerating charge increases current, because charge speed increases. Increasing current makes increasing magnetic field. Accelerating charge makes faster moving electric field. See Figure 3. (Decelerating charge decreases current, decreases magnetic field, and makes slower moving electric field.)
initiation
As charge accelerates, electric and magnetic fields accelerate, and magnetic field increases. See Figure 4.
propagation
Electric-field (and magnetic-field) change cause magnetic-field (and electric-field) gradient, by Maxwell's laws, so electric and magnetic fields interact. Interaction is at light speed. See Figure 5.
When induced electric field and magnetic field reach far-away test charge, electric-field vertical component accelerates test charge. See Figure 6.
When induced electric field and magnetic field pass far-away test charge, test charge continues at constant velocity. See Figure 7.
propagation: direction
Electromagnetic-induction is only at wave front, because all energy is there. Behind wave front, electric and magnetic fields return to zero, as fields, coming from many points with all phases, cancel. Waves propagate outward from accelerated charge, because electromagnetic-induction electric and magnetic fields behind have all phases and cancel.
propagation: no medium
Electromagnetic waves can propagate through empty space, because electric and magnetic fields are their own medium.
propagation: induction rate and wave speed
Electric-force strength determines electromagnetic-induction rate, which is light speed. Material electric charges, relativistic apparent electric charges, other electric fields, and other magnetic fields exert force on electromagnetic waves, and so reduce electromagnetic-wave speed.
Unaccelerated Charge Makes No Electromagnetic Wave
Unaccelerated moving charge makes moving constant electric field and constant concentric magnetic field. See Figure 4. No acceleration makes no force, so fields stay constant. Only radial force affects test charge, so it has no transverse motion.
Charge Acceleration Makes Traveling Electric Field
See Figure 5. Collision, gravity, or electric force can accelerate charge. Acceleration makes force, so fields change. Acceleration is transverse to radial electric-field line, so test charge has transverse motion. See Figure 6.
pure electric waves
There are no pure electric non-magnetic waves, because waves require electric-field changes, which always make transverse relativistic electric fields, which are magnetic fields. There are no pure magnetic non-electric waves, because waves require magnetic-field changes, which always make transverse relativistic electric fields.
Waves {light}| {electromagnetic wave} can begin by charge accelerations or electronic transitions and propagate by electromagnetic induction. Charge-acceleration or electronic-transition energy change determines electromagnetic-wave frequency.
Accelerating charge makes a photon field, which differs near source {near field} and far from source {far field}. Far field is what lenses, mirrors, and instruments see. Point charges or nearby detectors can examine near field.
Equations {Maxwell's equations} {Maxwell equations} can find all electric and magnetic properties. For stationary and moving charges, electric-field and magnetic-field relations are Gauss's law, Gauss's law for magnets, Faraday's law, and Ampere's law.
stationary
Partial derivative of electric field with distance equals negative of partial derivative of magnetic field with time. Partial second derivative of electric field with distance equals electric permittivity times magnetic permeability times partial second derivative of electric field with time.
tensors
Maxwell's equations are equivalent to two equations. For magnetostatics and magnetodynamics equations, exterior derivative of electromagnetic-field tensor F equals zero: dF = 0. Electromagnetic-field tensor is a linear operator on velocity vector. Electromagnetic-field tensor has covariant components. This tensor is equivalent to delta function. For electrostatics and electrodynamics equations, exterior derivative of electromagnetic-field-tensor dual F* equals four times pi times four-current dual J*: dF* = 4 * pi * J*. This tensor is equivalent to delta scalar product.
current
The four-current has one component for charge density and three components for current densities in three spatial directions.
duals
Rank-x antisymmetric tensors relate to rank 4 - x antisymmetric tensors {dual, tensor}. Dual of dual gives original tensor, if rank is greater than two.
invariant
Electromagnetism invariant is current squared minus light speed times charge density squared, which equals negative of momentum times light speed squared.
retarded and advanced
Electromagnetic-field changes follow charge accelerations {retarded solution}. However, field changes can happen before charge accelerations {advanced solution}, because equations are symmetric. Other solutions can be linear retarded-solution and advanced-solution combinations.
Light speed {speed of light} {light speed}| is the same relative to any observer, moving or not. Light speed is invariant and absolute, in space with no electric fields.
speed
Light speed depends on electric-force strength, which determines electromagnetic-induction strength. In vacuum, light speed is 3.02 x 10^8 m/s (Hippolyte Fizeau and Bernard Foucault) [1849]. Light speed is fast, because electric forces are strong. All zero-rest-mass particles travel at light speed, because added energy does not affect them. Gravity does not affect zero-rest-mass particles.
space
Light does not travel through time, because it has no medium and so no reference frame or space-time. Light only travels through space, not time. Light has zero time.
cause
Observers see invariant speed, because light never has time component and so cannot go slower. Light cannot go faster, because it uses all of space already. When observers see light, light length appears to be zero and time appears to be at maximum dilation. Observer motion does not affect light speed observed, because light has no medium. Observer motion contracts length and dilates time, but light already has maximum length and shortest time. If observer moves at higher velocity, both time dilation and space contraction happen, so light speed stays the same.
space-time velocity
All objects travel through space-time at light speed. Light travels only in space. Stationary objects travel only in time. Moving masses travel in space and time.
terminal velocity
Light speed is like terminal velocity through space-time. Electromagnetic induction pushes wave, and forces in universe retard wave. Resistance to light motion can come from effects of all universe masses and charges.
mass
No object with mass can go faster than light. For mass at light speed, stationary observers see infinite mass, zero length, and zero time. To make infinite mass requires infinite energy. Infinite mass exerts infinite gravitational force. Infinite mass attracts and red-shifts light, dimming universe. Infinite mass, moving at light speed, appears to have infinite frequency and zero wavelength.
phase velocity
Light pulses contain wave sets. Light-pulse envelope carries energy. Envelope speed {group velocity} must be light speed or less. However, individual waves can have speed higher or lower than light speed {phase velocity}. Negative refraction cannot exist.
Perhaps, light travels in a stationary medium {luminiferous ether} {the ether} {æther}, not vacuum. As such, because light has constant velocity in any reference frame, ether is an absolute reference frame. It is fluid but does not disperse, has no viscosity, and has high tension and is rigid. It has zero rest mass and is transparent, continuous, and incompressible. Perhaps, it appears rigid to high-velocity objects or high-frequency waves but fluid to low velocity objects or low-frequency waves. Michelson and Morley [1887] measured interference of light traveling in Earth-motion direction and in opposite direction {Michelson-Morley experiment} and found no interference and no Doppler effect, leaving no physical properties to ether and so indicating that there was no ether.
Light can carry enough electric energy to knock electrons out of atoms {photoelectric effect}|. If light frequency is below threshold for material, atoms emit no electrons, because photoelectric effect requires minimum energy. Light with higher frequency than threshold imparts more speed to liberated electrons but does not emit more electrons. Higher-intensity light, which has more photons with enough energy, makes more electrons leave.
Radiation has entropy {entropy, radiation} {radiation entropy}. If space is isotropic and unpolarized, entropy S equals four times energy U divided by three times temperature T: (4*U) / (3*T). If system has more wavelengths or more directions, radiation entropy increases. Universe can absorb radiation and everything else without limit, so entropy continually rises.
Light has subatomic particles {photon, light}|. Photon is like wave packet. Continuous light {light ray} {ray, light} is many wave packets.
straight
Light rays and photons travel in straight lines.
energy
Photon energy E is frequency v times Planck constant h: E = h*v.
observers
What do people see as photon goes past? In empty space, people see particle contracted to zero length, with no mass but with frequency and wavelength. People see time standing still on photon.
What does photon see? In empty space, photon travels at light speed. Other objects pass by at light speed, with infinite mass and zero wavelength. Photon sees time as standing still on other things. Photon sees only point straight-ahead, and sees nothingness on sides, so photon sees along one-dimensional line.
Light can travel in two dimensions {plasmon}| and so travel in plane. Photons that hit interface between conductor and insulator induce surface electrons to vibrate at same or similar frequency and cause traveling wave. Wave reflections make resonances. Plasmons {plasmonics} can have same or shorter wavelength as impinging light.
Light intensity {illuminance}| is light flux (in lumens) per area. Light intensity depends on amplitude squared, photon number, and frequency squared.
Light intensity {Poynting vector} has maximum of half times light speed times permittivity e times electric field E squared: 0.5 * c * e * E^2. Poynting vector equals half times electric field E times magnetic field H: 0.5 * E * H.
At high intensity, wave electric field can affect molecule electric fields {Kerr effect} {optical Kerr effect}.
Radiation has pressure {radiation pressure} {pressure, radiation} from photon flow. Pressure P equals energy U divided by three times volume V: P = U / (3*V).
Light meters {photometer}| can measure light intensity.
Radiation has frequency {radiation frequency}.
Deceleration as electrons hit metal makes radiation {Bremsstrahlung radiation}| with wavelength 10^-12 meters.
Beta-particle electrons, with velocity higher than light speed in water, emit blue light {Cerenkov radiation}| {blue glow} as shock waves when they enter water. Water surrounding nuclear-reactor cores, which emit high-velocity electrons, has blue glow.
process
Electrons traveling in water use some energy to polarize water molecules along travel direction. After electrons pass, polarized water molecules emit light. If electrons travel slower than light speed in water, emitted radiation appears low, because electromagnetic waves emitted by molecules along path are random and destructively interfere. If electrons travel faster than light speed in water, emitted radiation appears high because electromagnetic waves emitted by molecules along path are shock waves that constructively interfere.
Infrared-light rotational and vibrational energies cause differences in visible light reflected from molecules {Raman scattering}|.
In atmosphere, secondary cosmic rays {spallation}| arise if cosmic ray hits atomic nucleus.
Charged particles accelerated by spiraling in magnetic field can emit microwaves {synchrotron radiation}|. Synchrotron radiation happens when electric field is parallel to electron-orbit plane.
Electromagnetic radiation has frequency range and wavelength range {spectrum, light}|.
low frequency
electric wave. radio wave. short wave. very-high-frequency TV wave. ultra-high-frequency TV wave. microwave radiation. infrared ray.
visible
Visible light is 4 x 10^14 Hz with wavelength 6.8 x 10^-7 meters for red light, orange, yellow, wavelength 5.5 x 10^-7 meters for yellow-green, green, wavelength 4.4 x 10^-7 meters for blue light, indigo or ultramarine, and 7.5 x 10^14 Hz with wavelength 4.1 x 10^-7 meters for violet light.
Violet is 380 to 435 nanometer, with middle at 408 nanometer. Blue is 435 to 500 nanometer, with middle at 463 nanometer. Cyan is 500 to -520 nanometer, with middle at 510 nanometer. Green is 520 to 565 nanometer, with middle at 543 nanometer. Yellow is 565 to 590 nanometer, with middle at 583 nanometer. Orange is 590 to 625 nanometer, with middle at 608 nanometer. Red is 625 to 740 nanometer, with middle at 683 nanometer.
high frequency
near ultraviolet. ultraviolet. far ultraviolet. X ray. gamma ray. secondary cosmic ray. cosmic ray. primary cosmic ray.
Smallest frequencies and longest wavelengths {electric wave}| are 3 to 60 Hz and 10^8 to 5 x 10^6 meters.
Next smallest frequency and wavelength {radio wave}| are 10^3 Hz and 3 x 10^5 meters.
High-frequency radio waves {short wave}| are for global communication.
Typical TV frequencies and wavelengths {very high frequency TV wave}| (VHF) are 10^8 Hz and 3 meters.
higher TV frequencies and wavelengths {ultra high frequency TV wave}| (UHF).
frequencies below infrared {microwave radiation}|.
Heat-ray {infrared}| frequency is 10^12 Hz, with wavelength 3 x 10^-4 meters.
Light {visible light}| can have wavelength 400 nm to 700 nm. Visible light has same wavelengths as diameters of, and energy changes in, atoms and molecules. Matching diameters allows people to focus on objects, because light is not too diffracting or too strong. Matching energy changes allows absorption, emission, and chemical reactions.
Smallest visible-light frequency {red light}| is 4 x 10^14 Hz, with wavelength 6.8 x 10^-7 meters.
Highest visible-light frequency {violet light}| is 7.5 x 10^14 Hz, with wavelength 4.1 x 10^-7 meters.
higher frequency than violet {ultraviolet}|.
Light {far ultraviolet}| {black light} can have frequency 1.5 x 10^15 Hz and wavelength 2 x 10^-7 meters.
higher frequency than far ultraviolet {X ray}| {x ray}.
Next-to-highest frequency {gamma ray, spectrum}| is 10^23 Hz, with wavelength 3 x 10^-15 meters.
Highest frequency {cosmic ray}| {primary cosmic ray} is 10^25 Hz, with wavelength 3 x 10^-17 meters. Quasars and powerful energy sources make cosmic radiation.
Light {monochromatic light}| can have one wavelength.
Light {polychromatic light}| can have many wavelengths.
Magenta, yellow, and green pigments {primary pigment}| mix to make black.
Variations {dichroism}| in absorbed-light color can depend on light-polarization direction. Dichroism indicates molecule orientation, which can be linear, circular {circular dichroism}, or elliptical. Microvilli rhabdom can lie parallel, exhibit dichroism, and detect polarized-light polarization plane.
If one photon accelerates, light-wave electric field vibrates in one plane {plane polarized wave} {polarized light}, and light-wave magnetic field vibrates in perpendicular plane {polarization, wave}|. Typically, many charges accelerate in all possible planes, so there is no polarization.
materials
Materials can allow only light with one electric-field plane to transmit. Polaroid plastic and tourmaline can polarize light.
circular
Asymmetric-molecule electric forces cause substances to rotate electric-field planes {circularly polarized wave} around light travel direction.
Optical activity can vary with light frequency {dispersion, light}|. Higher frequencies cause more rotation, because photons have more energy.
If polarized light with different wavelengths passes through asymmetric medium, shorter wavelengths rotate plane more than longer wavelengths {optical rotatory dispersion} {Cotton effect}.
Materials with asymmetric-molecule electric forces can have refractive index different for left and right circularly polarized light {optical activity}|. Carbon can bond four different atoms, in two mirror-image forms.
Light takes shortest path, and so least time, between two points {Fermat's principle} {Fermat principle}.
Action is energy times time, or momentum times distance, or angular momentum times angle. Light uses path with least action between two points {Hamilton's principle} {principle of least action, Hamilton} {least-action principle, light}|. In quantum mechanics, action has quanta, which have size Planck constant h, so photons have energy quanta h * frequency, momentum quanta h / wavelength, and angular-momentum quanta h / 2 * pi.
Lasers produce light waves {coherent light}| that have same phase.
Light passed through consecutive slits {collimate}| has many light waves in phase.
Devices {laser}| {Light Amplification by Stimulated Emission of Radiation} can emit many photons in phase [1960].
light source
Flash tube excites atom electrons into highest orbital. Below highest orbital are one or two lower-energy levels, and below them is ground-state level.
light
Electrons spontaneously fall to intermediate-energy level by vibration, rotation, or radiation.
Then previous photon causes electron to fall to next-lower level {lase}, which simultaneously makes another photon, so both photons are in phase and photon number doubles. This process repeats to make many in-phase photons. Lasers can emit light axially or transversely.
collimation
Photons conserve momentum, so they have same direction.
amplitude
Mirrors can build power by repeated lasing and reflecting, until shutter opens {Q switching} and light releases. Shutter can be rotating mirror, Pockels cell, photochemical, or exploding film. Current modulation can modify laser amplitude. Lasers can pulse or be continuous. Laser can be tunable to different light frequencies.
materials
Lasers can use helium-neon, helium-cadmium, argon, krypton, carbon dioxide, and gallium arsenide. Ruby lasers emit red light. Gallium-nitride lasers emit blue light. Zinc selenide can also make blue light.
purposes
Lasers can align exactly, measure distances by reflection from corner reflectors, attach retinas by burning them on, weld, and make holographs. Lasers can separate atom isotopes, by exciting only one isotope. Lasers can measure thickness, drill holes, and carve miniature circuit blocks. Lasers can implode pellet to start nuclear fusion in tube {hohlraum}.
fiber optics
Laser light passed down non-linear optical fiber {microstructure fiber} broadens in wavelength {supercontinuum light}. Light can alter material, which then alters light {self-phase modulation}.
timing
Lasers {mode-locked laser} can make one-femtosecond microwave or light pulses at 1-GHz. Frequencies are visible light within 150-nm wavelength interval. Superposition makes pulses have few wavelengths. Phase {offset frequency} increases slightly with each pulse. Wave-train pulses have higher net frequencies until cycling again, with equal spacing. Pulses are beats, so pulse frequency is lower-frequency frequency difference. Given reference frequency, beat frequency can determine unknown frequency.
Storing light-wave interference patterns {hologram}| on photographic plates {holograph} allows display of three-dimensional images [Gabor, 1946].
production
Coherent light can shine directly on photographic plate and can reflect from static scene onto plate. Wave-front superposition makes interference pattern that photographic plate can record.
projection
Shining coherent light on or through photographic plate can project scene wave front into space. Plate positions contribute to all image points, whereas photograph points contribute to one image point. Observer sees wave front coming from three-dimensional space, rather than from surface. Observer can view image from different points to see image from different perspectives.
Shining coherent light on part of plate makes whole image but with lower resolution, because number of contributions is less, so standard error is more. Using longer-wavelength coherent light to reconstruct image can magnify image size. Using shorter wavelength coherent-light to reconstruct image reduces image size.
In space-time, observers calculate that observed object times, lengths, distances, and masses depend on observer velocity relative to object {relativity}.
space-time
Space has three dimensions, time has one dimension, and space and time dimensions combine into unified four-dimensional space-time.
Objects and observers move through events on a path (world-line) through space-time. Because light is self-propagating and has no medium, zero-rest-mass particles move through space at light speed. By experiment, all objects move through space-time at light speed, so space dimensions and time dimension relate by light speed. Therefore, rest masses move through time at light speed.
In space-time, time dimension and space dimensions have same units. If space dimensions have time units, their time is space-dimension length divided by light speed. If time dimension has space units, its distance/length is time-dimension time times light speed.
absolute space-time
Universe has absolute space-time (and absolute reference frame). Absolute space-time has local curvature depending on mass and energy space-time positions. Absolute space-time has global curvature depending on mass and energy space-time positions, and so has an overall shape.
However, observers move at light speed through space-time and can calculate only observed-object relative-motion properties. Because they have a reference frame relative to space-time, they have no direct knowledge of space-time. Special relativity describes how observers moving with uniform velocity in relation to objects calculate object motions and properties. General relativity describes how observers moving with non-uniform velocity, due to gravitational fields and/or accelerations, in relation to objects calculate object motions and properties.
space and time
In classical physics, space vs. time graphs show object movements through space locations over time intervals. Objects can have any position and time. Space and time are separate and independent variables. Distance vectors sum.
In relativity, space-time unifies space and time coordinates. Space-time graphs show object space-time events. Object events cannot be at all space-time points, because object maximum speed is light speed. Because some events cannot have space-time separations from other events, separation vectors do not sum.
measurement
Stationary observers measure object length using rulers, which are stationary objects with standard length {unit length} {length unit}, such as one meter. Measurements count the number of unit lengths between object ends. In space-time, because all observers calculate that light always travels at light speed through space, observers count light's time of flight between the two object-end space-time events, and then multiply by light speed to find length.
Stationary observers measure object time using clocks, which are stationary objects with standard time intervals {unit time} {time unit} between clock ticks, such as one second. Clocks have frequencies, such as one cycle per second. If time-interval unit increases, frequency decreases. Measurements count the number of unit times between two events. In space-time, because all observers calculate that light always travels at light speed through space, observers count time of flight between space-time events.
accuracy
Experiments show that stationary clocks and their time units, and rulers and their length units, maintain accuracy for stationary observers after movement over time and space. Clocks at different space locations can synchronize. Rulers at different times and space locations can coincide. Therefore, observers can standardize on the same time at different space locations, and standardize on the same length at different times and space positions.
reference frame
Observers and objects have space-time coordinate systems (reference frame) centered on themselves. Observers and objects traveling with same uniform velocity share the same reference frame and are stationary with respect to each other. Reference frames can be stationary, have uniform velocity, or accelerate in relation to other reference frames.
Observers and observed objects travel through absolute space-time. If they travel at same speed and same direction (same velocity), observed object appears stationary, and observers and observed objects have the same space-time and reference frame. If they travel at different velocities, observed object appears to move at velocity difference, and observers and observed objects have different reference frames. Different reference frames have different events in their space-times and so have different lengths and times.
Because space and time unify in space-time, and different reference frames differ by uniform velocity, reference-frame coordinates can linearly transform into each other.
object speed through space-time
By experiment, all observers always observe that electromagnetic waves travel at light speed through empty space, even if light sources have different uniform velocities. Moving light sources do not add to or subtract from light's observed speed.
By experiment, all observers always see that massless particles move through empty space at light speed, even if accelerations start the particles. Accelerations do not add to or subtract from massless-particle observed speed.
By experiment and calculation, all observers always see that stationary, uniformly moving, and accelerating objects move through space-time at light speed.
Universe objects are always moving through space-time. If they have no rest mass, they travel through space at light speed and so do not travel through time, and space-time interval has zero proper time (light-like). If they have positive mass, they travel through less space and more time, so space-time interval has positive proper time (time-like). If they are relatively stationary, they travel through no space and only time. If they have negative mass, they travel through space and backward through time, so space-time interval has negative proper time (space-like).
stationary observers and objects
Assume that stationary-observer reference frame has a positive-upward vertical time coordinate and a positive-right horizontal uniform-motion-direction space coordinate. See Figure 1.
Stationary objects do not move through space, so stationary observers see stationary objects move at light speed through time only. If stationary observers start at space-time coordinate origin, their space position stays at 0, and their events move along positive time coordinate. Their world-line is perpendicular to time coordinate. After one second, they have moved one light-second vertically. See Figure 2.
uniformly moving objects
Moving objects move through space and time at light speed. To stationary observers, if a uniformly moving object starts at space-time coordinate origin, object events move along a straight line up and to the right, at angle less than 45 degrees to time axis. For example, for velocity 0.5 * c, line has constant slope of 2. See Figure 3. After one second, object has moved one-light-second in that direction.
To stationary observers, because moving object has moved through space more, and object speed through space-time is constant light speed, object has moved through time less, object time has shortened, and object time-interval unit has increased (time dilation).
massless particles
Massless particles move through space at light speed. To stationary observers, if particle starts at space-time coordinate origin, particle events move along a straight line equally up and to the right, and so at a 45-degree angle to both vertical time coordinate and horizontal uniform-motion-direction space coordinate. See Figure 4. After one second, they have moved one-light-second in that direction. To stationary observers, because moving object has moved through space maximally, and object speed through space-time is constant light speed, object has moved through time minimally, object time is zero, and object time-interval unit is infinite.
accelerating objects
Accelerating-object velocities change. In space-time, object world-lines curve. See Figure 5.
stationary observers and moving objects, and time dilation
Stationary observers see that moving objects move through space over time. Because objects move through space-time at light speed, and relatively moving objects move through some space, stationary observers calculate that moving objects move less through time than stationary objects do. Stationary observers calculate that moving objects have shorter times. All observers see that light moves through space at light speed, and frequency times wavelength equals speed, so shorter times mean longer time-interval units (time dilation).
Because moving objects move along uniform-motion-direction space coordinate compared no movement for the stationary case, stationary observers calculate that moving objects have moved less along time dimension than stationary objects did. The number of time intervals is less, so time intervals have become longer (and clocks have slowed down).
To stationary observers, if uniformly moving objects start at space-time coordinate origin, their events move along a straight line through space-time in the positive space direction and in the positive time direction. For example, for relative velocity half light speed, straight line has constant slope 2. After one second, objects have moved one-light-second along straight line.
Because moving objects move along positive uniform-motion-direction space coordinate, stationary observers calculate that times are the same (clocks synchronize) along a different space coordinate than for stationary objects. Because light-travel distances are less, observers calculate that times are less. If clock signals come to observer from earlier in time, then they have taken longer. If clock signals come to observer from later in time, then they have taken shorter. Therefore, stationary observers calculate that moving objects receive clock signals that came from later in time. The motion-direction space coordinate has time zero at all events. To stationary observers, the motion-direction space coordinate points in the positive time direction and positive motion-direction space direction. For example, for relative velocity half light speed, uniform-motion-direction space coordinate has constant slope 0.5.
To stationary observers, moving-object reference-frame time coordinate and uniform-motion-direction space coordinate rotate toward each other the same angle. To stationary observers, massless-particle reference-frame time coordinate and uniform-motion-direction space coordinate rotate toward each other until they coincide at angle 45 degrees to both stationary reference-frame time coordinate and stationary reference-frame space coordinate.
Scientifically, time is time interval required for a photon to return to the same position in a cycle. For example, a photon leaves a source, travels to a mirror, reflects back to the source, triggers a click or photon that goes to a detector (for counting), and causes a second photon to repeat the cycle. Time measures number of unit times between time points. Time measures clock oscillations between events. Unit time can use clock frequency or period. Unit time can be oscillation number as light travels unit length.
If observers and objects are relatively stationary, the photon leaves from, and returns to, the same space position, over a time interval. The observed time and scientific time are the same.
If observers move relatively to objects, the photon leaves from, and returns to, different space positions. Because distance is longer, time interval is longer, so time is slower, and frequency is lower. Because people do not see the same path as the calculated path, scientific time is shorter.
synchronized clocks
Stationary-observer reference frame has time coordinate perpendicular to uniform-velocity-direction space coordinate. Stationary observers at an event on the time coordinate can set distant clocks to the same value as at the event (synchronization).
Observers moving at positive uniform-velocity move through positive time and space. Therefore, the time coordinate is at an angle (between 0 and 45 degrees) away from the stationary time coordinate toward the space coordinate.
Moving observers move toward stationary-observer positive-space-direction synchronized clocks. Therefore, they receive signals from those clocks earlier than stationary observer does, so they calculate that those clocks have past/earlier times. To synchronize moving-observer distant clocks with the clock at moving-observer current event, those clocks must have later times than stationary-observer clocks at the same time as moving-observer current event. Therefore, uniform-velocity-direction space coordinate is at an angle (between 0 and 45 degrees) away from stationary-observer space coordinate toward time coordinate.
Observers moving at uniform velocity have an acute angle (not a right angle) between positive time coordinate and positive uniform-velocity-direction space coordinate.
If moving observer has light speed, time and space coordinates merge at the 45-degree line. Massless particles travel only through space. All clocks have same time. Time interval becomes infinite, and time slows to zero. Lengths shorten to zero. Massless particles travel toward synchronized clocks as fast as light from clocks travels toward massless particles, but light can only travel through space at light speed, not higher.
simultaneity
Using synchronized clocks and knowing light speed, observers can calculate that events occur at the same time (simultaneity). Both events are on one reference-frame line that is parallel to time coordinate.
space-time separation
Space-time space and time coordinates use same units, either length units or time units. For time units, space distance changes to time. Because distance traveled equals light speed multiplied by time in motion, space-coordinate time is distance divided by light speed. For example, if distance is 300,000,000 meters, space coordinate has time 1 second.
For space units, time changes to space distance. Because distance traveled equals light speed multiplied by time in motion, time-coordinate distance is time multiplied by light speed. For example, if time is 1/300000000 second, time coordinate has distance 1 meter.
Two space-time points (events) have space-time separation, measured using time units or space units. Space-time separation is not spatial distance and is not time interval, but depends on both time and space coordinates. If first space-time point is at coordinate origin, and second point is (t, x, y, z), space-time separation s = (t^2 - (x/c)^2 - (y/c)^2 - (z/c)^2)^0.5, using time units.
Because unified space-time has light speed as maximum speed, space-time coordinates are not like Cartesian coordinates with time on horizontal axis, distance on vertical axis, and separation equal to (s^2 + t^2)^0.5. Cartesian coordinates are about independent space and time, which allows infinite speed through space.
Simultaneous events have same time on time axis (no temporal separation), and so have only spatial separation. See Figure 6. s2, s3, and s5 have same time. Setting c = 1 for convenience, space-time separation between s2 and s3 is 1.
Stationary particles do not change position, so stationary-particle events have only temporal separation. See Figure 6. Setting c = 1 for convenience, space-time separation between s1 and s2 is (2^2 - 0^2)^0.5 = 2. Because motions shorten times and lengths, stationary particles have maximum space-time separation.
Moving objects have more spatial separation and less temporal separation than stationary particles. See Figure 6. Setting c = 1 for convenience, space-time separation between s1 and s4 is (3^2 - 1^2)^0.5 = 8^0.5, representing slower particles, which have greater separation. Separation between s1 and s3 is (2^2 - 1^2)^0.5 = 3^0.5, representing faster particles, which have lesser separation. Separation between s1 and s5 is (2^2 - 2^2)^0.5 = 0, representing massless particles, which have maximum speed and have no space-time separation.
Negative space-time separation means objects have moved backward in time. Because objects cannot travel faster than light, objects cannot move backward in time and so cannot have negative space-time separation.
time and separation
Space-time separation is time observed by moving object as it travels. Zero-rest-mass objects travel at light speed and feel no time change. Stationary objects feel maximum time. Massive objects travel at less than light speed and have slower time than stationary objects. Faster objects have slower time than slower objects.
time and space relation
Because space coordinates subtract from time coordinate, shortest spatial distance is longest space-time separation. Longest spatial distance is shortest space-time separation.
simultaneity is relative
Stationary observers can synchronize different-space-position clocks to the same time. For stationary observers, if time axis is vertical, simultaneous events are on horizontal three-spatial-dimension hyperplanes. In two dimensions, if time axis is vertical, simultaneous events are on a horizontal line. See Figure 7. Events along horizontal axis are simultaneous. Events at s2, s3, and s5 are simultaneous at later time.
Moving observers can synchronize different-space-position clocks to the same time. If moving observers compare their synchronized clocks to stationary-observer synchronized clocks at the same spatial positions, moving observer clocks have later time in the uniform-motion direction (and earlier time in the opposite direction). Because they are moving closer to the clocks all the time, so the signals arrive quicker, for the same information to arrive at the current space position for both moving and stationary observer, the time at the distant position must be later for moving-observer synchronized clocks. For moving observers, simultaneous events are on three-spatial-dimension hyperplanes at an angle to time axis. In two dimensions, simultaneous events are on angled lines. For stationary observer, moving observer clocks are simultaneous along a positively sloped line, at a less-than-45-degree angle. See Figure 8. Stationary observer sees events s2, s3, and s5.
Simultaneous events are not simultaneous for observers with different velocities.
The three-spatial-dimension hyperplane of simultaneous events, line s2-s3-s5 in the diagram, is at same angle as world-line angle.
To depict moving observer at actual world-line, rather than as stationary, three-spatial-dimension hyperplanes of simultaneous events must transform their coordinate axes. Hyperplanes of simultaneous events must change from right angles to angle between world-line and space axis (limiting angle is 45 degrees). The angled-line series in the diagram represents the hyperplanes. See Figure 9.
stationary observers and moving objects, and length contraction
Because time is shorter, observers must calculate that motion-direction object distances and lengths are shorter (length contraction) so that all observers still see that light moves through space at the same light speed. Stationary observers calculate that both times and motion-direction lengths (and distances) shorten in the same proportion to keep light velocity constant. Lengths shorten in inverse proportion that time dilates.
Scientifically, length is space-time spatial distance between endpoints at space-time same time. Length measures number of unit lengths between space locations. Unit length can be a ruler. Unit length can be how far light travels in unit time.
If observers move relatively to objects, those light rays must leave the endpoints at different times, and scientific length is shorter. Moving-observer simultaneous times are later than stationary-observer ones. Coordinate transformations can calculate scientific lengths.
distances and relativity
Time shortening and length contracting are ratios. Events near each other in space and time have small total shortening. Events far from each other in space and time have large total shortening. For example, people walking and people sitting perceive several-days time difference when observing events in Andromeda galaxy. At large distances, slow relative motions can have measurable relativistic effects. At short distances, only fast relative motions show measurable relativistic effects.
observations
People and cameras observing lengths see light rays that arrive at the same time at iris or aperture. People and clocks observing time see light-ray-path-endpoint light rays that arrive at the same time at iris or detector.
If observers and objects are relatively stationary, those light rays leave the endpoints simultaneously, and observed length and scientific length are the same. Observers see and measure events whose information simultaneously reaches their space-time events. For example, stationary observers receive information from both ends of a stationary ruler perpendicular to sightline between eye and ruler-center at same time. Stationary observers simultaneously see the whole stationary ruler, but light from some positions along the ruler left before other positions. Stationary observers simultaneously see the whole night sky, but light from nearby stars left those stars a few years ago and light from farther stars left those stars longer ago.
Relativity is not about actual observer observations but about calculations based on knowledge of light speed, space-time, and space-time curvature. Length contraction and time dilation involve simultaneous points in space-time at object, not at observer. In stationary-observer reference frame, moving lengths calculate to be shorter, but human observers and instruments do not actually see or measure shorter lengths.
light
All observers see that light moves at light speed through space, no matter if light source moves relatively toward or away from observer. Light has no medium and self-propagates at light speed through empty space. Charge acceleration starts electromagnetic induction. Electric and magnetic fields change. Changing electric and magnetic fields interact to send transversely-changing electric and magnetic fields perpendicular to charge acceleration and velocity direction. Therefore, light-source motion does not supply extra motion to light speed. Light speed is the same for forward, backward, or no source motion.
Observers moving relatively to light sources along light-ray direction see frequency changes, because relative motion causes observer to encounter light-wave peaks and troughs at a different rate than if light sources are relatively stationary. If observer velocity is toward light source, expected light speed is higher than observed light speed. To reduce speed to observed light speed, length must decrease (contract). By Doppler effect, wavelength is shorter, and light frequency becomes higher. If observer velocity is away from light source, expected light speed is lower than observed light speed. To increase speed to observed light speed, length increases. By Doppler effect, wavelength is longer, and light frequency becomes lower.
Observers moving transverse to light-ray direction see light traveling at light speed, because no motion component is along light ray. If observer velocity is transverse to light-ray light source, observer sees relativistic length contraction and time dilation, in the same proportion, so relative velocity remains constant.
Relativity is about relative motion transverse to observer sightline toward object center. If massless particles, such as light photons, travel transversely, they travel only through space, do not travel through time, have infinite time interval (and clocks have stopped), and have no length in transverse direction.
To stationary observers, if massless particles start at space-time coordinate origin, their events move along a straight line at a 45-degree angle to both stationary time coordinate and stationary space coordinate. After one second, they have moved one-light-second in that space-time direction.
From a space-time event, signals can go only to space-time events in positive time direction and positive or negative space direction {light-cone}. All observed events happen in the present, unaffected by time. Only influences from previous events that have simultaneously reached space-time event can affect event. At any instant, observers see a space-time spatial cross-section.
relative distances and times
Observers with different relative velocities see the same observed objects at different events in observer space-time, so objects have different measured/calculated lengths and times. Observations are relative to velocity differences.
relativity principle and invariance
In space-time, all observable physical laws are the same at any constant velocity {principle of relativity} {relativity principle}|. At different relative velocities, observers see different length contractions and time dilations, which cancel to maintain physical laws. Only relative times, masses, and lengths have meaning for physical laws.
Because physical events occur in unified space-time, and all objects move through space-time at light speed, for stationary and moving observers, and for uniform-velocity and stationary (zero uniform velocity) systems and reference frames, physical laws are the same (invariance). Kinetics and dynamics equations, Maxwell's electromagnetism equations, and Newton's and Einstein's gravitation laws are invariant for all reference frames with uniform velocity.
By relativity, stationary observers calculate shortened lengths and times for moving objects. Because both are shorter in the same ratio, velocity is constant, and system kinetics are the same.
Such systems have no forces or accelerations. All system parts and reference-frame points have the same motion.
Calculated length contraction, time dilation, and mass increase change in the same ratio, so physical laws are the same in all reference frames differing only in uniform velocity. Space-time always has energy-momentum conservation.
no absolute velocities
Because systems with different uniform velocities have the same physical laws, uniform velocity has no physical effects, and observers cannot determine their or other object's absolute uniform velocity through space-time. All velocities are relative to observers and reference frames.
no absolute lengths and times
Because systems with different uniform velocities have the same physical laws, observers cannot determine absolute lengths and times. All lengths and times are relative to observers and reference frames.
events and physical laws
Because physical laws are invariant for all uniform-velocity reference frames and under linear coordinate transformations, physical space-time location can have no influence on physical laws, so physical laws are the same at all universe space-time points (events). There are no preferred space-time events.
event order
Before and after are relative concepts, because different observers can see different event sequences. Different observers have different pasts and futures.
Moving and stationary observers see some events in different orders. Compared to stationary observers, for events {spatially separated events} far enough apart that light cannot travel from one event to the other, moving observers see stationary events later than faster moving events.
Moving and stationary observers do not see the same spatially separated simultaneous events. Compared to stationary observers, moving observers see stationary events later than faster moving events.
relativity and space
At large distances and times, by proportionality, small relative velocities have large time-dilation and length-contraction effects. The space of relativistic velocities is hyperbolic space.
hyperbolic space
Distance equals light speed times time: x = c*t. Product of position and inverse-time always equals light speed: x * t^-1 = c. On space-time coordinates, equation graph is a hyperbola. If position is far, time is far. If position is near, time is near. If objects are rulers or clocks, distance separation and time separation are directly proportional.
mass, momentum, energy
In space-time, mass, momentum, and energy unite in a vector (energy-momentum tensor) {4-momentum}. Momentum is movement through space. Energy is movement through time. Energy equals mass times light speed squared. Action relates momentum and space and relates energy and time.
mass increase
Stationary observers calculate that relatively moving objects increase mass.
gravity and acceleration
Relative acceleration and gravity change relative velocity, so observed lengths and times depend on relative acceleration and gravity [Greene, 1999] [Mach, 1885] [Mach, 1906] [Rees, 1997] [Rees, 1999] [Rees, 2001] [Smolin, 2001] [Weyl, 1952].
electric charge
Stationary observers calculate that relatively moving charged objects increase charge density.
electric field and magnetic field
In space-time, electric field and magnetic field unite. To stationary observers, motion makes relativistic charge that has magnetic field. To moving observers, the same motion stands still, and charges have electric field. In space-time, electric field is in space, and magnetic field is in time (electromagnetic-field tensor).
geodesic
Objects travel through space-time along shortest space-time separation, which is the straightest path (geodesic) through space-time. Objects travel through positively curved space-time along shortest spatial distance and longest time. Objects travel through negatively curved space-time along shortest time and longest spatial distance.
curvature
At space-time points, mass, energy, stress, and pressure curve space-time. Around masses, gravity-field gradient (gravitational potential) is space-time curvature. Central masses curve local space-time, and that curvature pulls adjacent space-time points to curve space-time at faraway points.
Alternatively, central masses cause field energy density at far space-time points, and that energy curves space-time. Farther away space-time points have less curvature, because same energy spreads over more surface area, so energy density is less.
Energy and curvature spread to infinity at light speed, because space-time has tension and propagation characteristics the same as light-propagation speed.
free fall
Freely falling observers see no space-time curvature, because they see no acceleration, because they are at rest in the free-fall reference frame. Stationary observers at earth's surface see no space-time curvature, because they see no acceleration, because they are at rest in their reference frame.
equivalence
To stationary observers, gravity and applied force accelerate mass. Acceleration indicates space-time curvature. For both gravity and applied force, stationary observers calculate that objects move along geodesics through space-time curvature. Gravity and applied force acceleration are equivalent.
Space-time curvature is relative to observer, and so is not absolute. Observers detect only accelerations and cannot detect space-time curvature per se. However, acceleration is real and can slow clocks.
curvature and acceleration
In curved space-time, objects increasingly travel through more space, and so decreasingly less time, which means that objects accelerate. Stationary observers see increasing time dilation and time unit, decreasing frequency and time, and decreasing length.
gravitons
Gravitons are quanta of space-time gravitational waves. Because gravitons have zero rest mass, gravity acts out to infinity.
Whereas photons propagate through empty space as traveling waves in field lines, gravitons propagate through empty space as traveling waves in field surfaces. Gravitational field lines and electromagnetic field surfaces have the same tension, which is the maximum tension that they can have. Because of that maximum tension, and because they have no mass, photons and gravitons travel at light speed.
Alternatively, all zero-rest-mass bosons propagate the same because they are really the same at high energy.
tidal effects
Gravity varies inversely with distance, so objects in gravitational fields that have non-negligible diameter feel different forces on near and far sides.
However, gravity does not have to have space-time tidal effects, because local fields can be uniform, two fields can cancel, and pressure can cancel gravity, so that local space-time curvature is zero or constant.
forward time
Spatial directions can go forward or backward, but the time dimension has one direction, forward. Entropy increase, system evolution, or memory can cause time unidirectionality.
universe and relativity
Perhaps, universe is infinitely old and large, with no expansion or evolution. Perhaps, curved time coordinate allows far clocks to run slower or faster. Perhaps, far clocks stop at infinite distance, where red shift is infinite, so red shift does not need universe expansion as explanation.
electromagnetic and gravitational forces
Electromagnetism and gravity have effects to infinity and so transmit force using zero-rest-mass particles, which travel at light speed for all observers. Electromagnetic-force changes and gravitational-force changes propagate through space at light speed. Those forces' strengths determine propagation speeds, which are both light speed. Perhaps, those strengths correspond. Force strength depends on charge and mass quanta, so space-time relates to quanta. Perhaps, space-time space and time dimension relations depend on electromagnetic-force and gravitational-force strengths.
Particle energy varies directly with mass and light-speed squared {E equals m c squared}: E = m * c^2.
gravity and electromagnetism
Gravitation energy relates to electromagnetism energy.
Both electromagnetic force and gravitational force exchange zero-rest-mass particles, so both forces have effects out to infinite distances.
Electromagnetism and gravitation are spatial fields. Electromagnetism makes radial electromagnetic-field lines. Gravity makes radial gravitational-field surfaces. Surfaces have more space than lines, so electromagnetism is stronger than gravity.
Electromagnetic and gravitational waves do not travel through a medium. They propagate by induction along field lines and surfaces. Wave-propagation speed depends on field strength and field type. Electromagnetism is stronger than gravity, but in same proportion gravity uses more space, making both field lines and surfaces have maximum tension, so both electromagnetic and gravitational waves travel at light speed.
energy
Object total energy equals rest energy plus kinetic energy plus potential energy: E = RE + KE + PE. Kinetic energy varies directly with mass and velocity squared: KE = 0.5 * m * v^2. Potential-energy change PE varies directly with local force-field force and position change d in field: PE = F * d. Rest energy is constant.
energy and momentum
In classical physics, particle energy and momentum are separate physical properties, with separate conservation laws. Energy conservation depends on time symmetry. Momentum conservation depends on space symmetry.
In relativity, space and time unite in four-dimensional space-time. By experiment and calculation, all particles and objects travel at light speed through space-time. Particle motion through space-time has momentum and energy, but energy is through time and momentum is through space. In space-time, momentum and energy unite into one four-dimensional vector {energy-momentum four-vector} (4-momentum). Energy is time-like component, and momentum is space-like component.
energy and momentum conservation
For constant particle rest energy, energy conservation means that potential-energy change equals negative of kinetic-energy change. In space-time, potential energy changes through space, and kinetic energy changes through time. Kinetic-energy change changes velocity and so changes momentum. Because energy and momentum stay constant, energy-momentum four-vector separation is invariant for any inertial space-time reference frame and under any linear coordinate transformations. For potential energy (including rest energy) and momentum changes, 4-momentum-vector space-time separation has equation, in space units: s^2 = E^2 / c^2 - p^2, where E is change in potential energy and rest energy, and p is kinetic-energy change in space units. (Dividing by c makes time units into space units.)
rest-mass energy
Resting masses {proper mass} (rest mass) have no speed through space dimensions, and so travel through time dimension at light speed c. Along time dimension, rest-mass 4-momentum-vector separation s is m0 * c, where m0 is rest mass. Because rest masses do not change space position, potential energy is zero, and rest mass is constant. Because rest masses have no velocity, kinetic energy is zero, and momentum is zero. Therefore, s^2 = (m0 * c)^2 = E^2/c^2 - (0)^2, so E = m0 * c^2.
Rest mass has available energy. Rest masses are like energy concentrations. Mass densities are like energy fields.
moving masses
Moving masses have increased positive kinetic energy. Increased kinetic energy is similar to concentrated mass, so stationary observers calculate that moving masses have mass increase (relativistic mass).
Moving mass goes through space-time separation m * c, where m is total mass (rest mass and relativistic mass). Moving mass has momentum total-mass times velocity. Rest-mass energy is rest mass times light speed. For example, if potential energy is zero (with no gravity), and velocity is 0.75 * c, - s^2 = - (m * c)^2 = - (m0 * c^2)^2/c^2 - (m * 0.75 * c)^2, then - m^2 * c^2 = - m0^2 *c^2 - (0.75)^2 * m^2 * c^2, and then - 0.25 * m^2 = - m0^2, and total mass m = 2 * m0.
In empty space, energy E depends on rest energy, in time dimension, and kinetic energy KE, in space dimensions. Rest energy = m0 * c^2. KE depends on momentum p: KE = p * c. Total energy sums rest energy and kinetic energy {relativistic energy-momentum equation}: E^2 = (m0 * c^2)^2 + (p * c)^2.
For zero-rest-mass particles, E = p*c.
For resting masses, p = 0, and E = m0 * c^2.
For moving masses, total energy is total mass m times light-speed squared: E = m * c^2. If velocity is near zero, total mass is almost the same as rest mass. If velocity is near light speed, total mass is very large, much greater than rest mass.
relativistic mass increase
Objects traveling through space have momentum and kinetic energy. Higher-velocity objects travel more through space and less through time, causing more time dilation and length contraction. Objects traveling more through space increase 4-momentum momentum and kinetic energy, in the same proportion that time dilates. Therefore, total mass m increases with velocity: m = m0 / (1 - (v^2 / c^2))^0.5, where c is light speed, v is velocity, and m0 is rest mass. For example, if velocity is 0.75 light speed, total observed mass is twice rest mass.
equivalence
Energy in time dimension can go into momentum in space dimensions, and vice versa {mass-energy equivalence}.
relativistic energy in series format
In series format, in empty space, total energy E = m0 * c^2 + m0 * v^2 / 2 + (3 * m0 * v^4) / (8 * c^2) + ...., where m0 is rest mass. See Figure 1. The first term is the rest energy. The sum of the higher-power terms is the kinetic energy. For slow particles, later terms are very small, so kinetic energy is m0 * v^2 / 2, matching the classical value of 0.5 * m * v^2.
mass and energy equivalence
Particle decomposition and composition experiments show that mass and energy are equivalent and depend only on reference frame.
cases
An unstable particle with mass can become two zero-rest-mass particles that travel at light speed in opposite directions from particle position. Zero-rest-mass particles have no potential energy. Moving particles have kinetic energy. Mass changes into kinetic energy, to conserve mass-energy. Momentum in one direction equals momentum in opposite direction, to conserve momentum.
An unstable particle with mass can become two particles, one with mass and one with no mass. Total mass is less than before, to conserve energy. The zero-rest-mass particle travels at light speed in opposite direction from new particle with mass, which travels at less than light speed. Momentum in one direction equals momentum in opposite direction, to conserve momentum.
Two particles with mass can collide to make one particle with mass and one zero-rest-mass particle. The zero-rest-mass particle travels at light speed in opposite direction from new particle with mass, which travels at less than light speed, to conserve energy. Momentum in one direction equals momentum in opposite direction, to conserve momentum.
Two equal particles can collide and stop, so kinetic energy becomes mass. Total mass is then more than sum of rest masses before.
Physical objects and events occur in unified space-time. Uniformly moving observers observe no forces and no accelerations and have space-time coordinates (reference frames) with no curvatures. All system parts and reference-frame points have the same motion.
Because uniformly moving reference frames can linearly transform into each other, and objects move through space-time at light speed, physical laws are the same {invariance, uniform motion} for all uniformly moving observers and objects. Locally, kinetics and dynamics equations, Maxwell's electromagnetism equations, and Newton's and Einstein's gravitation laws are invariant for all reference frames with uniform velocity.
conservation laws
Motion equations relate local momentum and energy exchanges between particles and fields. Energy and momentum conservation laws are examples of invariance. Energy conservation is about time symmetry. Momentum conservation is about space symmetries. Space-time unites space and time, so space-time has one energy-momentum conservation law.
cause
By relativity, stationary observers calculate shortened lengths and times for moving objects, in the same ratio. Therefore, velocity is constant, and system kinetics remain the same.
no absolute velocities
Because systems with different uniform velocities have the same physical laws, uniform velocity has no physical effects, and observers cannot determine their or object absolute uniform velocity through space-time. All velocities are relative to observers and reference frames.
no absolute lengths and times
Because systems with different uniform velocities have the same physical laws, observers cannot determine absolute lengths and times. All lengths and times are relative to observers and reference frames.
linear coordinate transformations
Uniform velocities relate reference-frame coordinates linearly. All uniform-velocity reference frames can transform to all other uniform-velocity reference frames by linear coordinate transformations. Therefore, physical laws are invariant for linear coordinate transformations. For example, linear coordinate transformations can derive Maxwell's equations from Coulomb's law.
events and physical laws
Because physical laws are invariant for all uniform-velocity reference frames and under linear coordinate transformations, physical space-time location can have no influence on physical laws, so physical laws are the same at all universe space-time points (events).
space-time separation
Space-time separation is invariant for linear coordinate transformations.
physical constants
Because physical laws are invariant for all uniform-velocity reference frames and under linear coordinate transformations, fundamental physical values are constant for all uniform-velocity reference frames and under linear coordinate transformations. For example, angular momentum and other quanta remain constant.
accelerating objects
In local space-time regions, and for small accelerations, reference frames can approximate uniform-motion reference frames, so physical laws are invariant over linear coordinate transformations. Large space-time regions and large accelerations break physical-law invariance.
Physical laws do not have action-at-a-distance. Physics laws are about what happens at space-time points {local interaction}. Except for large forces, local interactions are approximately linear, allowing linear coordinate transformations and energy-momentum-field tensors. Spaces with local metrics include Minkowski space, Einstein space, and Lorentzian space.
Global physical effects are typically non-linear.
Observers can measure relative speed and direction {relative velocity}| with respect to physical objects (and measurement reference frames), using clocks for time units, rulers for length units, and/or light signals for time and length units.
observers
Observers can measure space-time separation from current space-time event to future space-time event using light signals. Observer clocks measure time for light to go from observer to reflector and return from reflector to observer. Observer rulers measure length for light to go from observer to reflector and return from reflector to observer.
Observers and objects can be stationary or moving. Observers observe themselves as stationary and observe objects with no relative velocity as stationary, and relatively stationary things have no special-relativity effects.
Observers can move relative to objects, and objects can move relative to observers. If relative velocities are the same, the two situations are physically equivalent.
Relatively moving things have relativistic effects, which are space-time calculations about simultaneous distant space-time events. Relativity is about what must be true at distant events and so depends on calculations.
In relativity, only calculations change. Objects do not change, because observations do not affect their space-time event.
observations
People and instruments receive simultaneous signals at their current space-time event. Signals came from different points on lengths and from different phases of time. Their observations do not have time dilation or length contraction.
uniform velocity
Relative velocity measures proportion of light speed through space-time that is through space compared to that through time. Light has maximum speed through space.
Special relativity is about relative uniform velocity, with no acceleration. Special relativity is about empty space.
acceleration
Changing velocity speed or direction requires gravitational and/or electromagnetic force (including mechanical force) fields.
Zero-rest-mass particles, such as photons and gravitons, have no inertia and have no gravity, so they do not interact with other masses and gravitational energies. Adding gravitational or mechanical energy to zero-rest-mass particles does not increase or decrease their velocities. Zero-rest-mass particles travel at maximum-velocity light speed, cannot travel faster or slower than light speed, and cannot be at rest.
Zero-rest-mass and zero-charge particles, such as photons, have no inertia and have no electromagnetism, so they do not interact with other charges and electromagnetic energies. Adding electromagnetic energy to zero-rest-mass particles does not increase or decrease their velocities. Zero-rest-mass and zero-charge particles travel at maximum-velocity light speed, cannot travel faster or slower than light speed, and cannot be at rest.
direction
Two objects can move toward or away from each other (radial motion) and/or right-left and/or up-down with respect to each other (transverse motion). Special-relativity relative-measurement differences are about relative transverse motion. Motion toward or away does not change measured/calculated length or time.
observers and absolute velocity
Universe has absolute space-time. Space-time unites space and time symmetrically. Objects travel through space-time at light speed. Because light travels at light speed no matter observer or object motion, observers cannot observe absolute space-time or absolute velocity. Observers measure that moving objects have length contraction and time dilation, in the same proportion, so relative velocity stays constant, and light has constant light-sped velocity. Relative velocity does not affect physical laws, so observers cannot use experiments to find absolute velocity.
velocity, length, time
For relative velocity v, length is x * (1 - (v/c)^2)^0.5, and time interval is t / (1 - (v/c)^2)^0.5. For example, if relative velocity is half light speed, length is 0.86 * x, and time interval is 1.16 * t. If relative velocity is 99.9% light speed, length is 0.01 * x, and time interval is 99 * t. If relative velocity is light speed, length is zero, and time interval is infinite.
relative velocity maximum
If objects travel faster than light, relativistic length becomes less than zero, so physical objects cannot travel faster than light speed. If objects travel faster than light, relativistic mass becomes more than infinite.
If objects travel faster than light, relativistic time interval becomes more than infinite, and time goes backward. Traveling backward in time violates causality. Space-time events can only receive signals from finite space-time event regions, whose space-time events are near enough so signals from them can reach the space-time event. See Figure 1.
faster than light
Because light always travels at light speed, faster-than-light signals from stationary objects appear to go backward in time to observer moving away. See Figure 2. If signals continue to second stationary object, toward which observer is moving, second stationary object can reflect signal back to first stationary object. Faster-than-light signals from second stationary object appear to go backward in time to observer moving toward it. The signal returns to first stationary object before it sends original signal. If signal can travel faster than light speed, observer can have event knowledge before event happens.
See Figure 3. Moving observers have tilted hyperplanes of space-like space-time events relative to object going from s0 to s1 to s2. They see signals from s2 as going backward in time. Moving observers have different tilted hyperplanes of space-like space-time events relative to object going from s0 to s3. They see signals from s3 as going backward in time. Object going from s0 to s1 to s2 appears to receive reflected signals from s3 at s1, before signals left s2.
Relatively stationary objects have energy {rest energy}| due only to rest mass. Relative motion does not affect rest energy.
Universe has three continuous space dimensions and one continuous time dimension, and they unite symmetrically in four-dimensional space-time {space-time, relativity}|. Time and space are not separate and independent physical properties.
events
All particles and objects move through space-time at light speed. Observers and objects move through time as well as space.
Space-time has events, not independent times and spatial positions. Observers and objects move through space-time events. Objects travel through space-time along four-dimensional-vector paths {world line}, along geodesics.
units
Space-time coordinates have the same units for time and space. Because distance equals time times velocity, distance unit can be time unit times light speed: c*t. Because time equals distance divided by velocity, time unit can be distance unit divided by light speed: x/c. Time unit can use one oscillation over one distance unit: 1 / cm = cm^-1.
relativity
Though relatively moving observers calculate different lengths and times, because lengths and times shorten in same proportion, space-time separation between two space-time events is constant for all uniformly moving observers.
regions
Different space-time regions behave differently. Regions can be inside gravity or electromagnetic sources. Vacuum regions can be near sources. Regions, such as flat space-time, can have weak fields. Regions can have weak fields but have radiation.
time coordinate as imaginary numbers
Relatively moving observers have different reference frames and relative space-times. Relative space-times differ by relative velocity (boost), which causes three-dimensional rotations into time. The complex plane and space-time coordinate system have the same properties, so time coordinate is like imaginary-number coordinate.
Simultaneity requires an observer and two space-time events. By direct observation, two space-time events are simultaneous {simultaneity, relativity}| if light signals from both events reach observer at same time. See Figure 1. To another observer at a different space-time event, the two events are not simultaneous. (If an observer sees that two events happen at the same spatial location, relatively moving observers do not calculate that they coincide.)
synchronized clocks
Two space-time events can be simultaneous for an observer if they occur at the same time on observer's synchronized clocks. Because this simultaneity occurs at distant space-time events, observer can only measure and calculate this simultaneity.
Observer reference-frame space coordinates show all space locations, with clocks synchronized to same time. Events that happen on a space-time space coordinate are all at the same time. For example, at space-time origin, time is 0, and space coordinate shows all space locations, whose events have time 0.
space coordinates
Space coordinates for relatively moving observers are different. Compared to stationary observers, uniformly moving observers move toward stationary-observer synchronized clocks, and receive light rays sooner than stationary observers. Because they come sooner, relatively moving observers calculate that those light rays came from later-time events. See Figure 1.
examples
At s1, observer has time 0 and position 0. s2 and s3 happen at same time on time axis, and information about them reaches s4 at same time.
Information from s1 reaches s2 and s3 at same time on time axis, so they are simultaneous. s2 and s3 are two different observers.
Because signals travel at light speed, information from s1 can reach s6.
Information from s1 cannot reach s4, s5 or s7.
Information from s3 reaches s6 later.
At s4, observer is at time 2 and position 0. Information from s4 reaches s7.
s4, s5, and s6 happen at same time on time axis.
No information about s1, s2, s3, s4, and s6 reaches s5.
Information about s3 reaches s6.
Information about s5 comes to observer at position 0 sooner than information from s6, because s5 is closer in space.
s7 happens later than s4, s5, and s6 on time axis.
space-time separation
Observers and objects move through space-time events. Space-time events have space-time separations. In space-time, neither time separation nor space separation exists independently.
Simultaneous events for observer have same space-time separation from observer. Observers cannot detect events from space-time points outside their light cone. See Figure 2. s2 and s3 are simultaneous for observer at space-time position s4. s4 and s6 are simultaneous for observer at space-time position s7. s4, s5, and s6 are not simultaneous for any observer.
event order
Relatively moving observers do not agree on event locations or times, and can calculate that the same space-time events happen in different orders.
absolute time
Because relatively moving observers calculate different times, spatial positions, and event orders for the same space-time events, observers cannot detect absolute time (or absolute location).
Local space-time coordinate systems can transform by first-power functions into all other local space-time coordinate systems {Lorentz transformation} {linear transformation}.
Distinct space-time points have different pasts and futures {space-time point set}. Because maximum speed is light speed, space-time points have possible past points {past-set} and possible future points {future-set}. Point past-sets and future-sets are unique {indecomposable}. Indecomposable past-sets can affect the space-time point. The space-time point can affect indecomposable future-sets.
Geometries can have points at infinity (ideal point). Space-time should not have ideal points or singularities.
All space-time points {event} have past-set and future-set, so space-time has possible causes and effects {causal structure}. Space-time events change causal structure over time and space.
Space-time points have a space-time region {global causal structure} that light can reach in the future. A space-time point can only affect those events. A space-time point has a space-time region whose events can affect it.
Space paths cannot reverse time, so no event can happen at two times. Between past point and future point reachable from past point, all space-time points are reachable {hyperbolic space-time, global}, so space-time has no singularities.
All light rays from a space-time point make a space-time cone {light cone}|. All light rays to a space-time point make a light cone.
If light rays from a space-time point later converge {converging light cone}, convergence point is a singularity.
In zero gravity, object translations, rotations, vibrations, scale changes, and inversions in space do not change object geometric shape. Zero-gravity four-dimensional space-time has symmetry {conformal symmetry} {conformal symmetry group} that preserves geometric shape, because metric-scale changes {Weyl transformation} leave proper time and proper length unchanged. Mathematically, the Poincaré group, scale invariance (dilation or dilatation), and inversion-translation-inversion (special conformal transformation) have conformal symmetry and preserve geometric shape.
Observers moving uniformly in unified space-time in relation to objects calculate that object length in uniform-velocity direction is shorter than for relatively stationary objects {distance contraction} {length contraction}|.
relativity
Stationary observers calculate that moving objects have shorter lengths in movement direction than stationary objects. Moving observers calculate that stationary objects are moving and have shorter lengths in movement direction. In both cases, observer and object have relative velocity. See Figure 1.
direction
Length contraction happens only in movement direction. Length contraction depends on relative transverse velocity. The radial velocity component has no effect, and directions perpendicular to movement direction have no length contraction.
distance from observer
Because contraction direction is perpendicular to distance direction, distance away does not affect length-contraction ratio.
calculation
When stationary observers look at moving rulers, ruler points do not have same time. See Figure 1.
space-time reference frame
On space-time reference frames, moving events trace vectors. Stationary objects trace vectors parallel to time coordinate. See Figure 2.
space-time separation
Space-time events are separate in both time and space.
Compared to stationary rulers, moving-ruler leading end is earlier in time and behind in space. Trailing end is later in time and ahead in space. Observer calculates that object length and time are shorter. Length contraction and time dilation have same percentage, so physical laws do not change, and space-time separation is same as before. For all uniformly moving observers, physical laws are the same, and space-time separations are the same.
See Figure 3. In space-time, space gain causes time loss, so space-time separation s depends on space separation x and time separation t (ignoring y and z dimensions). Because distance x is light speed c times time t, s^2 = x^2 - (c*t)^2, using distance units, or s^2 = t^2 - x^2/c^2, using time units.
For constant motion, t = x/v, so s^2 = (x/v)^2 - (x/c)^2 = x^2 * (1/v^2 - 1/c^2) = (x^2 / v^2) *(v^2 / v^2 - v^2 / c^2) = (x^2 / v^2) * (1 - v^2 / c^2) = (x^2 / c^2) * (c^2 - v^2) / v^2. Therefore, s = (x/v) * (1 - v^2 / c^2)^0.5 or s = (x/c) * (c^2 - v^2)^0.5 / v. Stationary observers calculate that moving-object length is shorter than stationary length.
length-contraction percentage
If moving object has velocity 0.5 * c (half light speed), space-time separation s = (x/c) * (((c^2 - 0.5 * c)^2)^0.5 / (0.5 * c)) = (x/c) * ((c^2 - 0.25 * c^2)^0.5 / (0.5 * c)) = (x/c) * ((0.75*c^2)^0.5 / (0.5 * c)) = (x/c) * 0.865/0.5 = 0.43 * (x/c).
If moving object has velocity 0.9 * c (nine-tenths light speed), space-time separation s = (x/c) * ((c^2 - (0.9 * c)^2)^0.5 / (0.9 * c)) = (x/c) * 0.19/0.9 = 0.21 * (x/c).
If moving object has velocity 0.99 * c (99% light speed), space-time separation s = (x/c) * ((c^2 - (0.99 * c)^2)^0.5 / (0.99 * c)) = (x/c) * 0.02/0.99 = 0.02 * (x/c).
As moving object approaches light speed, stationary observer sees that length decreases toward zero. Stationary objects have maximum space-time separation.
maximum speed
Length less than zero is impossible. Therefore, nothing can go faster than light speed, and nothing can go backward in time.
length measurement
To measure stationary rulers, stationary observers at one ruler end can send signals to a reflector at other end. See Figure 4. Signal travels from end to end and back. Time to go is same as time to return. Travel time is directly proportional to length.
To measure moving rulers, stationary observers at one ruler end can send signals to a reflector at other end. Ruler reflector moves closer as signal travels and reflects earlier. Observer measures shorter time and measures that ruler has shorter length.
Stationary observers calculate that stationary rulers spread over space only. Stationary observers calculate that moving rulers spread over space and time. Stationary and moving rulers have same space-time separation. See Figure 5.
For moving rulers, for ends to seem simultaneous, ends lie along line tilted away from vertical, not on vertical. See Figure 5. Leading end is further along in space, and trailing end is behind in space. Middle moves toward where leading end was, and away from where trailing end was. For signals to reach middle simultaneously, leading end must signal later in time, and trailing end must signal earlier in time.
time
When moving object passes stationary observer, one end reaches observer before other end. Other end lags behind in time, because ends are traveling through time at less than light speed, and it takes time for other end to reach observer. When moving object moves through space faster, lengths appear shorter, and moving object moves through time slower, so time slows. See Figure 2. At less than light speed, angle is less than 45 degrees. At light speed, angle is 45 degrees.
analogies
Length contraction is like looking at rulers rotated away from perpendicular to sightline. For space-time, rotation is into time dimension.
Length contraction is like looking at rulers from farther away.
Length contraction is like light rays curving inward from both ruler ends, like a concave lens (opposite from gravitational lensing).
Because space-time separation has a negative sign under the square root, length contraction is like using imaginary numbers. Space-time time coordinate is like imaginary axis, so space-time is like complex plane.
Observers moving uniformly in unified space-time in relation to objects calculate that object time in uniform-velocity direction is shorter than for relatively stationary objects {time dilation}|, and that unit time interval takes longer, so time slows down.
relativity
Stationary observers calculate that moving objects have shorter times than stationary objects. Moving observers calculate that stationary objects are moving and have shorter times. In both cases, observer and object have relative velocity. See Figure 1.
direction
Time dilation depends on relative transverse velocity. The radial velocity component has no effect.
distance from observer
Because transverse relative velocity is perpendicular to distance direction, distance away does not affect time-dilation ratio.
observation
When stationary observers look at moving clocks, times are not at same positions. See Figure 1.
space-time reference frame
On space-time reference frames, moving events trace vectors. Stationary objects trace vectors parallel to time coordinate. See Figure 2.
space-time separation
Space-time events are separate in both time and space.
Compared to stationary clock, moving-clock first tick is behind in space and so earlier in time, and latest tick is ahead in space and so later in time. Observer calculates that object length and time are shorter. Length contraction and time dilation have same percentage, so physical laws do not change, and space-time separation is same as before. For all uniformly moving observers, physical laws are the same, and space-time separations are the same.
See Figure 3. In space-time, space gain causes time loss, so space-time separation s depends on space separation x and time separation t (ignoring y and z dimensions).
Because distance x is light speed c times time t, s^2 = x^2 - (c*t)^2, using distance units, or s^2 = t^2 - x^2/c^2, using time units.
For constant motion, x = v*t = (c*t)^2 - (v*t)^2 = t^2 * (c^2 - v^2) = c^2 * t^2 * (1 - v^2 / c^2). Therefore, s = c * t * (1 - v^2 / c^2)^0.5. s/c = t * (1 - v^2 / c^2)^0.5. Stationary observers calculate that moving-object time is shorter than stationary time.
time-dilation percentage
If moving object has velocity 0.5 * c (half light speed), time s = t * (1 - (0.5 * c)^2 / c^2)^0.5 = t * (1 - 0.25) = 0.75 * t.
If moving object has velocity 0.9 * c (nine-tenths light speed), time s = t * (1 - (0.9 * c)^2 / c^2)^0.5 = t * (1 - 0.81) = 0.19 * t.
If moving object has velocity 0.99 * c (99% light speed), time s = t * (1 - (0.99 * c)^2 / c^2)^0.5 = t * (1 - 0.98) = 0.02 * t.
As moving object approaches light speed, stationary observer calculates that time decreases toward zero. Stationary objects have maximum space-time separation.
maximum speed
Negative time is impossible. Therefore, nothing can go faster than light speed, and nothing can go backward in time.
time measurement
To measure stationary clocks, stationary observers observe light from clock when it reaches same resonating-wave oscillation, spin, or revolution phase. Clocks have wavelengths, frequencies, and periods. See Figure 4. Signal travels from end to end and back. Time to go is same as time to return for stationary observers. Travel time is directly proportional to length.
To measure moving clocks, stationary observers can send first-clock-beat signal to reflector at other end. Clock reflector moves closer as signal travels and reflects earlier. Observers measure that resonance cavity is longer and time interval is longer.
When moving object moves past stationary observer, one part reaches observer before other parts. Other parts lag behind in time, because they are traveling through time at less than light speed. It takes time from other parts to reach observer. Moving object goes through space faster, so lengths appear shorter. Moving object goes through time slower, so time slows. See Figure 2. At less than light speed, angle is less than 45 degrees. At light speed, angle is 45 degrees.
Frequencies are clocks. Time interval unit is time between beats or ticks, such as one second. Time is number of beats or ticks, such as 60 cycles. When time slows, frequency decreases, wavelength increases, time unit increases, and cycles decrease. Time dilation makes time unit become longer, so number of ticks is fewer, so time passes more slowly.
analogies
Time dilation is like looking at a repeating process (clock) rotated away from perpendicular to sightline. For space-time, rotation is into space dimension.
Because space-time separation has a negative sign under the square root, time dilation is like using imaginary numbers. Space-time time coordinate is like imaginary axis, so space-time is like complex plane.
Observers moving uniformly in unified space-time in relation to objects calculate that object mass is greater than for relatively stationary objects {mass increase}| {apparent mass} {relativistic mass}.
relativity
Stationary observers calculate that moving objects have greater masses than stationary objects. Moving observers calculate that stationary objects are moving and have greater masses. In both cases, observer and object have relative velocity. See Figure 1.
direction
Mass increase depends on relative transverse velocity. The radial velocity component has no effect.
distance from observer
Because transverse relative velocity is perpendicular to distance direction, distance away does not affect mass-increase ratio.
measurement
Observers measure mass using standard mass {unit mass} {mass unit}, such as one kilogram. Mass measurements use forces, energies, distances, and times. To count mass, observers measure number of unit masses.
comparison to length and time
Stationary observers calculate that length contracts, time dilates, and mass increases. See Figure 2. See Figure 3.
cause
Stationary mass (rest mass) travels only through time and has no kinetic energy or potential-energy change. Moving mass travels through space (and time) and so has kinetic energy and may have potential-energy change.
Because space and time unite in space-time, momentum and energy unite. Momentum and energy both vary directly with mass. Momentum is along space coordinate, and energy is along time coordinate. As velocity increases, object moves more through space and less through time, so relative momentum increases more than velocity, so mass increases.
zero-rest-mass-particle relativistic mass and frequency
For zero-rest-mass particles, rest mass stays zero, but relativistic mass increases. Zero-rest-mass-particle energy E is directly proportional to frequency v: E = h * v, where h is Planck's constant. Zero-rest-mass-particle energy is E = m * c^2. Therefore, relativistic mass is m = h * v / c^2. Adding energy to zero-rest-mass particles increases frequency. Removing energy from zero-rest-mass particles decreases frequency.
non-zero-rest-mass particle relativistic mass
Particles with mass move through space and time, so length contracts, and time dilates. See Figure 4. Relativistic mass m is rest mass m0 plus space-dilation mass mr due to kinetic energy: mr = m0 / (1 - v^2 / c^2)^0.5. As relative velocity increases, stationary observers calculate mass increase.
Relative speed greater than 80% light speed makes object relativistic-mass kinetic energy exceed object rest-mass energy: E = m * c^2 = 0.5 * (3*m) * (0.82 * c)^2.
maximum speed
As objects approach light speed, mass increases toward infinity. As mass increases, inertia resists further acceleration, so nothing can have infinite mass or energy. No object with mass can move at light speed.
Space-time curvature describes motions of accelerating objects and objects in gravitational fields {general relativity}| (geometrodynamics).
space-time
Three spatial dimensions and one time dimension unify into space-time. Space-time has no preferred time direction, no preferred spatial direction, and no handedness.
local space-time
Physical laws are about what happens at space-time points. With small gravity and/or acceleration, space-time-point reference frames locally approximate uniform-velocity reference frames, which have linear coordinate transformations. Their space and time coordinates are straight lines.
Distant galaxies have negligible gravitational effects on local space-time, so empty space has no gravitational fields and no space-time curvature.
Observers traveling with relative uniform velocity to objects calculate that objects shorten time and contract length, whose amount corresponds to angle between time coordinate and motion-direction space coordinate. Angle varies directly with relative velocity.
space-time curvature
Objects accelerate by mechanical force or by gravitation. Observers accelerating with respect to objects increase relative velocity, so length contraction and time dilation change. When they change, reference-frame space-time coordinates change angle between time coordinate and motion-direction space coordinate. This coordinate angle change is space-time curvature. Space curvature alone and time curvature alone cannot happen, because curvature is the angle change between space and time coordinates.
Therefore, models using circle curvature (1/r), sphere curvature (1/r^2), or 4-sphere curvature (1/r^3) do not show the essence of the story. Neither do models showing a flat surface with curvature in the middle, for example, a trampoline with a weight on it.
In space-time, all objects move at light speed. Objects at rest move through time only. Objects moving at light speed move equally through time and space. (Objects cannot move only through space, because motion requires time by definition. Objects cannot move through space more than time, because experiment shows that light speed is maximum speed.)
Space-time plots for motions through flat space-time have object trajectories that are straight lines. Coordinates show equally spaced units of space and time. Coordinate positions are number of space units (meters) and number of time units (seconds or light-seconds).
If coordinates show equally spaced units of space and time, space-time plots for motions through curved space-time have object trajectories that are curved lines, because the relation between space and time is always changing. Note: Using log-log plots, with ln (y) and ln (x), makes power law functions, y = a * x^b, become straight lines. Using semi-log plots, with ln (y) and x, makes exponential functions, y = a * e^(b*x), become straight lines. However, the relation between space and time coordinates is not a power law or exponential function.
Space-time curvature is not about changes to coordinate units. Time dilation and length contraction are about simultaneity relations between objects and observers in different coordinate systems (reference frames). Space-time curvature is about intrinsic properties of space, and how motion partitions between time and space. In curved space-time, motion cannot be purely through time, because the time and space coordinates are not orthogonal, so motion must have both time and space components. Objects originally at rest in a gravitational field must move through space, since all objects move through space-time at light speed. The more space-time curves, the more the space component increases compared to the time component, so objects move faster through space the closer they get to a (larger) mass. A ball thrown upward slows down as space-time curvature decreases, until it is at rest at the top of its trajectory, where upward and downward motions are equal.
global
Non-locally, time coordinate and motion-direction space coordinate angle changes make global reference frames non-linear.
non-linearity
Objects with mass have gravitational fields and curve space-time. Because the objects pass through this curved space-time, their own gravitational field affects their motions. In general relativity, mass acts on itself through its gravitational field. In general relativity, therefore, total force is not the vector sum of forces. Non-local motions are non-linear. Non-local curved space-time is non-linear.
absolute effects
Objects start with no acceleration and in negligible gravitational fields. After objects mechanically accelerate and/or pass through gravitational fields, they return to no acceleration and negligible gravitational fields. Stationary observers calculate that objects have permanently shorter times, so passing through curved space-time has absolute physical effects for stationary observers.
energy-momentum tensor
Energy conservation is due to space-time time symmetry. Momentum conservation is due to space-time spatial symmetry. Angular-momentum conservation is due to space-time right-left symmetry. Because space-time unifies distance and time, space-time unifies energy, momentum, and angular momentum into an energy-momentum tensor.
relativity tests
Relativity tests have all proved that general relativity is correct, and other metric and non-metric theories are not correct. Measurements agree with general-relativity theory to within 10^-12 percent.
For example, the sun bends light rays that come from stars behind Sun at calculated rate.
Uniform-velocity observers calculate that accelerated and then decelerated clocks have lost time and aged less at calculated rate.
Mercury's perihelion precesses around Sun at calculated rate.
Earth and Moon change separation distance periodically at calculated rate.
Distant-star spectral lines red-shift at calculated rate.
Spectral lines red-shift as they pass through Earth gravity at calculated rate.
Accelerating masses, and objects changing mass, make gravity waves. Gravity-wave emission causes binary pulsars to have smaller orbits and shorter orbital periods at calculated rate.
other physics theories
Besides gravity and accelerations, general relativity applies to thermodynamics, hydrodynamics, electrodynamics, and geometric optics.
Because of the Big Bang, universe space is expanding uniformly and linearly {space expansion} {expansion, inflation}. For example, twice as far away, space expands twice as fast. Far enough away, space can expand faster than light.
relativity
Spatial expansion is about space itself expanding. Spatial expansion is not movement through space, so relativity does not apply.
effects on objects
Space expansion is less strong than electromagnetic and nuclear forces, so objects only stretch slightly. Space expansion is less strong than gravity, except between galaxies.
universe inflation
Universe began with low entropy. Before cosmic inflation, universe had little mass-energy, only 10 kilograms in 10^-28 meter diameter sphere, or 10^-8 kilograms in 10^-35 meter diameter sphere if minimum entropy. Only one trapped inflaton can start inflation. Cosmic inflation expanded space faster than light. During cosmic inflation, inflaton field gained potential energy, because space expansion reduces kinetic energy and increases potential energy. Energy density was constant, because energy grew equally with volume.
Perhaps, there are many inflated universes.
Stationary observers in gravitational-force fields calculate the same local object motions that stationary observers calculate for local accelerating objects. Accelerating observers feel the same effects as if they were stationary in gravitational-force field. Local accelerations and gravitational-field effects curve space the same. Uniform-velocity observers cannot distinguish whether object motions are due to gravitational force or acceleration {equivalence principle}| {principle of equivalence} {background independence, acceleration}.
To observers, accelerations caused by gravitation and accelerations caused by mechanical forces are equivalent. Observers cannot distinguish between gravity-caused accelerations and rocket, elevator, or collision accelerations. For example, people inside an elevator cannot distinguish if elevator has accelerated upward or gravitational field is greater, because locally they feel the same stress on their feet.
non-local
Except for high gravity and/or acceleration, space-time points approximate reference frames with linear coordinate transformations. Observers apply special relativity.
Over space-time regions, gravitational fields vary with distance and accelerations vary, so reference frames have non-linear coordinate transformations. Over space-time regions, because space-time curvature differs over space-time points, observers can distinguish object motions due to accelerations or to gravitational-force fields. Observers apply general relativity.
inertial and gravitational mass
Mass has two properties. Mass (gravitational mass) causes gravity. Mass {inertial mass} resists acceleration. Because space-time unifies space and time, gravitational mass is the same as inertial mass, because they both curve space-time the same amount. For example, in gravitational fields, all objects, no matter what their mass, accelerate (free fall) at same rate. Same-diameter lead balls fall at same rate as cloth balls. Object acceleration depends only on gravitational-field strength, not on object mass. This is because, gravity from object and object resistance to motion are equal. Objects in free fall feel no force. Observers in free fall observing objects in free fall see no relative motion. Space-time curvature is not an outside force but sets the field of motion.
Particles and objects have gravitational interactions with universe (fixed) distant galaxies. Particles and objects resist accelerations because of these gravitational interactions. Accelerations are absolute (not relative) with respect to the fixed distant galaxies {Mach's principle} {Mach principle}. Universe distant galaxies make an absolute reference frame, and gravitational mass and inertial mass are equivalent because of these interactions.
However, general relativity does not use Mach's principle. In general relativity, gravitational mass and inertial mass are locally equivalent to observers, because they both curve space-time the same.
By special relativity, object-observer relative motion causes observers to calculate that object has time dilation and motion-direction length contraction. For uniform-velocity observers and objects, time-dilation (and length-contraction) ratio does not change. Reference-frame time coordinate and motion-direction space coordinate maintain same angle to each other. Because coordinates maintain same relation, observed space-time does not curve.
Observers accelerating at same rate and direction as accelerating objects have no relative motion, so space-time time coordinate and motion-direction space coordinate are the same for both observer and object. Observed space-time does not curve.
acceleration
Observers accelerating in relation to objects change relative velocity. Observers calculate that time-dilation and motion-direction length-contraction ratio changes. If relative velocity increases, observers calculate that positive space-time time coordinate rotates toward positive motion-direction space coordinate, and motion-direction space coordinate rotates toward positive time coordinate. (The two space coordinates perpendicular to the motion-direction space coordinate have no changes.) Because the angle between the two coordinates changes, space-time curves {curvature, space-time}. Space-time curvature means that objects traveling along space-time events change relative travel amounts through time and space. If space-time curvature changes, outside observers see acceleration along geodesic direction.
space-time
Because space-time unifies space and time, space-time curvature is not about space curvature or time curvature separately. Coordinates do not curve. Only angle between coordinates changes.
gravity
In classical physics, masses have gravitational fields around them and attract each other by gravity. Gravity varies inversely with squared distance from mass.
Energy conservation is about time symmetry. Momentum conservation is about space symmetry. Energy and momentum vary directly with mass. In general relativity, because space and time unify into space-time, mass, energy, and momentum unify into momentum-energy. Mass-energy curves space-time over all space and time, making a field. General relativity is a field theory.
Because field varies inversely with distance squared, both a time-dilation gradient and a length-contraction gradient are at every space-time point. Space-time curvature is the unified time-dilation and length-contraction gradient. Gradients, curvatures, and accelerations are larger nearer to mass-energies.
tidal force
Objects moving in gravitational fields feel different forces at different distances from central mass. Object near side has more force than object far side (tidal force). Space-time curvature and object acceleration differ at different distances from central mass-energy.
no torsion
Space-time curvature fields have time coordinate, radial space coordinate, and two space coordinates perpendicular to radial coordinate. Because general relativity has no torsion, mass-energy does not affect the two space coordinates perpendicular to the radial space coordinate.
light
Photons and massless particles move through space at light speed. Because all observers calculate constant light speed, observers calculate no space-time curvature along light-ray direction. However, curved space-time can make light rays move transversely to light-ray direction, so photon trajectories bend toward mass-energies.
congruency
Spaces with constant curvature allow congruent figures.
universe curvature
Riemann geometry models spherical, hyperbolic, and no-curvature (flat) space-times. If universe space-time has no overall curvature, universe average mass-energy density and local space-time curvature are everywhere the same. Average mass-energy increases as distance cubed. Space and time coordinate relations do not change.
Euclid's postulates apply to flat space. 1. Only one straight line goes through any two points. Unified space has no curvature. 2. Straight lines can extend indefinitely. Space is continuous and infinite. 3. Circles can be anywhere and have any radius. Space is continuous and infinite. 4. All right angles are equal. Figures can be congruent, and space is homogeneous and isotropic. 5. Two straight lines that intersect a line, so that interior angles add to less than pi, will intersect. Space has no curvature, and parallelograms can exist. Playfair's axiom is another way of stating the fifth postulate.
If universe space-time is hyperbolic {concave space-time}, universe average mass-energy increases more than distance cubed, and average mass-energy density increases with distance. Universe has a saddle-shaped surface, with constant negative curvature, on which geodesics have infinite numbers of parallels. Initially parallel motions and so geodesics diverge.
If universe space-time is spherical {convex space-time}, universe average mass-energy increases less than distance cubed, and average mass-energy density decreases with distance. Universe has a spherical-shaped surface, with constant positive curvature, on which geodesics converge. In spherical space-time, because universe is like a lens, objects halfway around universe appear focused at normal size, and objects one-quarter around spherical universe appear minimum size.
Elliptic geometry is for ellipsoids, including spheres, which have positive curvature and on which geodesics have no parallels. Initially parallel motions and so geodesics converge.
universe shape
Because space is homogeneous, universe shape must be completely symmetric. Possible symmetric shapes are Euclidean, torus, sphere, or hyperboloids. Because universe has mass and energy, it has space-time curvature. Infinite three-dimensional space can have zero curvature, with all three spatial dimensions equivalent. Three-dimensional torus has zero curvature with no boundary. Sphere has positive curvature. Hyperboloid has negative-curvature "saddle". Hyperbolic "torus" has negative curvature "saddle" with no boundary. Universe average mass-energy density determines overall universe shape.
infinite or finite universe
If space is infinite, as it expands, it stays infinite. If space is infinite, as it contracts, it becomes finite and changes shape.
If space is finite, as it expands, it stays finite. Expanding space changes average mass-energy density and changes universe shape. If space is finite, as it contracts, it stays finite. Contracting space changes average mass-energy density and changes universe shape.
universe maximum density at origin
Perhaps, universe started with maximum mass, minimum volume, and maximum mass-energy density.
expansion or contraction with no equilibrium
Even if gravity exactly balances universe space expansion, so space neither expands nor contracts at that time, space cannot stay in that state. Because particles always travel at light speed through space-time, system always has perturbations, and perturbations decrease or increase gravity and space expansion. Because decreased gravity makes more expansion and decreases gravity more, and increased gravity makes less expansion and increases gravity more, non-equilibrium states always continue to expand or contract. Therefore, universe must always expand or contract. There is no steady state or equilibrium point.
Star masses make universe gravitational field, which is an absolute reference frame for accelerated motion, including rotational motion. Water in spinning buckets is concave because it rotates with respect to universe, not with respect to bucket {bucket argument}.
Newton imagined a water bucket {bucket experiment} [1689]. On Earth, bucket hangs on a rope and spins. At first, bucket rotates, but water does not, and water surface is flat. Then water rotates, and water surface becomes concave. If bucket slows and stops, water first rotates faster than bucket but then becomes less concave, and then becomes flat. What will happen if bucket rotates in outer space? What will happen if bucket rotates in empty space?
Universe absolute curved space-time shape can be a 4-cylinder {hypercylinder}, with time as cylinder axis and space as cylinder three-dimensional cross-section.
Space-time surfaces and hypersurfaces have a path {geodesic} between two space-time points (events) that has shortest separation {space-time separation}. For no-curvature space-times (planes and hyperplanes), geodesics are straight lines. For no-curvature space-times, separation has shortest distance and shortest time. On spheres and saddles, shortest space distance between two points is great-circle arc. See Figure 1.
spheres
Convex, positive-curvature space-times include spherical surfaces, which have two dimensions, have centers, and have same constant curvature for both coordinates. Starting from nearby points, parallel geodesics converge. Geodesics have shortest-distance and longest-time trajectory.
saddles
Concave, negative-curvature space-times include saddle surfaces, which have two dimensions and have no center or two centers. Coordinates have constant opposite curvature. Starting from nearby points, parallel geodesics diverge. Geodesics have longest-distance and shortest-time trajectory.
geodesics
Experiments show that particles and objects always travel at light speed through space-time, along shortest-separation trajectory (geodesic) between two space-time points, whether or not matter and/or energy are present. Masses free fall along space-time geodesics. Observers and objects traveling along geodesics feel no tidal forces.
object mass
All objects and particles follow the same geodesics. Because inertial mass and gravitational mass are the same, object mass does not affect trajectory. Gravity is not a force but a space-time curvature field.
In a metric field with isometry, vector fields {Killing vector field} can preserve distances. In relativity, translations, rotations, and boosts preserve space-time separation.
Convex surfaces have two points {conjugate point} through which many geodesics have same distance, so geodesics are not unique. For example, Earth North Pole and South Pole have many equivalent geodesics (longitudes).
Curved space-time can have discontinuities {singularity, relativity}|, when geodesics are not continuous and/or points do not have neighborhoods. Those space-time events have no past or no future points, and so start or stop world-lines.
gravity
If gravity is high enough to prevent light from exiting a space region, space-time curvature becomes so great, with curvature radius equal Planck distance, that space closes on itself. The space region has a surface from which nothing can escape. As orthogonal light rays converge, spatial surface {trapped surface} has decreasing area. Space-time geodesics do not continue infinitely in space-time but stop at space boundary.
causes
Stellar and galactic-center collapse can make singularities, such as black holes.
Perhaps, Big Bang, white holes, Big Crunch, and/or black hole are space-like or light-like singularities. Perhaps, universe beginning was a singularity and began time. For black holes and Big Crunch, tidal distortions can be large. For Big Bang, at low entropy, tidal distortions (described by Weyl curvature tensor) are small. Perhaps, white holes violate the second thermodynamics law.
physical law
At space-time singularities, all physical laws break down, so field equations do not hold. Because space-time has high curvature, singularities violate CPT symmetry. Space-time-curvature radius is approximately Planck length, so space-time separations are approximately zero.
physical law: quantum mechanics
Quantum-mechanical-system states develop in unitary, deterministic, local, linear, and time-symmetric evolution in Hilbert configuration space. By Liouville's theorem, phase-space volumes are constant. However, "reduction of state vector" is asymmetric in time, and "collapse of wave function" adds phases and information, so phase-space volumes are not constant, and past and future have different boundary conditions, just as singularities have discontinuities between space-time pasts and futures. Quantum-mechanics measurements cause wave-function collapse.
Perhaps, quantum-mechanics measurements and wave-function collapse relate to general-relativity singularity space-time points and their formation. Perhaps, general relativity disrupts, or makes unstable superpositions of, quantum states and breaks equilibrium at measured states (objective reduction). General relativity has non-local negative-gravity potential energy and has positive-energy gravity waves, while state-vector-reduction time depends on inverse diameter and energy.
Singularities {naked singularity} can have high density but not enough gravity to form event horizons. Space-time paths that go through time can enter and leave naked singularities (but cannot leave other singularities). For example, spindle-shaped singularities have spindle ends that are naked singularities. Objects with spin faster than mass-determined rate are naked singularities. Objects with electric charge higher than mass-determined rate are naked singularities.
Perhaps, some or all singularities {thunderbolt} go to infinity and have no confinement, thus removing their space-time points from space-time.
General relativity is about gravity {gravitation, relativity}| and accelerations. For small gravity, observers calculate that gravitation and acceleration have the same local effects on space-time curvature. Because gravitational field strength varies inversely with distance, observers calculate that gravitation and acceleration have different global effects on space-time curvature.
time
Because stationary observers calculate that gravity rotates space-time time and radial-space coordinates toward each other, clocks in gravitational fields, or undergoing accelerations, run slower. People age slightly more quickly on Moon than on Earth, because Moon has smaller gravitational field. People age more slowly on accelerating rockets than on Earth.
object length
Observers calculate that accelerating massive objects decrease length. After accelerating finishes, observers calculate that length returns to previous amount.
object mass
Observers calculate that accelerating massive objects increase mass. However, mass increase increases inertia and resists further acceleration. After accelerating finishes, observers calculate that mass returns to previous amount.
energy
Gravity depends on mass, directed potential energy, directed kinetic energy, and random-energy temperature. Mass and random energy are always positive. Gravity fields cannot cancel, because they are only positive. Because gravity is infinite and is only positive, gravity can have unlimited energy amounts.
sources
General-relativity stress-energy tensor has ten independent gravitational-field sources and ten independent internal-stress sources. Sources all conserve energy and momenta. Field equations are d'Alembert potential equations.
physical-law invariance
Gravitation and acceleration curve space-time, so non-locally physical laws vary under coordinate transformations.
uncertainty principle
Gravitational-field values correspond to position. Field-value-change rates correspond to momenta. Therefore, uncertainty principle applies to gravitational-field values and value-change rates.
black holes
Because gravity is unlimited, gravity can become strong enough to overcome all object accelerations, so even light cannot escape the space region. Outgoing geodesics converge. Space curves so much that it closes on itself, forming a region separate from space-time, not observable from outside. Only gravity can cause space-time singularities, because it is never negative.
gravitational entropy
Spaces have entropy that depends on topology (Euler number). Gravity curves space-time and creates different topologies, so gravity has entropy. Because only gravity is always positive, only gravity has entropy. Other forces cannot curve space-time, because they are not infinite and/or are both positive and negative.
gravitational entropy: black hole
Because gravity has entropy and forms black holes, black holes trap entropy. Black-hole trapping amount depends on event-horizon radius, so black-hole entropy depends on event-horizon spatial area.
Because black holes have entropy, they have surface temperature at event horizon. At event horizon, virtual-particle creation can allow one virtual-pair member to tunnel through event horizon to space, causing black hole to lose matter and eventually dissipate. Entropy decreases, rather than always increasing. Black holes disrupt quantum-state deterministic development and mix states {mixed quantum state}.
repulsion
Perhaps, gravity can temporarily repulse, and cause universe origin. Exotic particles can have negative pressure, causing repulsion. Larger spaces have more repulsion because pressure is in space, not in ordinary particles.
Objects with mass have gravitational forces {gravitational pressure} on top, bottom, middle, and sides, all pointing toward mass center. See Figure 1.
Imagine that object is fluid. Gravitation pulls all points straight down toward mass center. Pull is directly proportional to mass m and inversely proportional to distance r squared: m / r^2. Pull is least at farthest points least and most at nearest points.
Volume reduction changes mass density and energy density, and so changes pressure. Gravity tends to reduce volume, and increase pressure, until outward pressure force per area balances inward gravitation force per area.
Gravitational-field accelerations make waves {gravity wave}| {gravitational wave}. Gravitational waves make space-time curvature oscillate in two dimensions.
speed
Gravitational waves travel at light speed.
frequency
Gravitational-wave frequencies are about 1000 Hz.
medium
Gravitational waves oscillate gravitational-field surfaces. Gravity waves need no other medium.
quadrupoles
Gravity waves have two orthogonal linear-polarization states, at 45-degree angle, making field surfaces (not just lines). Gravity waves are quadrupole radiation. Because mass can only be positive (unlike electromagnetic positive and negative charges), no mass-dipole or gravitational-dipole radiation can exist.
At peaks, potential energy is maximum, and kinetic energy is minimum. As they pass, gravitational waves stretch and compress (vibrate) objects with mass.
spin
Gravitational waves can rotate. Primordial gravitational waves have different spin {polarization, gravity} than current ones.
graviton
Gravitational-force exchange particles are gravitons and have no mass. Gravitons have spin 2, which is invariant under 180-degree rotation around motion direction.
sources
Gravity waves come from oscillating and/or accelerating masses, such as pulsating stars, irregularly rotating stars, collapsing stars, exploding stars, or interacting star clusters.
superposition
Because masses are always positive, gravitational fields cannot cancel each other. However, locally, accelerations and/or decelerations can cancel gravitational fields. Because gravitational waves are non-local and have components in more than one direction, and accelerations are in only one direction, accelerations and/or decelerations cannot cancel gravitational waves.
comparison with electromagnetic waves
Gravitational fields have advanced and retarded solutions and their equations are similar to those for electromagnetic waves.
renormalization
Gravitational waves are infinite and require renormalization for gravitational-wave calculations.
Pressure measures momentum exchange. System external pressure puts force per area on system-boundary surfaces. It is due to kinetic energy, which increases with temperature.
internal pressure
System internal pressure {internal pressure}| puts force per area on system particles. It measures system potential energy changes as system expands or contracts while keeping temperature constant. Internal pressure is positive for attractive forces and negative for repulsive forces.
Vacuum has no forces, so its internal pressure is zero. Particles have no internal forces, so their internal pressure is zero. Solids have attractive forces, but particle distances do not change at constant temperature, so internal pressure is zero.
positive internal pressure
Gas particles slightly attract, and system volume can change at constant temperature, so particle distances can change at constant temperature, and gases can have positive internal pressure. Hotter gases push particles farther apart against attractive forces, increasing positive potential energy, so hotter gases have more internal pressure than cooler gases. Photons have radiation pressure that pushes against electromagnetic forces, increasing positive potential energy, so photon "gases" have positive internal pressure.
negative internal pressure
Systems that have internal repulsive (negative) forces have negative potential energy and negative internal pressure. For example, if external force compresses rubber membranes, rubber has repulsive forces that tend to push particles apart. The internal restoring force is negative, so internal potential energy is negative, with negative internal pressure.
gravity
At space-time points, gravity G depends on mass-energy density M and on internal pressure P: G ~ M + 3 * P. Hotter gas has more positive internal pressure than cooler gas and so more positive gravity. Photon "gas" has positive internal pressure that is one-third of energy density, so gravity doubles: M + 3 * (M/3) = 2 * M.
Quantum vacuum has negative (repulsive) force that expands space, increasing negative potential energy (dark energy) by subtracting universe positive kinetic energy, and so cooling the universe. Quantum vacuum has negative internal pressure between one-third and one of mass-energy density, so repulsive antigravity is between zero and negative two times mass-energy density: M + 3 * -(M/3) = 0 and M + 3 * (-M) = -2*M.
Gravitational fields have different strengths at different distances from mass-energy. In gravitational fields, objects have different forces {tidal force} on side nearest to mass-energy, side farthest from mass-energy, and middle. Tidal distortions depend on gravitational-field strengths at different space points.
Gravity varies inversely with distance squared {inverse square law}, so tidal effects vary inversely with distance cubed (by integration). Therefore, tidal effects can measure gravitational-field strength.
See Figure 1. The larger object is denser and has much more mass than smaller object. The smaller object is fluid. The objects are not far apart.
near and far
Gravitation pulls smaller-object nearer side, farther side, and middle straight toward larger-mass center. Nearer side feels strongest gravity, and its particles accelerate most. Middle feels intermediate gravity, and its particles accelerate intermediate amount. Farther side feels weakest gravity, and its particles accelerate least. Along vertical, small object tends to stretch out from middle, keeping same volume.
left and right
Gravitation pulls left and right sides toward larger-mass center diagonally, straight down along vertical component and across inward along horizontal component. Left and right sides feel slightly less gravity than middle, because they are slightly farther away from larger-mass center. Those particles accelerate downward slightly less than middle does. Left and right sides also accelerate small amount horizontally toward smaller-mass center. This pushes other molecules equally up and down and contributes to vertical stretching out.
waves
Changing gravity changes tidal forces and can cause mass oscillations. Mass accelerations make gravitational waves.
Rotating objects with mass pull space-time around {frame dragging}| {Lense-Thirring effect} {gravitomagnetism}. An analogy is rotating masses drag viscous fluid around them. For particles orbiting around rotating masses, relativity causes orbit-plane precession, because rotation and angular momentum couple.
Objects can move forward and backward in space, and physical laws have no preferred space direction. Objects cannot move forward and backward in time, though physical laws have no preferred time direction. In space-time, can objects move forward and backward in time {time travel}|?
If time travel is possible, people can deduce what has happened from knowledge of the future. Past-time observation affects past. Because contradictions violate causation, nothing can communicate or transport backward through time. The meaning of space and movement prevents moving forward and backward in time. Relative velocity is moving through space over time. Movement is always in space-time spatial dimension. Moving forward and backward in time cannot separate from moving forward and backward in space.
Space paths must not reverse time, so nothing can happen at two times. Between a past point and future points reachable from the past point, along geodesic, all space-time points must be reachable {hyperbolicity} {hyperbolic space-time, relativity}. If geodesics exist, space-time has no singularity.
One twin stays on Earth. The other twin takes a high-speed trip, traveling to a space point and then back to Earth. Second twin must accelerate to leave Earth and travel in space, must accelerate to round point in space, and must decelerate to land on Earth. Traveling twin's clocks appear to run slower to Earth observer. Second twin is younger than first twin on return to Earth. Traveling twin ages more slowly than Earth twin {twin paradox}|.
length contraction
Traveling at almost light speed, people can cross universe in 86 years of their time, because universe lengths contract greatly. People on Earth age 13 billion years during that time.
space-time graph
Space-time graphs {Minkowski diagram} can show travel effects. The diagram assumes first twin is stationary and is observer. First twin has vertical world-line on space-time graph. Second twin has angle to right, away from Earth as twin leaves Earth, and angle to left, toward Earth as twin returns to Earth.
At beginning, twin accelerates to leave Earth and has curved world-line, with greater angles to time axis. At turning point in space, twin changes direction and has curved world-line, with lesser angles to time axis, reaches vertical, then has curved world-line, with greater angles to time axis. At landing, twin decelerates to stop on Earth and has curved world-line, with lesser angles to time axis. See Figure 1.
space-time trajectory
The shortest path is the longest time. Traveling twin has longer path and shorter time.
universe
If second twin is observer, twin on Earth travels, relative to second twin, with same motions and accelerations as described above. However, first twin does not undergo acceleration relative to universe masses, as second twin does. To second twin, universe masses have same speeds and accelerations as first twin. During acceleration relative to universe masses, time slows, because mass curves space-time. Curved space-time makes longer path and shorter time.
Permanent aging happens only during accelerations and decelerations. Uniform-velocity time dilations are symmetric between observers, are momentary, and are reversible.
Masses have gravitational fields, which have energy, and energy has mass and makes gravitational fields. Masses interact with their gravitational fields to make energy {self-energy}| {matter field}. Because mass is only positive, gravity has interaction energy greater than zero.
renormalization
Mathematical renormalization adjusts values to prevent infinities.
electromagnetism
Perhaps, if charge moves (and external electric field is zero), charge gains velocity by interacting with its electric field, because energy in point-charge field is infinite (by Maxwell's equations or quantum electrodynamics). However, because positive and negative charges can induce each other, electromagnetism has no interaction energy.
Wheeler-Feynman theory
In Wheeler-Feynman theory, universe particles absorb moving-charge electric field, so field at large distances is zero, and system has no advanced solutions and no infinities. However, this theory is not correct.
perfect absorption
In perfect absorption, electric field is relativistically invariant, so all force-induced fields, including reaction forces, form other particles using photon exchanges and go to zero. Perfect absorption has only retarded solutions, because advanced solutions are improbable by thermodynamic laws. In expanding universes, absorption happens at low frequency for retarded solutions and at high frequency for advanced solutions. However, this theory is not correct.
Time-like space-time paths have points where space-time curvature and path curvature are not the same, and net gravity is zero {general energy condition}.
For classical matter, energy density is greater than or equal to zero in all reference frames {weak energy condition}. However, weak energy condition is false for quantum-mechanical scales.
For classical matter for long enough distances, energy density is greater than or equal to zero for all time-like paths {strong energy condition}. However, strong energy condition is false for quantum-mechanical scales.
Geodesics have space-time separation {geodesic deviation} along (straight) line perpendicular to geodesics. An equation {equation of geodesic deviation} calculates separation: (D^2)r / Ds^2 + G * r, where D^2 is second partial derivative, r is curvature radius, D is first partial derivative, s is space coordinate, and G is Gaussian curvature. In empty space-time, geodesics are parallel straight lines. Empty space-time has no curvature, so r is zero, and geodesic deviation is zero.
Geodesics converge along tangent vector to hypersurface path. Geodesic-convergence rate relates to shear and gravitation {Newman-Penrose equation} {Raychaudhuri equation}.
In special and general relativity, field equations {d'Alembert equation, relativity} describe how masses, their gravitational fields, and space-time gravitation potentials determine object motions.
Similar field equations describe how charges, their electrostatic fields, and space-time electrostatic potentials determine object motions. Such equations {electrodynamics} are similar to curved-space-time special-relativity equations.
Equations {Einstein field equation} describe how mass-energy affects space-time geometry, and how space-time geometry affects mass-energy motions. Local-space-time average curvature tensor G {Einstein tensor} is proportional to mass-energy tensor T {stress-energy tensor}: G = 8 * pi * T.
Einstein tensor has six components for tide-producing acceleration: particle position, particle velocity, field amplitude, field-change rate, geometry, and geometry-change rate. Einstein tensor has four components for space-time coordinates.
Stress-energy tensor has components for stresses, momentum densities, and mass-energy density.
Einstein tensor G relates to local-space-time curvature tensor R (Riemann curvature tensor): G = R - gamma * R/2. Stress-energy tensor T relates to Riemann curvature tensor R: R - gamma * R/2 = 8 * pi * T. Riemann-curvature tensor has 20 components. In empty space-time, stress-energy-tensor gradient is zero, so Einstein-tensor gradient equals zero, and Riemann curvature tensor is zero.
Surfaces have Gaussian curvature. Tensors {Riemann curvature tensor} represent space-time curvature using geodesic separation. Riemann curvature tensor represents total curvature. It adds tidal distortions (Weyl curvature tensor) and volume changes (Ricci curvature tensor).
Two-dimensional space requires one curvature component, curvature radius. Three-dimensional space requires six curvature components, three for each dimension's curvature and three for how dimensions curve in relation to each other. Four-dimensional space requires 20 curvature components, four for each dimension's curvature, twelve for how pairs of dimensions curve in relation to each other, and four for how triples of dimensions curve in relation to each other.
invariance
Curvature is invariant over linear space-time-coordinate transformations.
electromagnetism
Like gravity, electromagnetism exerts force that decreases with distance squared {Lorentz force equation}. Lorentz force equation and Riemann curvature tensor are equivalent. At low velocity, because relativistic effects are negligible, only the nine Lorentz-equation electric-field components, and the corresponding Riemann-curvature-tensor mass components, are significant.
Curvature tensors {Ricci curvature tensor} can describe space volume changes, which is local curvature caused by local matter.
Perhaps, at one second after universe origin, thermal variations in Ricci curvature tensor formed particles and black holes.
Curvature tensors {Weyl curvature tensor} can describe tidal distortions, which is non-local curvature caused by non-local matter.
At Big Bang, quantum fluctuations and damping cause small variations. At Big Crunch, variations have no damping and can be large. Perhaps, this asymmetry causes time to have direction. Alternatively, past and future singularities can be different.
Space-time theories {Kaluza-Klein theory} can use four space dimensions and one time dimension. Fourth space dimension is only several Planck lengths long, has curvature so high that it makes a circle, and is unobservable. A small fourth space dimension allows the vacuum to have higher energy density than three space dimensions have.
Time relativistic theories {kinematic relativity theory} describe finite expanding universes.
Gravitation theories can use metrics {metric theory} or be non-metric. A ten-parameter general metric theory {parametrized post-Newtonian formalism} can model all metric gravitation theories, which then differ only in parameter values.
parameters
The ten parameters model: How mass causes space curvature. How gravity-field superposition is non-linear. If space has preferred reference frame, or all spatial directions are equivalent. If all four space-time components have momentum conservation. If distant galaxies affect local interactions. If general metric theory does or does not include gravitational-radiation effects or other gravity-strength changes.
types
Metric theories include general theory of relativity, scalar-tensor theories, vector-tensor theories, tensor-tensor theories, conformally flat theories, stratified theories, and quasi-linear theories.
non-metric
Non-metric gravitation theories violate completeness, consistency, relativity, and/or Newtonian limit.
Abstract spaces {superspace} can have approximate three-dimensional space by tetrahedron skeletons and have tetrahedral edge lengths. They can have space dynamics, change over time, and represent different geometries.
Riemann surfaces are Riemann sphere, torus, and pretzel-shaped surface. Their angles are the same as in Euclidean space. Riemann surfaces can define field theories {conformal field theory} that pair with string theory.
General-relativity dynamics {geometrodynamics}| is three-dimensional Riemann-space dynamics, using a method {ADM formalism} {canonical quantization} developed by Paul Dirac and later Richard Arnowitt, Stanley Deser, and Charles Misner.
Geometric optics {geometric optics}| models plane waves in flat space-time. Geometric optics applies if wave-packet wavelengths are much less than wave-front space-time curvature radius. Wave photons have same momentum and polarization. Photon number determines ray amplitude. Like adiabatic flow, photon number conserves. Light rays are null geodesics. Polarization vector is perpendicular to rays and propagates along rays.
Quantum general-relativity gravitation theories {relational quantum theory} have different observers whose calculations are the same at corresponding space-time points.
In one renormalization, electric field is relativistically invariant, so all force-induced fields, including reaction forces, form other particles using photon exchanges and go to zero {perfect absorption}. Perfect absorption has only retarded solutions, because advanced solutions are improbable by thermodynamic laws. In expanding universes, absorption happens at low frequency for retarded solutions and at high frequency for advanced solutions. However, this theory is not correct.
In a renormalization theory {Wheeler-Feynman theory}, universe particles absorb moving-charge electric field, so field at large distances is zero, and system has no advanced solutions and no infinities. However, this theory is not correct.
If general relativity has canonical quantization, Wheeler-DeWitt equation has no time coordinate {frozen time problem} {problem of frozen time} {problem of time} {time problem}.
In empty space, space-time can have many equivalent reference frames {covariance, relationalism} {covariance, relativity} {general covariance, relationalism}.
In relativity theories {relationalism}, mass-energy determines space-time curvature and shape, and space and time are not absolute or real but differ for different observers.
Perhaps, space and time are real and absolute {substantivalism, relativity}, and mass-energy alone does not determine space-time curvature and shape.
Vectors can equal another vector plus a scalar term {gauge, relativity}|. Scalar gauges can change with position. For example, space-time curvature can change with position, and gauges can represent linear curvature changes with position.
Using linear transformations {gauge transformation}, gauges can relate vectors expressed in different coordinate systems. Gravitation, electromagnetism, and chromodynamics use gauge transformations to model infinitesimal, finite, scalar-coordinate transformations. For local space-time regions, general relativity is invariant under finite coordinate transformations, and a generalized gauge transformation represents general relativity. Using gauge scalars can simplify differential equations.
Because derivatives of scalars equal zero, gauge changes do not affect physical measurements, motion differential equations do not change, and gauge transformations preserve invariants.
In gravitational fields so weak that space-time has negligible curvature, gravity does not move gravitational-field-source masses and does no work on them, so masses have no self-energy. For this case, theories {linearized theory of gravity} represent space-time-coordinate changes as infinitesimal gauge changes, which change space-time-metric coefficients.
Space has properties {space, physics}.
measuring
How can instruments measure or perceive space or time, or changes to space or time?
origin of space
How did space arise? What causes number of spatial dimensions? Perhaps, space and time result from object interactions. Perhaps, motion necessity and nature create and require three long-range spatial dimensions. More than three spatial dimensions provide too many possibilities to be stable. Fewer than three spatial dimensions cause immobility. Perhaps, space and time result from induction.
universe
How is matter and energy universe related to space or time or space or time changes?
lattice
Perhaps, space is like lattice, with particles at nodes. Lattice has different spatial frequencies and wavelengths. Lattice diffracts light and matter, making quantum waves, with no wave interference.
Perhaps, double-slit or beam-splitter interference experiments are not about wave interference from two sources but are only about diffraction. Perhaps, entangled particles are actually always beside each other and so affect each other immediately. Perhaps, if entangled and they move apart, system wavelength increases and energy goes down, so as they move apart self-disturbance is low. Perhaps, they have continuous interaction. Perhaps, all waves are the same or share constant. Perhaps, wave amplitude depends on diffraction type.
dimensions
Space-time has three infinite space dimensions and one infinite time dimension. Spaces can have any number of dimensions. Dimensions can be not only straight and infinite, but also curled-up into circles and finite (compactified). Dimensions can be orthogonal and independent or can have relations. Dimensions can be continuous or discrete. Perhaps, space has imaginary number dimensions, and particles are local in that dimension while apart in real dimensions.
space geometries
Perhaps, space geometry started with zero dimensions and evolved to be stable space-time, with unified space and time dimensions. After zero dimensions came non-metric geometries and ordered geometries. After that came projective and/or affine geometries. (Projection uses ideal points that curl dimensions, leading later to compact dimensions with greatest curvature.) After that came metric geometries. (Metric dimensions have number, magnitude, orientation, and direction sense. Only metric dimensions can be infinite. Space-time time metric dimension is inverse space metric dimension.) After that came parabolic metric geometry (and Euclidean geometry), non-Riemannian geometry, single elliptic geometry, double elliptic geometry, and hyperbolic geometry. After quantum foam, space metric geometries evolved to be differential pseudo-Riemannian geometry.
Perhaps, only space-time can have high energy, and other metric geometries can have only empty space.
Perhaps, space-time results from hybridization of three space dimensions and one time dimension.
Perhaps, space and time dimensions cannot exist separately.
Space {Newtonian space} and time can be separate, unchanging, passive, and absolute backgrounds for matter and motions. Leibnizian space and time are object and event relations. Machian space and time are relative to matter and energy amounts and relative positions. Einsteinian space and time interrelate, change, are active, and have relative backgrounds for matter and motions. Quantum-mechanical space and time are discrete, and entangled objects stay in direct contact over any space and time amount.
If Lagrangian has continuous symmetry, it makes conservation law {Noether's theorem} {Noether theorem}.
Perhaps, space has no vacuum but only continuous objects {plenum, space}. However, space really has empty spaces.
Physical non-quantum systems with finite energy in finite volume always return to almost-similar state from any state {Poincaré recurrence theorem}.
Physical processes can turn things inside out {version, physics}|. Fluid and flexible things can turn inside out. Rigid things can turn inside out in the imagination. When right-handed glove turns inside out, it becomes left-handed glove. When bowl turns inside out, inside becomes outside and curvature reverses. Can fundamental particles turn inside out? Can universe turn inside out?
Physical-system evolution depends on point relative-distance changes, kinetic energy, and angular momenta {relative configuration}.
Given four relative configurations, find kinetic energy and angular momenta {Tait's problem} {Tait problem}.
Change shows time {time, physics}. Time orders changes.
direction
Time flows forward, not backward. Time changes are never symmetric. Locally, time changes can be almost symmetric if no change happens.
isosynchrony
Because universe is homogeneous, time flows almost the same everywhere.
physical laws
Physical laws are time-symmetric now, except for neutral kaon decay. Physical laws are always parity-charge-time symmetric. Reasoning works the same in both time directions.
imaginary time
Space-time has real time, which has direction. Space-time can have imaginary time {imaginary time}, which has no direction. If time has imaginary-number component, time is complex number and can have more than one dimension. Real-number time always increases, but imaginary-number time can be decreasing or increasing, just like spatial dimension. Real-number time is always positive, but time measured by imaginary numbers can be negative or positive. Having imaginary-number time dimension does not change physical laws. Having imaginary-number time dimension allows time to stand still. It also allows space not to have singularities.
origin
How did time arise? What causes number of time dimensions? Perhaps, time and space result from object interactions. Perhaps, motion necessity and nature create and require one long-range time dimension. More than one time dimension provides too many possibilities and cannot be stable. Less than one time dimension causes immobility. Perhaps, space and time result from induction.
origin: symmetry
Perhaps, the only allowable or most probable universes are asymmetric in time, though physical laws are symmetric in time. Perhaps, only asymmetric universes support life or intelligent life.
Spatial dimension can contain time {intrinsic time}.
Objects with no forces move at uniform speed, and observers can compare motions to other-object motions using object clocks {inertial clock}.
Events happen during times {duration}. Displacement can be over different paths, so distance between two points can differ. Similarly, time interval can be over different paths, so interval between two time points can differ. Perhaps, different paths solve paradoxes of time in quantum mechanics and relativity.
Moments {moment, time} cannot move in time and so cannot change, just like position cannot change. The present moment does not fade into the past or become the future. Just like all space exists everywhere, all time exists always, and all space exists always and all time exists everywhere. Time does not flow.
Time always flows forward, not backward {asymmetry, time} {time, asymmetry}.
agency
People can deliberately perform previous action to alter later event {agency, time}. By definition, agency is asymmetric in time. Agents cannot know everything about the past but only about accessible past. In particular, limited access to past can allow backward causation and backward dependence.
antiparticle
Antiparticles travel backward in time.
collisions
Particles collide and spread out. Entropy increases. Perhaps, time relates to spreading caused by collisions. One particle has no time, because time is relative.
behavior correlation
After two objects interact, their activities correlate at all future times. Before two objects interact, do object actions correlate or not? In quantum systems, correlations are not observable, but non-correlations are also not observable. In classical systems, correlations are not observable. Classical case does not necessarily derive from quantum case.
dependence
Later events depend on previous events. Perhaps, physical-law asymmetry can mediate dependent temporal asymmetry {third arrow strategy, time}. However, no physical-law asymmetry mediates temporal asymmetry.
dissipative structures
Larger-system subsystems {dissipative structure} can reduce entropy, if energy is available and systems use only their own processes. Perhaps, time is only about whole systems.
dark energy
In the past, all matter and energy distributed evenly, though gravity makes masses group together. Perhaps, space has repulsive property. How can universe contract to similar state, or how can time-reversed processes happen?
flow
Time is not flow but dimension, though dimension does have direction. Time is relative with space. Perhaps, time has more than one dimension, and time can take different paths.
phase transition
Universe phase transitions are about symmetry changes. All physical laws reflect symmetries. Time symmetry makes energy conservation.
rotating universe
Rotation drags light and space-time and so allows travel into the past. However, it is unlikely that universe rotates.
Time has direction, preferred series, orientation, and order along dimension, as does order in space. Events move forward from past to present to future {direction, time} {arrow of time}| {time's arrow}. The present contains records of the past. Time arrows are thermodynamic, electromagnetic, cosmological, or psychological.
Advanced causation cannot happen, because a later cause can happen when an earlier effect does not happen, and an earlier effect can happen when a later cause does not happen {bilking argument}|.
Radiation {coherent radiation}| travels outward from source, but coherent radiation does not travel inward to source, macroscopically. At classical levels, emitters add but absorbers cancel. Why is there radiation gradient, radiation flow, with respect to time? At quantum level, this condition does not necessarily hold. Perhaps, quantum-level emitters and absorbers can emit and receive coherent radiation, or coherence concept does not apply. Why are there coherent-radiation sources?
When isolated systems change state, they are more likely to change to state with higher-probability energy distribution. Statistically, motions become more random, and objects become more evenly dispersed {time, entropy} {entropy, time}. In isolated systems, entropy can decrease only temporarily and locally.
questions
Why was entropy low in the past? If time reversed, how can entropy decrease toward the past? Why is there entropy gradient, entropy flow, with time?
symmetry and heat
Higher heat means more symmetry. Heat release makes less symmetry.
curvature
Smooth curvature has less entropy than jagged curvature.
gravity
If no gravity, entropy is proportional to volume. If gravity, entropy is proportional to surface area. Perhaps, surface has no gravity, and gravity adds extra dimension.
addition
Entropy is an extensive quantity, because it is arrangement-number logarithm. If two systems merge, arrangement number is arrangement-number product, and total entropy is entropy sum.
low entropy in past
System is more likely to have higher entropy in the future. System is more likely to have had lower entropy in the past. If system had higher entropy in the past and low entropy now, intermediate steps were jumps down to lower entropy, which are unlikely. Universe entropy was lowest at first.
entropy change with time
Perhaps, at universe origin, everything was evenly distributed, with only one particle type, with lowest entropy. During inflation, there was negative gravity, making more even dispersal, with higher entropy, and less clumpiness, with low entropy. When inflation ended, many particles appeared, making more entropy. Particles spread through space, making lower entropy. Primordial gas had low entropy, and then gravity decreased entropy as it clumped matter, but overall entropy increased because potential energy changed into heat kinetic energy. Sun original gas cloud had medium-low entropy and temperature, and then Sun had higher entropy and temperature. Sunlight has medium entropy, and then heat has higher entropy. Food has medium-high entropy, and then waste has higher entropy.
black hole
Black holes have maximum entropy density: Boltzmann constant times surface area in Planck units divided by 4. Surface unit {Planck square} has one unit of entropy flux.
Two spatially separated correlated events can both be correlated with third event {fork asymmetry}.
Dynamical mechanics methods {quantum mechanics}| determine particle momentum through space {particle trajectory} or particle energy through time. In quantum mechanics, physical systems have states of objects, their properties, and events. Quantum-mechanics states are in phase space (coordinate space), rather than physical space. Phase space includes all particle positions and momenta and so includes physical space. Relativistic phase space includes physical space-time. Phase-space states represent discrete energy, momentum, angular momentum, length, time, and mass quanta. Adding energy, momentum, angular momentum, length, time, or mass increases energy, momentum, angular momentum, length, time, or mass by quantum leaps, not continuously.
particles and fields
Without quantum mechanics, in continuous space and time, particles have properties that cause forces, which make continuous force fields. Particles have continuous energies, times, positions, and momenta. Fields have potentials at all space locations and at all times. Particles have zero energy. By statistical mechanics, with or without quantum mechanics, energy tends to spread to all positions equally (energy equipartition). Because continuous fields have many more locations than particles, energy tends to go from particles into fields over time.
Without quantum mechanics, in continuous space and time, electrons orbiting atomic nuclei interact with other nuclei and electrons and their orbits decay, As they spiral into nucleus, they emit electromagnetic waves to maintain conservation of energy. As electrons become closer to nucleus, they emit higher-frequency electromagnetic waves because force is stronger. By energy equipartition, electrons eventually fall into nucleus, ending all atoms. However, the universe has stable atoms.
As they absorb electromagnetic waves from space and emit them to space, hot objects in cooler space tend to transfer heat to space, because absorption concentrates on one location but emission spreads to all other locations and so never returns to the location. By energy equipartition, all frequencies have equal probability. Without quantum mechanics, higher frequencies carry away most energy, and hot objects cool quickly. However, experiments show a Boltzmann distribution of frequencies, with higher frequencies having lower probabilities. Hot objects cool slower.
In quantum mechanics, particles have discrete states and discontinuous fields. In quantum mechanics, length, time, and mass have quanta, with non-zero minimum (Planck) length, time, and mass. Minimum length makes maximum momentum. Minimum time makes maximum energy. Minimum mass makes maximum frequency. For each property, particles have one quantum plus a number of quanta, up to a maximum number. Atom electrons must have minimum energy and so do decay into atom nuclei. Higher states have lower probabilities, because higher states are harder to reach. High-frequency electromagnetic waves are fewer. Quantum mechanics results in Boltzmann frequency distribution and observed slower cooling. Quantum mechanics matches atomic- and subatomic-particle behavior.
energy quanta
Particle-collision, light-absorption, and light-emission experiments show that particles absorb or emit energy in non-zero minimum amounts (quantum). Charge accelerations differ by a number of quanta. Minimum electromagnetic-wave energy varies directly with frequency. Higher energy means more electromagnetic waves. Particle and particle-system energy is one quantum plus a number of energy quanta. Energy levels differ by quanta and are discrete, not continuous.
particle wave
Simultaneously sending particles through two slits makes almost the same target patterns as sending electromagnetic waves, suggesting that particles behave like waves. Large particles at ordinary energies have very high-frequency waves with imperceptible wavelengths. Atomic- and subatomic-particles have low-frequency waves with observable wavelengths. Particles have waves in phase space, not in space-time. Particle waves have fundamental frequency, at lowest energy level, and higher harmonic frequencies at higher energy levels. Particle waves have discrete, not continuous, frequencies at overtones of the fundamental frequency. Wave energy varies directly with frequency, so particle energy levels are discrete.
Quantum-mechanics wave equations describe potential and kinetic energies in particle and force-field systems. Equation solutions are periodic wavefunctions that describe particle and field positions and momenta.
ground state
Particles cannot have zero energy, because they cannot have zero motion, because they have phase-space waves and waves propagate. Particles have a lowest energy state (ground state), which corresponds to the lowest-frequency (fundamental) particle wave. All other energy states are quantum amounts higher than ground state.
determinism
Previous particle positions and momenta determine future phase-space states. Quantum wave and particle mechanics is deterministic and follows normal causality.
observation
Observations interact with particle to put particle in one observed state. Observation immediately and discontinuously selects observed state from among possible states. From that moment, phase-space again follows determinism and causality. Old system wavefunction "collapses" to nothing, and new system wavefunction, about both system and measuring apparatus, begins.
Scientific experiments try to isolate observer from experimental system to prevent interactions. However, in experiments involving small things, observation has to cause disturbance and perturb observed system. Observer and observed become one new system. Observation causes measurement uncertainty about observed system.
Perception theory is about physical events that cause state perception. Quantum mechanics has no perception theory. Quantum theory does not describe observer, only observed system. Quantum mechanics leaves open the possibility of perceiving state superposition. However, people and instruments detect only states and never observe state superpositions. Quantum-mechanical waves are in phase space, not physical space.
particle systems
Without quantum mechanics, particle systems follow classical statistical Markov processes, such as diffusion and Brownian motion, in space-time. Diffusion and Brownian motion apply energy and momentum conservation to many particles. Many particles follow many paths (path distribution), completely determined by previous positions and momenta. Particles tend to have the highest-probability energy distribution.
Quantum-mechanics equations (Schrödinger equation) also derive from energy and momentum conservation and are similar to diffusion and Brownian-motion equations. Quantum-mechanics equations are about coordinate space or phase space, rather than physical space. Quantum-mechanics functions (wavefunction) are complex-number functions, which relate trigonometric and exponential functions. Because particles can be anywhere along infinite dimensions, wavefunctions are over infinite space and number of possible system phase-space states is infinite. In phase space, particles have possible trajectories (path distribution), each with different probability. Previous positions and momenta determine only probabilities of later paths. The sum of all probabilities equals 1 = 100% that the particle is somewhere. All possible states exist simultaneously and evolve independently. In quantum mechanics, particle measurement causes only one phase-space state/path to be observed. That state tends to have the highest probability.
particle systems: superposition
In quantum mechanics, particles have wavefunctions. Particle systems have wavefunctions that are particle-wavefunction linear combinations (superposition), just as electromagnetic waves superpose. Particle wavefunctions evolve independently, just as electromagnetic waves are independent. Waves do not have multiplicative or dependent effects on each other.
Because wavefunctions are complex-number functions, wavefunctions can add in two ways, constructive interference, A + B, and destructive interference, A - B = B - A. It is like positive and negative momentum, as in a reverberating system. Both superpositions are possible phase-space states, with probabilities.
particle systems: normalization
For linear equations, dividing all terms by any number results in equivalent equations, with same equation solutions. Dividing by any number only changes term coefficients/weights. Therefore, only coefficient/weight ratios determine equation meaning. Making sum of squared coefficients/weights equal one makes total probability 1 = 100% (normalization). Normalizing weights reflects the physical meaning of quantum-mechanics linear equations, that all state probabilities add to 100%.
particle systems: action at a distance
Physical processes can create two particles simultaneously, making two-particle systems. Both particles share one system wavefunction and have related energy levels. Measurement on one particle immediately affects system wavefunction and, by conservation laws, determines states of both particles, even if other particle is far away. State determination happens faster than light speed (action at a distance), appearing to send information faster than light and so violate relativity. However, observer knowledge of newly determined state happens only after information travels at light speed back to observer, so relativity is intact.
particle systems: energy partitioning
As in classical mechanics, for a specific total energy, quantum-mechanics phase-space wave interactions transfer total energy among wave frequencies so that energy distribution (Boltzmann distribution) is the wave-frequency distribution with maximum number of system states. Only waves with frequencies that make their energy less than half total energy can be in the distribution.
wave-particle duality
Quantum mechanics combines ideas about particles and waves (wave-particle duality). Particles have energies. Waves have wavelengths and positions. To calculate energies, quantum mechanics uses particle properties. To calculate positions, quantum mechanics uses wave properties.
waves
Quantum-mechanical particle phase-space waves extend infinitely in space and time. Wave equations have no initial conditions or boundaries.
waves: wave packets
Superposing many similar-frequency waves cancels amplitudes in most places but increases amplitude in a small space-time interval (wave packet). Wave packets are particles. Particles have many similar-frequency waves. Wave-packet frequency varies directly with particle energy. Wave-packet amplitude varies directly with particle-wave phase range, because narrower phase range makes higher amplitude.
waves: particle energies
Waves have frequency and wavelength. Quantum-mechanics wave-equation solutions are particle wavefunctions. Periodic solutions are true at lowest frequency and at all integer multiples (harmonics/overtones) of that frequency, because those waves have the same phase. Other frequencies have different phases and are not solutions. Wavefunction frequency varies directly with particle energy. Waves with higher frequency have higher energy, because field change is more when wavelength is less, so momentum change is more. Therefore, wavefunctions represent a series of possible particle energies. Energy levels differ by energy quanta. Particle-wave frequencies and energies are not continuous but discrete.
waves: position
Waves have frequency and wavelength and occupy space and time intervals. Waves cannot be at points. Particles and fields are waves and so have no definite position.
waves: shapes
Reverberations cause resonance and standing waves. Standing waves have different shapes depending on space boundaries and spin and orbit rotations. String waves have nodes at ends. For overtones of fundamental frequency, string waves have nodes at regular intervals. Molecule electrons have spherical s orbits and p, d, and f orbits with nodes. Wave shapes reflect average field density and probabilities that particle is in those space regions. In dynamic systems, wave shapes can vary over time.
waves: uncertainty
If wave has higher wavelength, position interval is wider and is less certain, but momentum and energy change, measured as less steep wave slope, is slower and so more certain. If wave has lower wavelength, position interval is narrower and is more certain, but momentum and energy change, measured as steeper wave slope, is faster and so less certain. Therefore, both position and momentum cannot be specific, and one or the other, or both, are uncertain.
waves: complex-number exponential functions
Wavefunctions are sine or cosine (trigonometric) functions. For frequency f, amplitude A, and position x, field = A * sin(2 * pi * f / x). For period t, field = A * sin(2 * pi * x / t). For wavelength l, field = A * sin(2 * pi * x / l).
Because e^(i*a) = cos(a) + i * sin(a), where a is angle in radians and is real, wavefunctions are complex-number exponential functions. sin(a) = (e^i * a - e^-i * a) / (2*i). cos(a) = (e^i * a + e^-i * a) / 2. Phase-space particle wavefunctions can superpose constructively and destructively, because they are complex-number functions.
quantum mechanics and projective geometry
Quantum mechanics has elements of projective geometry, which account for its non-local properties because projective geometry has no distance or between-ness. (General relativity is about metric geometry. Quantum mechanics is relativistic, so it also has metric properties.)
Particles have finite discrete physical-property values {quantum, quantity} {quanta, quantity}. Physical-property values do not vary over a continuous range but have definite values. For example, particle energies have discrete levels and do not have intermediate energies. Discrete physical-property values differ by an amount.
particle properties
Masses are aggregations of particles and so have quanta. As they change speed, masses add or subtract relativistic mass by quanta. Charges are aggregations of electrons and protons and so have quanta. As they change speed, charges add or subtract relativistic charge by quanta. Colors are aggregations of quark colors and so have quanta. As they change speed, colors add or subtract relativistic color by quanta. Strangenesses are aggregations of strangeness and so have quanta. As they change speed, strangenesses add or subtract relativistic strangeness by quanta.
Photons, gluons, bosons, and gravitons (exchange particles) have discrete energies, momenta, and angular momenta (such as spin). Light does not change frequency or wavelength as it travels in vacuum, or as it encounters other electric charges or magnetic fields. Gluons, bosons, and gravitons do not change as they travel or encounter fields. Therefore, forces, energies, and momenta have quanta.
maximum value
Relativity limits values to below maximum, because only infinite energy can make massive particles reach light speed.
Doppler effect
Because light speed is always constant, light sources moving toward or away change light frequency and wavelength. The change occurs at the source, so light does not change frequency and wavelength. as it travels or as it encounters other electric charges or magnetic fields.
Light travels at constant speed. If wavelength decreases, frequency increases. If wavelength increases, frequency decreases. If object is moving away, Doppler effect makes wavelength increase and frequency decrease. If object is moving closer, Doppler effect makes wavelength decrease and frequency increase. Faster motions make greater Doppler effects.
Time dilation is not about Doppler effect, because light is not clock, and light travels at light speed, not lower speed.
ground-state energy
Particle energies have a minimum value (ground state) above zero, because particles have phase-space waves, and waves propagate and so have minimum motion. Particles must move so they cannot have zero energy. Propagating waves have frequency and wavelength. Waves cannot have zero frequency, so waves have a lowest frequency (fundamental frequency) and so lowest possible energy. Electromagnetic-wave energy is frequency times Planck constant.
Because waves have wavelengths, they have uncertain position. Because waves have frequencies, they have uncertain momentum, and uncertain momentum requires minimum energy (uncertainty principle).
energy levels
Waves with the same phase satisfy the Schrödinger wave equation. Therefore, particles can have phase-space waves with harmonic frequencies. Fundamental-frequency harmonics determine allowed energy levels. See Figure 1. Higher frequencies have more energy.
Adjacent wave frequencies differ by fundamental frequency. The energy quantum varies directly with a function of particle phase-space wave fundamental frequency. As frequencies increase, energy differences decrease.
frequency
Wavefunctions with harmonic frequencies solve wave equation. Waves that solve the wave equation resonate in the system, like standing waves that constructively superpose to have net amplitude. Non-standing waves have zero amplitude. Possible standing waves have harmonic frequencies.
quanta
Particle energies, momenta, orbital and spin angular momenta, masses, forces, fields, velocities, accelerations, orbital radii, orbital periods, orbital frequencies, and properties have discrete levels separated by quanta. See Figure 2.
amplitude
Quantum-mechanical waves have amplitude. For any frequency, amplitude relates to probability that particles currently have that wave frequency.
system size
High-energy systems follow quantum mechanics, but phase-space wave wavelengths are too small to detect, so such systems do not appear to have quanta. Physical systems with very small energy or momentum differences, such as subatomic particles, atoms, and molecules, have measurable phase-space wave wavelengths, and such systems require quanta to describe their behavior correctly. See Figure 3. Some quantum-mechanical systems have large space and time differences.
In quantum mechanics, fields {quantized field} have quanta. Particles are like field singularities, vortexes, or discontinuities.
In quantum mechanics, particle and field quanta are at the lowest reductionist level. There is no subquantum world {subquanta}. Subquanta are smaller than Planck time, distance, charge, and mass. Subquantum interactions occur within Planck time, distance, charge, and mass. At subquantum sizes, space, time, forces, and energies do not exist or are indistinguishable. There is no gravity, electromagnetism, stromg or weak nuclear force, distance, time, or mass.
Particle energy E and particle phase-space wave frequency f (in cycles per second) are directly proportional by constant h {Planck constant}| {Planck's constant}: E = h * f, so h = E / f. Planck constant unit is energy times time, and action in physics is energy times time, so Planck constant is quantum of action {quantum of action} {action quantum}. h = 6.626 * 10^-34 Joule-seconds or 4.136 * 10^-15 eV-s.
Particle momentum p and particle phase-space wave wavelength w are inversely proportional by Planck constant: h = p * w. For light, E = h * f = h * c / w, so h = p * w. Momentum times distance is action in physics.
For angular frequency, radians per second, Planck constant divides by 2 * pi {reduced Planck constant} {Dirac constant} {h-bar}: h-bar = h / (2 * pi). h-bar is the quantum of angular momentum.
In quantum mechanics, phase space includes particle positions and momenta and so includes physical space. Particle systems have phase-space waves that determine probabilities of particle positions and momenta at times. In bounded space regions, such as atoms, molecules, and boxes, particles have resonating phase-space waves, with stationary points at boundaries, whose frequencies are harmonics. For example, a particle in a box has phase-space waves, with stationary points at box walls, which have fundamental frequency, twice fundamental frequency, thrice fundamental frequency, and so on. Phase-space wave frequencies determine energies, so system energies are discrete and in series: E0, E1, E2, and so on. Energy-level differences are quanta that are functions of fundamental frequency.
Because energy has quanta, momentum and angular momentum (including spin) have quanta. Electron experiments have determined the angular-momentum quantum unit to be h-bar / (2)^0.5. Momentum has quantum: h / (phase-space wave wavelength). Energy has quantum: h * (phase-space wave frequency). Electron experiments have determined that action has quanta, so energy times time, and momentum times distance, have quanta.
Because a continuous quantity times a discontinuous quantity would make a continuous quantity, for action to have quanta, time and length must have quanta. The quantum-mechanical uncertainty principle depends on particle-wave properties, relates indeterminacies in particle energy and time (or momentum and position), and so relates energy uncertainty to time uncertainty: dE * dt >= h. In space-time, maximum particle energy is where particle gravity has quantum effects and makes space-time discontinuous: 1.22 * 10^19 GeV. By the uncertainty principle, minimum time is then 10^-43 seconds (and minimum length is 10^-35 meters).
Maximum particle energy, 1.22 * 10^19 GeV, is where gravity has quantum effects and makes space and time discontinuous. Field theory no longer applies. Space is foam-like and loops and distorts, due to spin, and has no dimensionality.
Particles have phase-space waves. Particle momentum varies directly with particle-wave wavelength. Wavelength varies directly with time. Because momentum uncertainty times length uncertainty must be less than Planck constant, by the uncertainty principle, at maximum particle energy, quantum length unit {Planck length} is 1.6 * 10^-33 centimeters (1.6 x 10^-35 meters). Because space-time is no longer continuous, phase-space waves cannot have frequency greater than 10^43 Hz and wavelength less than 10^-35 meters.
Planck length depends on gravity strength and so gravitational constant g, electromagnetism strength and so light speed c, and action quantum Planck constant h: (h-bar * g / c^3)^0.5, where h-bar is Planck constant h divided by (2 times pi). h is the quantum of action, and h-bar is the quantum of angular momentum, so Planck length is the quantum of length. Planck length is distance light travels in Planck time.
Planck area quantum is 10^-66 cm^2. Planck volume quantum is 10^-99 cm^3.
Planck-length-diameter black-hole mass {Planck mass} is 10^-5 gram. Particle gravity has quantum effects and makes space-time discontinuous. Because particles are waves, if position uncertainty equals Planck length, gravity uncertainty is highest. Field theory no longer applies. Space is foam-like, due to spin, and has no dimensionality.
At universe origin or soon after, universe had Planck-length diameter. Space-time was discontinuous. Field theory no longer applies. Space is foam-like, due to spin, and has no dimensionality. When universe grew larger than Planck-length diameter, space became continuous, and temperature {Planck temperature} was 10^32 K.
Maximum particle energy, 1.22 * 10^19 GeV, is where gravity has quantum effects and makes space and time discontinuous. Field theory no longer applies. Time loops and distorts, due to spin, and has no dimensionality.
Particles have phase-space waves. Particle energy varies directly with particle-wave frequency. Frequency varies inversely with time. Because energy uncertainty times time uncertainty must be less than Planck constant, by the uncertainty principle, at maximum particle energy, minimum time unit {Planck time} is 5.391 * 10^-44 seconds. Because space-time is no longer continuous, phase-space waves cannot have frequency greater than 10^43 Hz and wavelength less than 10^-35 meters.
Planck time depends on gravity strength and so gravitational constant g, electromagnetism strength and so light speed c, and action quantum Planck constant h: (h-bar * g / c^5)^0.5, where h-bar is Planck constant h divided by (2 times pi). h is the quantum of action, and h-bar is the quantum of angular momentum, so Planck time is the quantum of time. Planck time is time light travels Planck length.
Time {chronon}| {time quantum} for light to travel (classical) electron radius is 10^-24 seconds.
Event-quantum time intervals {instanton}| are non-linear waves, lasting for one electronic transition or one quantum tunneling.
Past events determine, or at least constrain, future events. Because other events are too far away in space-time, only a subset of past events affects an event. In phase spaces, past states determine, or at least constrain, future states {consistent histories}. Because other states are too far away in phase space, only a subset of past states affects a state. In most systems, most states do not affect a future state. In quantum mechanics, systems have an infinite number of different consistent histories, each with a probability. Without quantum mechanics, completely determined systems have one consistent history.
In quantum mechanics, particle-system phase-space states have probabilities. States that do not happen have as much information as states that do happen. In quantum mechanics, because they had probability to happen, states that did not happen {counterfactual, quantum mechanics} can cause physical events/states on the same or other particles, because they collapse the wavefunction without interacting with the particle property/event/state. Measuring for a particle state that does not happen {null measurement} {interaction-free measurement} can gain information about another system particle or state without affecting that particle or state.
Particles simultaneously try all possible phase-space trajectories. Trajectories go directly (direct-channel) or indirectly (cross-channel) from one system energy level to another. The probability that the system reaches an energy level is the sum {sum over paths} {sum over histories} of renormalized path probabilities for direct-channel and cross-channel paths to that energy level.
Because bosons have integer spins, when previously independent identical-state bosons interchange, their wavefunctions stay the same as the other. Bosons have Bose-Einstein statistics. Therefore, interactions can bring two now-interdependent bosons to the same state. In a system, two bosons can be in the same state.
Because fermions have half-unit spins, when previously independent identical-state fermions interchange, their wavefunctions become the negative of the other. Fermions have Fermi-Dirac statistics. Therefore, no interaction can bring two now-interdependent fermions to the same state. In a system, no two fermions can be in the same state {exclusion principle} (Pauli exclusion principle).
To show directly that physics is non-local, measure entangled-electron spins {Bell experiment}. Electrons are indistinguishable. Around any measuring axis, electron spins have only two, clockwise or counterclockwise, angular-momentum states. For systems with zero total angular momentum, one electron has spin +1/2 and the other has spin -1/2. Experimenters can only measure one electron's spin, after wavefunction collapse, so system wavefunction before collapse had both electrons having both spins in superposition. Electrons 1 and 2 have spins along axes x, y, and z. If axes are indistinguishable and electrons combine randomly, states are 1x+2x-, 1x-2x+; 1x+2y-, 1x-2y+; 1x+2z-, 1x-2z+; 1y+2y-, 1y-2y+; 1y+2z-, 1y-2z+; 1z+2z-, 1z-2z+, so 6/12 of states involve x-axis, and 6/12 do not. If axes are distinguishable and electrons combine randomly, states are 1x+2x-, 1x-2x+; 1x+2y-, 1x-2y+; 1x+2z-, 1x-2z+; 1y+2x-, 1y-2x+; 1y+2y-, 1y-2y+; 1y+2z-, 1y-2z+; 1z+2x-, 1z-2x+; 1z+2y-, 1z-2y+; 1z+2z-, 1z-2z+, so 10/18 of states involve x-axis, and 8/18 do not. However, system has equal probability to start with 1x+2x- and 1x-2x+, so 2 x-axis states must be left out, making 9/18 of states involve x-axis, and 9/18 do not. If axes are indistinguishable and electrons entangle, states are xx, xy, xz, yy, yz, zz, so 3/6 of states involve x-axis, and 3/6 do not. If axes are distinguishable and electrons entangle, states are xx, xy, yx, xz, zx, yy, yz, zy, zz, so 5/9 of states involve x-axis, and 4/9 do not. Bell experiment result confirms the last conditions, so, if there are no hidden variables, electrons entangle, and physics is non-local.
In hypothetical experiments with closed boxes, particle decay triggers processes that kill cats {Schrödinger's cat} {Schrödinger cat}. If decay probability is one-half, is cat half-alive and half-dead inside box until observed? Is cat dead when particle decays, observed or not? In which state is cat after event and before observation? Less and less classical objects and events can replace cats, until replacements are quantum events and particle decay triggers a quantum state, so when does quantum-event wavefunction collapse?
In classical physical space, particles have definite positions and momenta, not probabilities of positions and momenta. If physical space has no external forces, positions and momenta are independent. If physical space has force fields, position change changes momentum in only one way, according to energy conservation. Because particles have definite positions and momenta, and classical configuration space has only real numbers, classical configuration space has no real-number/imaginary-number interactions and so no waves. The Hamiltonian function represents energy as a function of momentum (kinetic energy) and space (potential energy) coordinates.
In quantum-mechanical physical space, particles have probabilities of positions and momenta. Quantum-mechanical physical space has energy conservation, but positions and momenta are not independent, so energy-conservation equation (Schrödinger equation) and S-matrix theory, which relate kinetic-energy change and momentum to potential-energy change and position, have complex numbers. Exponentials with complex-number exponents represent cosine and sine waves. (Maxwell's equations relate kinetic-energy change and momentum to potential-energy change and position, and solutions are electromagnetic waves.) Frequency is time derivative. Wave number is spatial derivative. The time derivative introduces an imaginary number to multiply the time derivative to give a real number.
Quantum-mechanical configuration space (phase space) has complex-number particle position and momentum coordinates. Along each configuration-space dimension, real and imaginary numbers interact to make helical scalar waves {matter wave}| {de Broglie wave} {probability wave}.
scalar
Electromagnetic waves are vector waves, because electric and magnetic forces and fields have direction, electromagnetic waves propagate in a direction, and energy travels in that direction. Matter waves are scalar, because they are not about forces or fields, have no energy, and do not propagate and so do not travel and are standing waves. Scalar waves have amplitude but no direction.
phase space
Matter waves are not in physical space.
wavelength
Wavelength determines possible particle positions and momenta, at maximum-displacement positions. Frequency and phase affect amplitude.
amplitude and probability
Amplitude determines probability that particle is at that position or momentum.
frequency and kinetic energy
Particle kinetic energy E determines matter-wave frequency f: E = h * f, where h is Planck constant. For higher energies, matter waves have higher frequencies and lower wavelengths. Particle momentum p determines matter-wave wavelength w: h = p * w. Theoretical matter-wave velocity v increases with particle kinetic energy: v = f * w = (E/h) * (h/p) = E/p.
transverse wave
Real and imaginary number interactions make transverse waves around each phase-space dimension.
length
Matter waves are in configuration space, which has infinitely-long dimensions, so matter waves are infinitely long. By uncertainty principle, matter waves extend through all space, but with low amplitude outside physical system.
no propagation and no energy
Because they are infinitely long, matter waves do not propagate, are standing waves, and have no travel, no velocity, no energy, and no leading or trailing edge. Matter waves resonate in phase space.
positions, points, and intervals
Waves require one wavelength to be a wave, so there is no definite position. For waves, positions cannot be points but are one-wavelength or half-wavelength intervals.
solidity
Matter waves have width of at least one wavelength, so they cause matter to spread over space, not be at points. Matter waves make matter have area, and matter appears solid.
momentum and position
In quantum mechanics, unlike classical mechanics, momentum and position are not independent, because amplitude relates to position, frequency relates to momentum, wave amplitude-change rate relates to wave frequency-change rate, wavelength relates to position uncertainty, and amplitude-change rate relates to momentum uncertainty.
particle sizes
Large objects have high matter-wave frequencies. At high frequencies, matter-wave properties are undetectable, because wavelengths are too small, so classical mechanics applies. Small objects have low matter-wave frequencies, so atomic particles have detectable quantum properties.
waves and quanta
Resonating waves have fundamental frequency and harmonic overtones. Particles have matter waves with harmonic frequencies. Harmonic frequencies correspond to a series of positions or energy/momentum levels, separated by equal amounts (quantum).
Waves change frequency without passing through intermediate frequencies. No intermediate frequencies means no intermediate positions or energies/momenta. Matter waves explain why particles have discrete energy levels, separated by quanta, and why, during energy-level transitions, particles never have in-between energy levels. Particles also have discrete locations, separated by quantum distances.
physical systems
In free space, particle matter waves have a small range of frequencies and superpose to make a wave packet. Particle systems superpose particle matter waves to make system matter waves. Non-interacting particles have dependent matter waves that add non-linearly (entangle). In atoms and molecules, electrons, neutrons, and protons have phase-space matter waves that represent transitions among atomic orbits.
Electrons cannot be near nucleus, because then electron matter-wave interacts with proton matter-wave, and atom collapses.
philosophy
Perhaps, matter waves are particles, only associate with particles, are mathematical descriptions, or are all that observers can know.
Matter-wave wavelength equals Planck constant divided by momentum {de Broglie relation}|.
In quantum mechanics, matter-wave amplitude determines probability that particle is at that position. Matter waves are infinite and so have positive amplitude at all space points. Therefore, unlike classical mechanics, particles have a probability of being outside potential-energy barriers {tunneling}|.
At barriers, particle waves reflect back or refract through. Particles with higher matter-wave frequency and more energy have more refraction. As difference between barrier potential energy and particle energy increases, reflection {anti-tunneling} increases.
Matter waves are infinitely long. Because particle matter waves have fundamental frequency and its harmonics, particles have an infinite number of different-frequency matter waves. Because particles interact with other universe masses and charges, particles have matter waves differing in wavelength by infinitesimally small amounts. Superposition of an infinite number of infinitely long waves, differing in wavelength by infinitesimally small amounts, makes significant amplitude {wave packet}| in one region and insignificant amplitudes in all other regions. Particles are matter-wave packets.
time
Over time, as superposition makes different results, wave packets can disappear and reappear. Wave superposition can narrow or broaden wave-packet duration. and broadening frequency range.
size
Wave packets have three to ten oscillations, with maximum amplitude in center and no amplitude at edges. Longest wavelengths are in middle and smallest wavelengths are at edges. If wavelength range is small, packet is wide. If wavelength range is large, packet is narrow.
speed
Wave packets travel at particle speed, but wave-packet component waves travel at slower and faster speeds.
frequencies
Matter-wave-packet frequency varies directly with particle energy. Wave superposition can narrow or broaden wave-packet frequency range. If wave packet has many frequencies, volume is small, but energy is big. If wave packet has few frequencies, volume is large, and energy is small.
dispersion
Due to dispersion, wave packets spread out lengthwise and transversely.
In classical mechanics, positions and momenta (and energies and times) are independent variables, but in quantum mechanics, they are dependent variables and interact in wavefunctions. In classical mechanics, when two or more particles interact, system properties sum particle properties. In quantum mechanics, when two or more particles interact, system properties multiply and sum particle properties, and particle wavefunctions combine constructively and/or destructively to make a system wavefunction {entanglement}|. If two (indistinguishable) particles entangle, they both travel together on all possible state paths available to them, and they interfere with each other's independent-particle wavefunctions along each path. For example, two particles created simultaneously form one system with one wavefunction.
Entanglement does not put particles into unchanging states (that observers measure later). Neither do particle states continually change state as they move through space-time (not like independent neutrinos, which change properties as they travel). Therefore, observation method, time, and space position and orientation do not determine observed particle state. In quantum mechanics, particles have probabilities, depending on particles and system, of taking all possible space-time and particle-interaction paths, and measurement finds that the particle has randomly gone into one of the possible particle states.
system wavefunction
When two particle wavefunctions add, system-wavefunction frequency is the beat frequency of the two particle-wavefunction frequencies, and is lower than those frequencies. System wave packet has smaller spatial extension than particle wave packets, and has higher amplitude (more energy) at beat-frequency wavelengths. Quantum-mechanical particle and system wavefunctions have non-zero fundamental frequency and its harmonic frequencies and have non-zero amplitudes over all space and time. Systems spread out over space and time.
system wavefunction decoherence
After entanglement, system wavefunction lasts until outside disturbances, such as measurement, particle collision or absorption, and electromagnetic, gravitational, or nuclear force field, interact with one or more particles. At that definite time and position, system wavefunction separates into independent particle wavefunctions (decoherence). Whole system wavefunction ends simultaneously over whole extent.
measurement
By uncertainty principle, experimenters can precisely measure either particle energy or particle time (or momentum or position) but not both. After two entangled particles separate, separate instruments can measure each particle's energy (or momentum) precisely and simultaneously and then communicate to determine the exact difference.
measurement: speculation
Perhaps, unobserved particles and systems are two-dimensional (but still in three-dimensional space). Observation then puts particles and systems into three dimensions. People observe only three-dimensional space. For example, observers see that gloves are right-handed or left-handed. Perhaps, unobserved quantum-mechanical-size gloves actually have no thickness and so have only two dimensions, so unobserved right-handed and left-handed gloves are the same, because they can rotate in three-dimensional space to superimpose and be congruent. Perhaps, unobserved clockwise and counter-clockwise particle spins are two-dimensional and so are equivalent. (Note that a two-dimensional glove appears right-handed or left-handed depending on whether the observation point is above or below the glove.)
Perhaps, unobserved particles and systems randomly, continually, and instantaneously turn inside out (and outside in), in three-dimensional space. Observation stops the process. For example, turning a right-handed glove inside out makes a left-handed glove, and vice versa. Perhaps, unobserved quantum-mechanical-size gloves continually and instantaneously turn inside out in three-dimensional space and so are equally right-handed and left-handed. Perhaps, unobserved clockwise and counter-clockwise particle spins continually interchange. (Note that a glove appears right-handed or left-handed depending on when the process stops.)
Perhaps, unobserved quantum-mechanical-size particle and system states are indeterminate and follow quantum-mechanical rules because space-time is not conventional four-dimensional space-time. Observation requires conventional three-dimensional space, and randomly makes definite three-dimensional particle and system states, with probabilities. Perhaps, time is not real-number time, but complex-number or hypercomplex-number time. Real-number times are separate, but imaginary-number times are not. Perhaps, space is not real-number space, but complex-number or hypercomplex-number space. Real-number distances are separate, but imaginary-number distances are not.
Observations measure real-number part of complex-number variables. Perhaps, wavefunction imaginary-number part continues after observation.
Perhaps, Necker cubes illustrate the effects of observation. Observer angle to Necker cube determines whether observer sees right-facing or left-facing Necker cube. Effects may be linear with angle or depend on cosine of angle.
interacting electrons and spin
If a process creates two electrons, momentum sum is the same before and after creation, by momentum conservation, and electrons move away from each other at same velocity along a straight line. Angular-momentum sum is the same before and after creation, by angular-momentum conservation. (If two separate electrons entangle, momentum sum and angular-momentum sum are the same before and after interaction.)
By quantum mechanics, measured spin is always +1/2 or -1/2. Because the electrons are in a system, one cannot know which has +1/2 spin and which -1/2 spin. Both electrons share a system wavefunction that superposes the state (wavefunction) in which first electron has spin +1/2 unit and second has spin -1/2 unit and the state (wavefunction) in which first electron has spin -1/2 unit and second has spin +1/2 unit, with zero total angular momentum in any direction. Two wavefunctions can superpose constructively (add) or destructively (subtract). Because two electrons are distinguishable, the two wavefunctions add, so system wavefunction is anti-commutative.
One possibility is that one particle has positive 1/2 unit spin along z-axis (motion line), and other particle has negative 1/2 unit spin along z-axis. See Figure 1.
After two particles interact and move apart, separate spin detectors can measure around any axis for first particle and around any axis for second particle, simultaneously or in succession. For example, the axes can be z-axis (motion line), x-axis, and y-axis. See Figure 2. Measuring spin around an axis fixes one electron's spin at +1/2 (or -1/2) and fixes the other electron's spin around an axis at -1/2 (or +1/2), to conserve angular momentum.
spin: possible axis and spin combinations
By quantum mechanics, left electron has spin +1/2 half the time and spin -1/2 half the time, around any axis, say z-axis. Around same z-axis, right electron always has opposite spin: left=z+ right=z- or left=z- right=z+. Around x-axis, right electron has opposite spin (while y-axis has same spin), same spin (while y-axis has opposite spin), opposite spin (while y-axis has opposite spin), or same spin (while y-axis has same spin): x-y+z-, x+y-z-, x-y-z-, x+y+z-; x-y+z+, x+y-z+, x-y-z+, x+y+z+. For right-electron z-axis compared to left-electron z-axis, spins are opposite all of the time: z+z-, z+z-, z+z-. z+z-. For right-electron x-axis or y-axis compared to left-electron z-axis, spins are same 1/2 of time and opposite 1/2 of time: z+x-, z+y+; z+x+, z+y-; z+x-, z+y-; z+x+, z+y+. See Figure 3. Because quantum mechanics has random probabilities, left and right electrons have same spin half the time and opposite spin half the time.
However, quantum mechanics with non-randomness (due to local real hidden factors) makes a different prediction. Non-random hidden factors correlate right and left spins, to conserve angular momentum. If left=x+y+z+, right=x-y-z-. If left=x-y+z+, right=x+y-z-. If left=x+y-z+, right=x-y+z-. If left=x-y-z+, right=x+y+z-. If left=x-y-z-, right=x+y+z+. If left=x+y-z-, right=x-y+z+. If left=x-y+z-, right=x+y-z+. If left=x+y+z-, right=x-y-z+. See Figure 4. For right-electron z-axis compared to left-electron z-axis, spins are opposite all the time. For right-electron x-axis or y-axis compared to left-electron z-axis, spins are same 4/9 of time and opposite 5/9 of time, higher than the 1/2 level for quantum mechanics. Local hidden variable theories correlate events through hidden variable(s), making probabilities non-random. Quantum mechanics has no more-fundamental factors and introduces uncertainties, and so is random. Therefore, correlated outcomes in classical theories have different probabilities than in quantum mechanics. Experiments show that outcomes are random, so there are no local hidden factors and/or no real hidden factors.
if infinite light speed
Perhaps, entanglement over large distances and times has no non-locality problems if light speed is infinite, as in Newton's gravitational theory. Assume that relativity is true but with light speed infinite. Time is zero for light, and speed is always infinite for all observers, so all objects are always in contact. However, light speed is finite.
action at distance
Wavefunctions do not represent physical forces or energy exchanges, so space and time do not matter. If system wavefunction does not decohere, system particles and fields remain connected, even over long duration and far distances. Experiments that measure energy and time differences, or momentum and position differences, show that particles remained entangled over far distances and long times, and that wavefunction collapse immediately affects all system particles and fields, no matter how distant (action at a distance). Seemingly, new information about one particle travels instantly to second particle. See Particle Interference, Scientific American 269(August): 52-60 [1993].
However, information about collapse only travels at light speed, preserving special relativity theory that physical effects faster than light speed are not possible. Observers must wait for light to travel to them before they become aware of information changes. All physical laws require local interaction through field-carrying particle exchanges, which result in space curvatures. All physical communication happens when particles are in contact and interact, so there is no actual action at a distance.
teleportation
After particle entanglement, particle wavefunctions have specific relations. By manipulating particle properties at interaction and at wavefunction collapse, experimenters can transfer particle properties from one particle to another particle, even far away, though the particles have no physical connection at collapse time.
Bombs can have photon or light pressure triggers. Bombs explode if trigger does not jam, but jamming happens often. How can testers find at least one working bomb without exploding it {Elitzur-Vaidman bomb-testing problem} [1993] (Avshalom C. Elitzur and Lev Vaidman)? Using photon entanglement can find good bomb without triggering it.
Particles can seemingly move from one place to another without ever being between the two places {teleportation}|. Teleportation requires that both locations share a particle pair {EPR pair}. Particles are identical, with entangled properties. For example, if one photon splits into two photons, new photons can be same-state superpositions. If instrument observes one particle's state later, it then knows other particle's state. If EPR pair exists, putting one pair member into one state can result in property disappearance at one location and other-pair-member property appearance at another location.
Instruments can measure momentum, position, energy, and time by absorbing energy and using clocks and rulers. However, instruments cannot simultaneously or precisely measure both particle momentum and position {uncertainty principle}| {Heisenberg uncertainty principle} {indeterminacy principle}, because measuring one alters information about the other. Instruments cannot simultaneously or precisely measure both particle energy and time, because they relate to momentum and position.
situation
The uncertainty principle is about measurement precision on one particle at one time and place. The uncertainty principle does not apply to different measurements on same particle over time. The uncertainty principle does not apply to simultaneous momentum and position, or energy and time, measurements on different particles.
wave packet
Particles have wavefunctions, so measurements are about wave packets. As particle moves through time and space, total uncertainty increases, because wave packet spreads out.
wave properties
Uncertainty follows from wave properties, because wave position and momentum, or time and energy, inversely relate. Energy and momentum depend on wave frequency. Position and time depend on wave amplitude. Measuring wave frequency or wavelength precisely prevents measuring wave amplitude precisely. Measuring wave amplitude precisely prevents measuring wave frequency or wavelength precisely. If momentum or position is specific, position or momentum must be uncertain. If energy or time is specific, time or energy must be uncertain.
At space points, wavefunctions that have high amplitude have precise position and timing. However, wavefunction slope is steep, so amplitude change between nearby points is large, so velocity change, momentum change, and energy change are large and so uncertain at that position. See Figure 1.
Wavefunctions with wide wave packets have large uncertainty. Wavefunction slopes are not steep, and amplitude change at nearby points is small, so velocity change, momentum change, and energy change are small in that region. Momentum is precise, while position is imprecise. Alternatively, energy is precise, while timing is imprecise. See Figure 2.
Waves that have just one frequency and wavelength have one momentum and energy. Only one wave can have no superposition and no cancellation anywhere in space or time, making wave equally present throughout all space and time, and so completely uncertain in position and time. See Figure 3.
Wavefunctions that have almost all frequencies and wavelengths have precise position and time, because waves cancel everywhere, except one space or time point. Wavefunctiond that have almost all frequencies and wavelengths have almost all momentum and energy levels, making wave momentum and energy very uncertain. See Figure 4.
Waves that have some frequencies and wavelengths have moderate uncertainty in momentum and energy and moderate uncertainty in position and time, because waves cancel, except at moderate-size wave packet.
Waves with two or three frequencies and wavelengths have beat frequencies where waves superpose. Beat frequency makes precise momentum and energy, but time and position are uncertain. See Figure 5.
Waves with harmonic frequencies and wavelengths have beat frequencies where waves superpose. Beat frequencies make precise momentum and energy, but time and position are uncertain.
measurement processes
Besides wavefunction effects, physical processes limit precision. To find precise frequency for energy and momentum takes time and space, so position and time information are uncertain. To find precise position and time takes high amplitude, so position and time information are uncertain. Uncertainty's physical cause is discontinuity, whereas uncertainty's quantum-mechanical cause is wave-particle duality, because particles are about momentum and energy and waves are about position and time, as shown above.
mathematics
Quantum of action is h, and energy over time is action. Therefore, energy uncertainty dE times time uncertainty dt equals at least Planck constant divided by 4 * pi: dE * dt >= h / (4 * pi).
dE = F * dx = (dp / (4 * pi * dt)) * dx, so dE * dt * (4 * pi) = dp * dx. Position uncertainty dx times momentum uncertainty dp equals at least Planck constant: dx * dp >= h.
dx = 4 * pi * dF, and dp = dN / 2. Phase uncertainty dF times phonon number uncertainty dN equals Planck constant divided by 2 * pi: dF * dN = h / (2 * pi).
energy levels
Electrons in lower atomic orbitals have higher frequency, kinetic energy, and angular momentum and lower time period and orbital diameter. Electrons in higher atomic orbitals have lower frequency, kinetic energy, and angular momentum and higher time period and orbital diameter. Therefore, higher orbitals have higher position uncertainty and lower momentum uncertainty.
For low-orbital and high-orbital electrons, photon absorption can cause electronic transition to adjacent higher energy level, increasing position uncertainty and decreasing momentum uncertainty. For low-orbital and high-orbital electrons, photon emission can cause electronic transition to adjacent lower energy level, decreasing position uncertainty and increasing momentum uncertainty.
For low-orbital and high-orbital electrons, photon absorption can cause electronic transition to non-adjacent higher energy levels, increasing position uncertainty and decreasing momentum uncertainty. For low-orbital and high-orbital electrons, photon emission can cause electronic transition to non-adjacent lower energy levels, decreasing position uncertainty and increasing momentum uncertainty.
Besides fundamental Heisenberg uncertainty, electron, proton, and neutron configuration changes affect measured amounts. Electronic transitions conserve energy, momentum, and angular momentum, so absorption and emission do not necessarily have the same photon frequency. Electrons cannot transition to same orbital.
two particles
Though instruments cannot measure either's time or energy, instruments can measure two particles' energy difference and time difference precisely and simultaneously. Such measurement can define one-ness and two-ness.
confinement
By uncertainty principle, particles confined to smaller regions or times have greater momentum and energy. In confined regions, even in vacuum, energy is high, allowing particle creation and annihilation.
matrices
In quantum mechanics, particle position and momentum are quantized and so are matrices (not scalars or vectors), with complex-number elements. Because particles have probabilities of being anywhere in space, matrix rows and columns have infinite numbers of elements, and matrices are square matrices. In quantum mechanics, position and momentum are not necessarily independent, but depend on the whole particle system.
Matrices represent electronic transitions between energy levels. Matrix rows are one energy level, and matrix columns are the other energy level. Matrix elements represent the probability of that electronic transition. Matrix elements are periodic to represent the possible quanta. The diagonal represents transitions between the same energy level and so has value zero. Near the diagonal represents transitions between adjacent energy levels and so has higher values. Far from the diagonal represents transitions between non-adjacent energy levels and so has lower values. Energy levels have ground state and no upper limit, so the matrices have infinite numbers of elements. There is no zero energy level.
For non-infinite-dimension square matrices with real elements, PQ = QP (commutative). For infinite-dimension and/or non-square and/or complex-number-element matrices, PQ <> QP (non-commutative). Matrix multiplication is typically non-commutative.
In quantum mechanics, particle action is the product of the momentum P and position Q matrices: action = PQ. For infinite-dimension square matrices with complex-number elements, PQ - QP = -i*h*I, where I is identity matrix and h is Planck constant, because action has Planck-constant units and complex number multiplication rotates the axes by pi/2 radians.
Though electrons and protons have strong electrical attraction, and outside electrical attractions and repulsions can disturb atom orbitals, electrons do not spiral into protons and collapse atoms. Because particles have matter waves, by the uncertainty principle, orbiting electrons cannot spiral into atomic nucleus {atom, stability}. See Figure 1.
waves
Particles have matter waves, whose harmonic frequencies relate to particle energy levels.
uncertainty
Waves by definition must be at least one wavelength long. Therefore, particle waves have location uncertainty of at least one wavelength. Particle waves have time uncertainty of at least one period, which is one wavelength divided by light speed. Particle waves have momentum uncertainty of at least Planck constant divided by wavelength. Particle waves have energy uncertainty of at least Planck constant divided by period. Particle waves make the uncertainty principle.
energy
By uncertainty principle, particles must move, and so they cannot have zero energy. Particles cannot have zero energy because they cannot have zero motion, because that violates conservation of both energy and momentum. Lowest particle energy is first-quantum-level ground-state energy.
orbits
Electron orbits have quantum distances from nucleus and take quantum durations to orbit nucleus. In lowest orbital, electron position uncertainty has same diameter as orbital. Electron can be anywhere in that region around nucleus. In lowest orbital, electron time uncertainty is same period as orbital rotation. Electron can be anywhere in that interval. In lowest orbital, electron is already at closest possible distance and smallest possible time.
transitions
From lowest orbital, electrons cannot go to lower orbits, because there are no lower energy levels. They cannot lose more energy, because if energy decreases then time increases, by uncertainty principle, making orbital go higher. They cannot lose more distance because if distance decreases then energy must increase, by uncertainty principle, making orbital go higher. Therefore, lowest orbital has lowest energy, smallest distance, and shortest time. Lowest orbital already includes nucleus region, so it cannot be smaller.
kinetic and potential energy
In quantum mechanics and classical mechanics, electric-field positions relate to potential energies. In quantum mechanics, unlike classical mechanics, kinetic energy cannot completely convert to potential energy, and vice versa. Kinetic energy and potential energy have minimum energy level and cannot be zero.
energy quantum
First energy quantum is difference between ground-state energy and next-highest-orbital energy. Second energy quantum is difference between next-highest-orbital energy and third-orbital energy. Energy quanta are not equal. Energy quanta decrease at higher orbitals. Energy quanta relate to wave harmonic frequencies. Higher adjacent wave frequencies have smaller energy differences.
atom nucleus
Atomic nucleus occupies only 10^-5 volume inside lowest-electron-orbital volume. Nucleus protons and neutrons have energy, momentum, position, and time uncertainty and so have ground-state energies. Nucleus protons and neutrons have quantum energy levels.
Lowest-orbital electrons and highest-orbital neutrons and protons never collide, because electrons have lower orbiting energies, and higher orbital radii, than neutrons and protons.
electron-proton collision
At high-enough energy and beam collimation, electrons can collide with atomic nuclei, because increased energy can narrow position, by uncertainty principle. Such electrons are not orbiting, so this situation is not about atom stability.
Particle in enclosed space {particle in box} must have velocity, because particle has fixed position, so uncertainty is in momentum. If enclosed space is smaller, velocity must be more.
Electric field and magnetic field cannot be at rest {quantum fluctuation}, because then they have precise position and precise zero momentum and so violate uncertainty principle. All fields have random motion, even in vacuum where net energy is zero.
At quantum level, empty-space field fluctuation {vacuum polarization}| is infinite.
Two parallel uncharged metal plates attract each other by reducing vacuum-energy fluctuations and number of wavelengths between them {Casimir effect} {Casimir force}: energy density = c / d^4, where c is constant and d is plate distance. Energy at plate is zero. Interior energy density decreases, so exterior energy density increases and pushes plates together. Fewer particle histories with closed time-like loops are between plates.
Particles cannot be at rest {zero point motion}|, because then they have precise position and precise zero momentum and so violate uncertainty principle. All particles have random motion, even in vacuum where net energy is zero.
For energy transfers, particles act like particles. For determining locations, particles act like waves {wave-particle duality}|.
Matter waves have spatial/momentum effects and time/energy effects, which instruments cannot detect simultaneously {complementarity, quantum mechanics}. Particles have energy, and waves have positions. Instruments cannot determine particle properties and wave properties simultaneously. Experiments can be only complementary, because particles always have both wave and particle properties.
Wave, photon, or particle sources can send collimated beams through one or two slits, to a measuring surface {slit experiment, quantum mechanics} {two-slit experiment}. For one slit, beam makes medium intensity line across from slit. For two slits, beam makes line with four times medium intensity across from slit. It makes alternating intense and clear lines on both sides. First intense line to side has two times medium intensity. Second intense line has medium intensity. Third intense line has lower intensity, and so on. Beam waves constructively and destructively interfere.
quanta
Particles sent through two consecutive pinholes create concentric rings on screen, as waves do. Particles sent through two adjacent pinholes make stripes perpendicular to line between pinholes on far screen, as waves do. If one slit closes, ring pattern appears. If slits alternate between closed and open, two ring patterns appear. If detector is at one slit, ring pattern appears. If detectors are poor, feeble stripe pattern appears. If half-silvered mirror is after one slit in particle-stream path, and both paths reflect from mirrors, stripe pattern appears.
wave
Particle motions are not single trajectories but diffract, as waves do. Wave theory accounts for all results. Matrix theory can account for results if slits act together to make periodicity.
Paths entangle, so electrons that pass through beam splitter and go past solenoid coil have quantum interference {Aharonov-Bohm effect}, though no electromagnetic field is outside solenoid coil.
Detectors can be after location at which particles must choose which path to take and can turn on after particles pass decision point {delayed-choice experiment} (Wheeler) [1980].
In two-slit experiments (Scully and Drühl) [1982], tagger {quantum eraser} can be in front of each slit to make spin clockwise or counterclockwise along axis. Screen can detect particle location and spin. There is no interference. Waves are present, but they cancel. Before screen, place spin tagger that always results in same spin. There is interference. Waves do not cancel.
down conversion
A photon can become two photons, each with half the energy {down-converter}. In beam-splitter experiments (Scully and Drühl) [1982], a down-converter can be on each path, to make one photon that continues on that path {signal photon} and one photon {idler photon} that is detected {delayed-choice quantum eraser}. Waves do not interfere.
When information about idler photon is random, because idler photon splits and goes on ambiguous paths, waves interfere. Instruments can receive the information before or after signal photons hit, by any amount of time or space. Waves are always present, but they can cancel.
detector
In two-slit experiments, particles make interference pattern when observed. If detector capable of knowing if particle went through left, right, none, or both slits is after slits, and it indicates that each particle goes only through either left or right slit, never both or none, there is no interference pattern.
If detector can operate without affecting particle in any way, and observer observes it, there is still no interference pattern.
If observer does not observe detector, there is interference pattern, even if detector puts the information in memory awhile and then deletes memory. This suggests that just gaining information is enough to end interference [Seager, 1999].
The quantum mechanics wave equation, which relates kinetic and potential energy to total energy, has complex-number, single-valued, continuous, and finite solutions {wavefunction}|. The wave equation, and its wavefunction solutions, are about abstract phase space, which includes space-time and describes system momenta and position or energy and time states. Wavefunctions represent possible physical-system energy levels and positions, and their probabilities. Wavefunctions correlate particle energies and times or particle momenta and positions. Wavefunctions typically depend on position, because energy includes potential energy. Wavefunctions typically depend on time, because energy includes kinetic energy. Wavefunctions are not physical waves and have no energy or momenta, but mathematically represent system properties.
Wavefunction is about infinite-dimensional abstract Hilbert space, in which wavefunction rotates as a unitary function and is deterministic.
energy and frequency
Because particle matter waves resonate in physical systems, wavefunctions have fundamental frequency and harmonics of fundamental frequency. System energy levels depend on wavefunction frequencies. System energy levels are discrete, and quanta separate energy levels. High-frequency waves have high energies. System boundary conditions set used or injected energy and wave fundamental frequency and harmonics of fundamental frequency.
amplitude, intensity, and probability
Wavefunction amplitudes are complex numbers that reflect physical-system position, time, energy, or momentum relations. Probabilities that particles are at locations depend on wavefunction amplitude for that location. Probabilities are linear and add, so probability of a set of states is sum of state probabilities. Wavefunction amplitudes can normalize, so sum of all state probabilities is one.
Intensity is absolute value of wavefunction-amplitude squared: wavefunction complex conjugate times position vector times wavefunction. Squared amplitude eliminates imaginary numbers and so is only real numbers. Absolute value makes only positive numbers. Intensities and energies are only discrete real positive numbers (eigenvalue). Amplitude squared absolute value relates to particle cross-section, collision frequency, and scattering-angle probabilities, and so to state probabilities.
wavelength
Waves have wavelength and so cannot be at a point but must spread over one wavelength. Particles have wave properties and can be at any point in region one-wavelength wide. Regions have wave amplitudes and so probabilities that particle is there.
resonance
In systems, reflected matter waves add constructively, and superpositions make standing-wave harmonic frequencies. Other frequencies cancel. Resonating fundamental wave has wavelength equal to system length or diameter and lowest-frequency. Fundamental-wave harmonic frequencies determine discrete possible particle energy levels.
deterministic
Wavefunctions are deterministic.
one particle
A one-particle system has a fundamental matter wave and its harmonics that determine possible particle positions and momenta. Harmonic wavefunctions are orthogonal/independent and linearly superpose. For particles with small momentum range and small position range moving along a straight line, wavefunctions are helices around the line with almost no amplitude at line ends and rising amplitude then falling amplitude near particle location.
one particle: definite momentum
For definite particle momentum along a straight line, position wavefunctions are helices around the line. If particle is at a well-defined position, helical waves have short wavelengths. If particle is at widespread positions, helical waves have long wavelengths.
one particle: no momentum
If particle has no momentum, momentum wavefunction is a straight line, and position wavefunction is constant.
one particle: definite position
For definite particle position along a straight line, momentum wavefunctions are helices around the line. If particle has high momentum, helical waves have short wavelengths. If particle has low momentum, helical waves have long wavelengths.
one particle: no position
If particle can be anywhere along a straight line, position wavefunction is a straight line, and momentum wavefunction is constant.
If energy times wavefunction, minus potential times wavefunction, is greater than zero, wavefunction oscillates {unbound state} {continuous spectrum}. Wavelength and quantum energy levels are too small to detect.
If energy times wavefunction, minus potential times wavefunction, is less than zero, wavefunction goes to zero {bound state} {discrete spectrum} only at special eigenvalues or else goes to infinity. At special eigenvalues, wavelength and quantum energy levels are large enough to detect.
Wavefunctions are complex-number functions with complex-number solutions, but intensity has positive real values {eigenvalue, quantum mechanics}.
Wavefunction amplitudes can adjust {renormalization, probability} {normalization, wavefunction}, so sum of all amplitude-square absolute values, or all energy-level probabilities, is 1 = 100%. Because systems have an infinite number of harmonic wavefunctions, without renormalization the sum of probabilities is infinite.
In quantum field theory, generalized wavefunctions {wavefunctional} are about higher spaces {field space}.
To find electronic-transition-series {series of electronic transitions} {electronic transition series} probability, multiply wavefunction complex-number amplitudes and then square product absolute value.
Abstract phase space describes system particle momenta and positions. Wavefunctions describe possible system particle positions and momenta states {state vector} {quantum state}. For example, in a system, a single particle has constant momentum and two possible positions. System has two (non-interacting) state wavefunctions, S1 and S2, with different probabilities depending on wave amplitude at the state, c1 and c2. Wavefunction W is sum of each state's amplitude times state wavefunction: W = c1 * S1 + c2 * S2. System wavefunction is a superposition of weighted state wavefunctions.
multiple particles
Particles have state wavefunctions at all possible positions and momenta. Particles can be independent or interact. If they are independent, particle wavefunctions multiply to make (linear) tensor products. Phase is not important for bosons, and tensor product commutes. Phase is important for fermions, and tensor product does not commute. If particles interact, system has entangled wavefunction.
Over time, Schrödinger-equation wavefunctions can change deterministically {unitary evolution, wavefunction}, as position, time, energy, or momentum change.
Isolated wavefunctions deterministically calculate future possible states. However, observing a particle measures particle position or momentum, putting particle into a definite phase-space state, and so cancels particle wavefunction {wavefunction collapse} {collapse of wavefunction}| {reduction of wave packet} {wave-packet reduction} {collapse of the wavefunction} {state vector reduction}. Wavefunction collapse is a discontinuity in physics. Collapse is time asymmetric. After observation, particle again has a wavefunction, until the nect observation.
observation and measurement
Observers and measuring instruments are too large to have observable wavefunctions, matter-wave wavelengths, matter-wave frequencies, or energy quanta. Observing and measuring cause particle interaction with a macroscopic system and make a new macroscopic system that includes the particle. Observers and instruments put particle wavefunctions into definite phase-space states {state preparation}, ready for measuring. Macroscopic systems have definite object positions and momenta.
Measuring requires that observer or instrument has definite phase-space state, and particle has definite phase-space state. Observers and instruments measure along one direction and detect particle position, time, momentum, angular momentum, or energy. Therefore, position, time, momentum, angular momentum, or energy observation/measurement operates on particle complex-number wavefunction and transforms it into a position, time, momentum, angular momentum, or energy real positive value. The value is any one of the set of possible different-probability quantum values (operator eigenfunction) described by the observer/instrument/particle wavefunction. State selection is completely random. Measurement results in a single value, not value superpositions or multiple values. The observer/instrument/particle wavefunction collapses to zero {measurement problem}. At measurement, particle phase-space state no longer exists, because particle wavefunction no longer exists.
operators
Measuring wavefunctions mathematically uses linear differential Hermitean operators.
causes
Measurements, absorptions, collisions, electromagnetic forces, and gravitational forces collapse particle wavefunctions. Gravitational effects can be gravitational waves, mass separation changes, gravitational self-energy changes, or fixed-star gravitational-field disturbances. Perhaps, measuring equipment is large and so affects wavefunction drastically (Bohr). Perhaps, collapse is large information gain (Heisenberg).
Perhaps, wavefunction collapse is due to particle and wavefunction properties. Perhaps, previous states have lingering wavefunctions that affect later wavefunctions. Perhaps, Gaussian wavefunction distributions coincide at random. Perhaps, wavefunctions have continual operators. Perhaps, wavefunctions are unstable every billion years {Ghirardi-Rimini-Weber} (GRW), so large masses collapse immediately (Giancarlo Ghirardi, Alberto Rimini, Tullio Weber).
Perhaps, wavefunction collapse is due to quantum mechanics. Perhaps, quantum fluctuations average {quantum averaging} to make definite energy states and space and time. Perhaps, cosmic inflation caused macroscopic-size quantum uncertainty and fluctuations {quantum uncertainty}.
wavefunctions and reality
Are wavefunctions just calculating devices, or do they exist in physical reality? Why do physical laws follow mathematical laws? How does perception relate to physical laws, mathematical laws, and material world? How does wavefunction collapse relate to physical laws, mathematical laws, and material world? How does wavefunction collapse relate to wavefunction time and space changes? How can observation/measurement and wavefunctions unify into a continuous explanation, rather than a discontinuous one?
alternatives: real wavefunctions
Perhaps, classical potential and quantum-mechanical potential both exist, so wavefunction is real. Measuring real wavefunction releases energy, starts wave fluctuations, and collapses wavefunction.
alternatives: undefined and defined states
Perhaps, particles have no wavefunction, so there is no collapse. Instead of wavefunctions, particles have only defined and undefined states. Undefined states can become one defined state. For example, particle density matrices represent possible different-probability physical states. Particle moves from undefined states to one state on the matrix diagonal. However, particles can be in superposed states, which matrices cannot represent. Particles can have only one or two possible states, which matrices cannot represent.
alternatives: subquanta
Perhaps, quantum levels involve even smaller properties, or quantities that cause them. However, particles have no hidden variables and so no subquanta.
alternatives: larger whole
Perhaps, physics has another conservation law about a larger whole. Observers and instruments measure only observable parts, while other parts are not observable. Whole system, observable and not observable, is deterministic, continuous, and time symmetric. For example, objects always travel at light speed, but some are time-like, and some are space-like. However, particles have no hidden variables and so no larger whole.
alternatives: two state vectors
Perhaps, quantum states have two phase-space state vectors, one starting from last wavefunction collapse and going forward in time and the other starting from next wavefunction collapse and going backward in time (Yakir Aharonov, Lev Vaidman, Costa de Beauregard, Paul Werbos) [1989]. Before and after phase-spaces are different. At events, forward-state vector happens first, and then backward-state vector happens. Their vector product makes density matrices, allowing smooth transitions between wavefunctions and collapses. This theory gives same results as quantum mechanics with one state vector. Forward and backward effects allow consistency with general relativity. However, time cannot flow backward, by general relativity.
alternatives: positivism
Perhaps, only measured results count, and wavefunctions are non-measurable things. However, experiments involving primitive measurements demonstrate that quantum state is deterministic and unique, so wavefunctions seem to have reality.
Entangled particle wavefunctions depend on each other, maintain phase relations, and have coherence. In isolated systems with entangled particle wavefunctions, system wavefunction continues to evolve deterministically. In non-isolated systems with entangled particle wavefunctions, measurements, absorptions, collisions, electromagnetic forces, and gravitational forces disturb particles and cause entangled superposed particle states to become independent {decoherence}|. System wavefunctions become non-coherent, and particle waves no longer interfere with each other {decoherent histories}, though observers only know this afterwards. System-state phase-space vector reduces to zero. Each particle is independent and has one position and one momentum.
Non-local large-scale gravitational processes eventually collapse all system wavefunctions {objective reduction}. Particle systems cannot remain isolated, because universe gravitation is at all space points.
For macroscopic systems without observers, macroscopic observation can separate states, so system states are distinct {state distinction principle} {principle of state distinction}.
Entangled particles stay in immediate and direct contact, by sharing the same system wavefunction, over any-size space or time interval {non-locality}|. Changes in one particle immediately affect the other particle, seemingly sending information faster than light speed. Conservation laws hold, because particle travels as fast as information, and same particle can go to both detectors. Perhaps, non-locality is due to quantum-mechanical space and time being discrete, foam-like, and looping.
Particles, energies, fields, and quanta are always in space-time. Physical objects and events happen only in space-time.
Wavefunctions are abstract non-physical mathematical objects that describe possible particle or system states and their probabilities. Particle and particle-system wavefunctions are not physical forces, are not energy exchanges, and are not objects in space-time. Wavefunctions describe all space-time points simultaneously. Waves have wavelength, and so are not about only one point, but all wave points at once. Wavefunctions account for and connect all space points, and so appear infinitely long.
As particles interact (and so form an interacting-particle system), the particle wavefunctions superpose to make a system wavefunction, in which all particle states depend on each other. Because wavefunctions connect all space, particles separated by arbitrary distances have states that affect each other. If one particle changes state, the other particle instantaneously changes state, no matter how far apart in space the particles are, because the system wavefunction (and all waves) collapse at all points simultaneously. Experiments that measure energy and time differences, or momentum and position differences, show that particles can remain entangled over far distances and long times, and that wavefunction collapse immediately affects all system particles, fields, and points, no matter how distant. (Because later times involve new wavefunctions, wavefunction collapse never changes particles at same place at different times.) State-vector reduction seemingly violates the principle that all physical effects must be local interactions, because coordinated changes happen simultaneously at different places.
Particle and system wavefunctions are about particles in indefinite states. Observation of one particle's definite state instantaneously collapses the system wavefunction and puts all system particles in definite states, no matter how far apart they are. No physical force or energy at the other particle causes the definite state, but the no-definite-state simultaneously changes to definite state {action at a distance}. The cause seemingly travels faster than light speed to make an effect. Therefore, the cause is non-physical.
Physical causes and effects must occur at one event in space-time. All physical communications, forces, and energies require local interactions through field-carrying particle exchanges in space-time. Physical interactions can have no action at a distance.
theories
Perhaps, wavefunctions reflect something physical that can account for action at a distance. Perhaps, particles can travel backward in time, from measured position to previous position, to make cause and effect at same space-time point. Perhaps, wavefunctions have retrograde wave components, so particles are always interacting at same space-time point. For example, in double-slit experiments, backward-flowing waves (from detectors to incoming particles) determine particle paths and explain whether wave or particle phenomena appear. However, general relativity does not allow time to flow backward. Furthermore, space-time points cannot have different times simultaneously.
theories: no-space-time
Perhaps, every space-time point touches an abstract outside-space-time structure. Perhaps, quantum foam has no-space-time in it. Perhaps, just as all sphere points touch sphere interior, all space-time points touch a no-space-time interior. By whatever method, every space-time point communicates with all others through no-space-time. No-space-time has no distances or time intervals, so space and time do not matter, and action at a distance can occur.
No-space-time is an abstract mathematical object, just as are quantum-mechanical waves. Perhaps, no-space-time carries quantum-mechanical waves.
Before measurement, particles can be said to be everywhere {Copenhagen interpretation}|, not necessarily close to the observed position. Because particle is everywhere, measured particle is always adjacent to other system particles, so there is no non-locality.
Spin-zero-particle decay can make two entangled coupled spin-1/2 particles, one +1/2 and one -1/2, which have one coherent system wavefunction {Einstein-Podolsky-Rosen experiment} {EPR experiment}. After particle-pair production, one particle always has spin opposite to the first, by conservation of angular momentum, but observation has not yet determined which particle has which spin. If an instrument detects one particle's spin direction and collapses the system wavefunction, the other particle immediately has the opposite spin, even over long distances. Einstein, Podolsky, and Rosen said instantaneous information transmission was impossible, so particles changed to the measured spins when the particles separated. Experiments showed that both particles have no definite spin until measured, so particles had superposed states until measured. By quantum mechanics, neither particle has definite spin-axis direction, so particles have superposition of +1/2 and -1/2 states until measured.
Experimenters must choose direction around which to measure spin and can measure in any direction. If they measure opposite direction, they can observe opposite spin. Therefore, particle production alone does not determine measured spin, and realism does not happen. Measuring system and particle together, as a new system, determine measured spin.
spin detection
If two spin-1/2 particles are in singlet state, three detectors oriented at -120, 0, and +120 degree angles perpendicular to moving-particle path can measure one particle's spin. Probability that both spins have opposite values is cos^2(A/2), where A is angle.
Named things have unique values {nominal level} {level of measurement} {measurement level} {absolute, measurement}. Name and value have one-to-one correspondence. Origin and units do not matter.
Different named things have value differences {interval level}. Affine linear transformations, such as t(m) = c * m + d, where m is value and c and d are constants, maintain differences.
For many named things, values have positions {ordinal level} in order. Monotone increasing transformations maintain order.
Values have ratios {ratio level} {log-interval level}. Power transformations, such as t(m) = c * m^d, where m is value and c and d are constants, maintain ratios. Linear transformations maintain ratio relations.
Interaction with matter collapses wavefunctions {measurement postulate}.
How do wavefunctions, such as electron fields, collapse everywhere simultaneously {quantum mechanical measurement problem}. Collapse is absolute, with no relativity.
Measurements can map directly to object properties {scale, measurement} {measurement scale}. Measurement relations can map directly to object-property relations.
Perhaps, measurement theory needs special prohibitions {superselection rules} on measurements.
For objective measurement, events {observable} must be independent of where or when they happen. Objective measurements cannot be functions of space or time coordinates.
Measurements need reference points, such as x=0, and measurement units, such as meter. By relativity, objective measurements cannot be functions of reference points or units.
Measured state is orthogonal to all other possible states, because if one state happens, others do not. Measured state can be along coordinate {primitive measurement}.
Measurements in systems with no waves, or with waves with no phase differences, can have any order {commuting measurement}. Primitive measurements commute, because they are not about phase, only about yes or no. Measurements in systems with waves and phase differences depend on sequence {non-commuting measurement}. Most measurements do not commute, because they find value or probability.
subjective measurement
In quantum mechanics, time and space are not continuous but have quanta. In phase space, momenta relate to positions, and energies relate to times, so events are functions of space and time coordinates. Because positions and lengths relate to momenta, events are functions of reference points and units. Objective measurement is not possible. Quantum mechanics has only subjective measurement.
interaction
To measure particle size, light must have wavelength less than particle diameter and so high frequency and energy. High energy can change particle momentum. Higher energy increases momentum uncertainty.
To measure particle momentum, light must have low energy, to avoid deflecting particle, and so long wavelength. Longer wavelength increases location uncertainty.
Measuring position requires different-frequency light wave than measuring momentum, so experiments cannot find both position and momentum simultaneously (uncertainty principle).
wavefunction collapse
Measuring disturbs particle and creates a new system of observer, instrument, and particle, with a new wavefunction. At actual measurement, the new system wavefunction collapses to zero. Measuring allows observing only one particle property.
operator
Momentum, energy, angular momentum, space, or time functions {operator, wavefunction} operate on wavefunction to find discrete positive real values (eigenvalue) of momentum, energy, angular momentum, space, or time, which are all possible outcomes, each with probability. Direct measurements project onto space or time coordinate or energy or momentum vector.
Projection operators operate on wavefunction and project onto space or time coordinate, or energy or momentum vector, to give discrete positive real values (eigenvalue) {direct measurement}, which are all possible outcomes, each with probability. Experimenters can know possible measured values and predict probabilities. However, values may be less than quantum sizes and so not measurable. Operators have same dimensions as particle.
Alternatively, experimenters can prepare a quantum system in a known initial state, have particle interact with prepared quantum system, separate particle and prepared quantum system, and then measure quantum-system state {indirect measurement}. Indirect measurements require entangling particle and prepared quantum system, to couple their states. Wavefunction collapse puts quantum system into a state that indirectly determines particle state. Quantum system can have same or more dimensions as particle.
Operators on wavefunctions produce discrete positive real values (eigenvalue) {positive-operator-valued measure} {positive operator-valued measure} (POVM).
Operators {projection operator, measurement} on wavefunctions can project values onto measurement axis.
For elastic collisions, discrete wavefunctions can find particle-energy probability, and continuous wavefunctions can find particle-position probability {theory of double solution} {double-solution theory}. Continuous waves guide discrete particles, with discrete energies and momenta, to positions. Double-solution theory does not account for inelastic collisions.
Fluids have density and flux. Quantum mechanics is like hydrodynamical density and flow (with no rotation, no mutual interactions, and no radiation absorption), with particles in continuous fluid streamlines {fluid of wave motion} {wave motion fluid}. Density is like state probability and matter-wave amplitude. Flux is like particle speed, momentum, and energy and matter-wave frequency.
Hypercomplex algebras {Jordan algebra} can be non-associative over multiplication and describe particle entanglement.
Perhaps, wavefunctions are real and have latent positions and momenta, which measurement makes definite {latency theory}.
Projective geometry can be equivalent to a hierarchical network {lattice theory}|. Lattice theory is similar to fiber-bundle theory and similar to set theory. Hierarchical networks {lattice network} have highest node and lowest nodes. Two nodes can connect through intermediate-level nodes. Two nodes can have no connections. Projective geometry uses complex continuous functions. Lattice networks use real discrete values at lattice nodes, so calculations are simpler.
Lattice-network operations are commutative and associative, and can be distributive or not distributive.
quantum mechanics
Lattice theory is like quantum mechanics. Both are discontinuous, have intermediate states between states/nodes, and have different paths from one state/node to another state/node.
types
Node subsets can have least upper bounds and greatest lower bounds (complete lattice). Lattices can be graphs, polyhedra, or simplexes. Lattices can be quasi-ordered lattices, oriented graphs, or semilattices. Lattices can have independent branches (modular orthocomplemented lattice). Higher-dimension lattices can have vector-space factors (one-dimensional subspace), finite Abelian-group factors {cyclic component}, or combinatorial topologies.
Particles try all possible phase-space trajectories simultaneously {many-paths theory}|. States have different probabilities. Large and/or many objects have no observable deviation from average trajectories and states.
Perhaps, wavefunctions do not collapse. Universe evolves all possible wavefunction states and keeps them orthogonal and independent {many worlds} {many worlds theory} {relative state}. Observations/measurements split universe wavefunction and, after that, many independent universes continue. Wavefunctions do not collapse but have disjoint parts in new universes. Universe beginnings have definite measurements. Many universes and/or many minds exist and account for all possible wavefunction states. However, conventional probability and frequency ideas are lost. This idea does not connect independent states to show how probabilities arise. It does not show how states are orthogonal, only always entangle. It does not allow measurements in transformed coordinates, which are in fact possible.
Measurement is a process separate from unitary wavefunction evolution {measurement theory}|, because measurement causes state-vector reduction. Instruments, such as photodetectors or charge sensors, are not quantum mechanical. They detect momentum, energy, position, and time real positive values. Wavefunctions discontinuously precede and follow measurements. Measurements set initial conditions for wavefunctions.
Perhaps, particles and quanta are moving singularities in wave fields {neoclassical radiation theory}. Linear classical operators describe particles. Quadratic-interaction Hamiltonians describe fields. Both operator types couple particles to fields, allowing energy exchanges.
Zero-rest-mass-particle random motion follows Brownian-motion trajectories {path integral}. Particle wavefunction is sum of path integrals over Brownian-motion trajectories, because Brownian motion is a Schrödinger equation if time is large compared to relaxation time. Zero-rest-mass-particle systems have Gibbs-ensemble average values.
Perhaps, particles have no wavefunction, so there is no collapse. Particles really always have definite positions and momenta, but waves {pilot wave} direct them immediately throughout time and space (Bohm). However, particles have no hidden variables and so no pilot waves.
Perhaps, for measurements on two entangled spatially-separated particles, random effects always cause time delay long enough to allow information from first-particle measurement to travel to second particle before second measurement {retarded collapse} (Euan Squires) [1992]. However, retarded collapse makes measurements independent, and entangled-spin experiments show that measurements are dependent.
Finite-effect change requires finite-cause change, so particle probability-amplitude superposition is equivalent to unitary-transformation metric {unitary particle interpretation}. Unitary particle interpretation has no waves {Duane quantum rule}. Unitary particle interpretation is similar to corpuscular diffraction theory.
Assume particle probability-density function and transition-probability function. An integral equation {Fokker equation}, with random position coordinates, defines a Markov process. Fokker equation transforms to particle Schrödinger equation and indeterminacy relations.
Quantum-mechanics equations {Hamilton's equations} {Hamilton equations} relate particle positions and momenta. Potential-energy change plus kinetic-energy change equals zero, by conservation of energy. Energy conserves between kinetic-energy and potential-energy exchanges, so potential-energy change and kinetic-energy change are equal and opposite. Therefore, potential-energy change equals negative of kinetic-energy change. Potential energy depends on field position. Kinetic energy depends on momentum. Potential-energy change is energy gradient. Kinetic-energy change is momentum-change rate. Hamilton equation states that energy gradient, dH / dx, equals negative of momentum-change rate (force), dp / dt. Partial derivative of potential-energy function (Hamiltonian) with position is negative of derivative of momentum with time: DH / Dx = - dp / dt, where D is partial derivative, H is Hamiltonian potential energy, x is position, p is momentum, and t is time. Hamiltonians are wavefunctions that solve Hamilton equation.
Rearranging makes Hamiltonian potential-energy change dH equal negative of momentum change dp times position change dx divided by time change dt: dH = - dp * (dx / dt) = - m * dv * v = - m * v * dv, where v is velocity.
Rearranging makes negative of first derivative of Hamiltonian with momentum equal position derivative with time: - dH / dp = dx / dt = v. Velocity v = dx / dt equals negative of derivative of potential-energy change with momentum change dH / dp.
comparison
Hamilton's method substitutes two first-order differential equations for Lagrange's one second-order differential equation.
time
If particles are stationary, so positions do not depend on time, derivatives with time equal zero, and energy gradient equals zero, so energy is constant over all positions.
If particles move, so positions depend on time, use angle instead of position, and action instead of momentum, to find particle matter-wave frequencies and particle energies. Physical action is energy over time, so momentum is energy gradient over time. Angle indicates phase which indicates frequency, and angle varies directly with position, so position is angle gradient over time.
Quantum-mechanics equations {Lagrange equations} relate positions and velocities. Lagrange equations depend on energy conservation. Potential-energy change plus kinetic-energy change equals zero. In one space dimension, m * D((d^2x/dt^2) * dx) / Dx + m * dv / dt = 0. Because they use acceleration, Lagrange equations are second-order differential equations. Lagrange equations have same form for all three (equivalent) spatial coordinates. Lagrange equations have same form in all transformed coordinate systems, because kinetic energy plus potential energy is constant for both old and new coordinate systems.
In classical mechanics, particles have definite physical-space positions and momenta (velocities) through time. Particles have trajectories through physical space-time. For one-particle systems in physical three-dimensional space, classical configuration spaces have six continuous, infinite, and orthogonal dimensions: three for position and three for momentum. Classical configuration spaces have trajectories of successive states.
In quantum mechanics, particles do not have definite physical-space positions and momenta through time. Particle positions and momenta are functions of system energy, momenta, position, and time. Particles do not have trajectories through physical space-time but can be at any position and any momentum in physical space-time. For one-particle systems in physical three-dimensional space, quantum-mechanics configuration spaces have six continuous, infinite, and not necessarily orthogonal dimensions. Quantum-mechanics configuration-space points have scalar displacements that can vary over time. Frequency varies directly with particle energy. Adjacent-point scalar displacements vary over a wavelength. Wavelength varies inversely with particle momentum. Matter waves do not propagate or travel and so have no energy and are scalar waves. Maximum displacements (amplitudes) differ at different points, varying with system energy, momenta, position, and time. Matter waves have complex-number amplitudes because space-time has time coordinate of opposite sign from space coordinates, because of energy and momentum conservation, and because complex-number amplitudes result in constant-amplitude waves. Real-number waves travel outward and lose amplitude with distance. In the complex plane, multiplying by i rotates pi/2 radians (90 degrees). Complex-numbers represent rotation, frequency, phase, and magnitude.
Constants can be matrices.
Quantum-mechanics complex-number wave equations {Schrödinger equation}| relate energies and times. Schrödinger equations are similar to diffusion equations, but with a complex-number term, which makes them wave equations. Schrödinger equations require an imaginary term because they are about space-time and time has opposite sign to space components. Hermitian operators act on possible system-state Hilbert space to define observable quantities. Operator eigenvalues are possible physical-quantity measurements. Hamiltonian is total system-energy operator.
Isolated systems have constant total energy. By energy conservation, Schrödinger equations set constant total energy equal to potential energy plus kinetic energy. Potential energy varies with position. Kinetic energy varies with momentum. For waves, kinetic energy E varies directly with frequency f, and momentum p varies inversely with wavelength l: E = hf and p = h/l. Potential energy Wavefunction solutions represent system energy-level probabilities.
phase space
Physical systems have particles within boundaries. Particles have positions and momenta. Abstract phase space represents all particle positions and momenta. Particles deterministically follow trajectories through phase space. Particles have a succession of states (state vector) in phase space.
matter waves
Particles have matter waves. Matter waves resonate in phase space with harmonic wavelengths. Matter waves describe particle trajectories through phase space.
matter waves and particle energies
Matter-waves have frequencies, which determine particle energies. Waves must have frequency to be waves, so wave energy cannot be zero. Lowest-frequency resonating fundamental wave has lowest ground-state energy. Resonating waves also have fundamental-frequency overtones. Wave frequencies are not continuous but discrete. Particle energy levels are not continuous but discrete and separated by energy quanta. Energy-level differences decrease with higher frequency. Higher frequency waves have higher energy and have lower probability. Wave frequencies can increase indefinitely.
transitions
Schrödinger equations describe conservation of energy in particle systems and phase spaces and relate particle energies and times. Schrödinger equations have wavefunction solutions that define possible different-probability particle energy levels over time.
Schrödinger equations are about particle energy-level transitions. Particle can go from one energy level to another along infinitely many paths. For example, particle can go directly from one energy level to another {direct channel} or go to higher energy level and then drop down to lower energy level {cross channel}. Particles have matter waves, and each transition changes matter waves to a different frequency. For cross channels, net transition is superposition of matter-wave transitions.
transitions: probability
Going from one energy level to another has a probability that depends on energy difference and starting energy. Schrödinger-equation wavefunction solutions have transition complex-number amplitudes. For cross channels, total amplitude is complex-number sum of all transition amplitudes. Transition probabilities are absolute values of squared amplitudes. Squaring complex numbers makes real numbers. Absolute values make positive numbers. Therefore, transition probabilities are positive real numbers.
transitions: renormalization
Because number of paths is infinite, transition-probability sum seems infinite. However, higher frequencies have lower probabilities, so amplitude renormalization can make probability sum equal 1 = 100%.
energy
Potential energy PE is force F from field E times distance ds: PE = F * ds = E * H, where H is wavefunction. Kinetic energy KE depends on mass m and velocity v: KE = 0.5 * m * v^2 = 0.5 * (1/m) * p^2, where momentum p = m * v. Momentum squared is (h / (2 * pi))^2 times second derivative of wavefunction, because momentum squared depends on velocity squared: KE * H = 0.5 * (1/m) * (h /(2 * pi))^2 * (d^2)H / (dx)^2, where H is wavefunction, (d^2) is second derivative, h is Planck constant, p is momentum, and m is mass. Schrödinger equation sets sum of wavefunction potential-energy and kinetic-energy operators equal to wavefunction total energy operator {Hamiltonian operator}.
operators
Momentum over position, or energy over time, is physical action. Momentum and position operators, or energy and time operators, are commutative.
time
Wavefunctions can change over time (time-dependent Schrödinger equation).
frequency
Frequency is partial derivative of wavefunction with time.
spin
Schrödinger equation does not include particle spin, because waves cannot account for spin.
relativity
Schrödinger equation does not include relativistic effects, because waves cannot account for relativity.
Schrödinger-equation time evolution equals difference pattern between two phase-locked static waves {semiclassical}. If Schrödinger equation does not change with time, difference is zero. If Schrödinger equation changes with time, difference is a wave at beat frequency.
If Schrödinger equation does not change with time, space wavefunctions have finite single values in Hilbert space of complex-valued square-summable Lebesgue integrals {wave mechanics}.
Quantum mechanics can be deterministic if nature has hidden particles {hidden particle}|. Measurable particles and hidden particles superpose. Such particle ensembles have zero dispersion. Current sensitivities detect no hidden particles.
Related variables describe world. Perhaps, some variables {hidden variable}| are not measurable.
process
Inputs go to both hidden and observable variables. Hidden and observable variables make outputs, each with conditional probability. Bayesian statistics can estimate optimal variable probabilities {maximal a posteriori estimate} {MAP estimate}. Experiments show that hidden variables do not exist.
non-locality
By GHZ, hidden variables cannot be local. Local hidden variables cannot predict quantum-mechanical events correctly (Bell's theorem). If some variables are hidden, quantum physics must be non-local. Classical physics is local, so classical physics has no hidden variables.
If quantum object actions do not correlate before they interact, Bell's theorem requires that quantum physics must be non-local.
local
If quantum object actions correlate before they interact, they can correlate in past or future, and quantum mechanics can be local. Past correlation means they had common cause, but then all tiny events must have common cause, making complex metaphysics. Future correlation means future interaction itself supplies correlation. Both these cases are unlikely. Therefore, quantum object actions do not correlate before they interact, and quantum physics is non-local. Experiments (Alain Aspect) [1981] on photon spin show non-local quantum-mechanical statistical distribution. Therefore, local realist theories are incorrect.
Local hidden variables cannot predict quantum-mechanical events correctly {Bell's theorem} {Bell theorem}. If quantum object actions do not correlate before they interact, Bell's theorem requires that quantum physics must be non-local. Coupled particles have properties as predicted by quantum-mechanic entanglement, not properties predicted by independent random sums.
Fokker-Planck differential operators {density matrix} represent quantum-measurement processes. Discrete phase-space states (eigenstate) are independent and orthogonal and have real-number probabilities. States are phase-space vectors (state vector). State vectors have complex-number amplitudes, and probabilities are positive real-number absolute values of amplitude squares. State probability is tensor product of normalized state vector with complex conjugate, which eliminates phase. Tensor products are planes through complex Hilbert space. Renormalization can make sum of state probabilities equal one, and density-matrix-trace sum is one.
measurement
Measuring instruments are density-matrix projectors with one state vector, with real-number probability 1 = 100%. Product of physical-system density matrix and measuring-instrument density matrix makes density matrix with one trace value, the measurement.
transformations
Coordinate transformations do not change density matrices, because they are linear.
Quantum-mechanics theories {matrix quantum mechanics} {S-matrix theory} can use linear-equation systems, with indexed terms, to model electronic-transition energies.
transition matrix
Square matrices can represent linear-equation systems. Infinite square matrices can represent Hilbert spaces with infinitely many dimensions. Matrix rows and columns represent the same energy levels. Matrices are infinite, because particles can go to any energy level, and energy levels can go higher infinitely. Matrix cells represent possible particle-energy-level transitions and their probabilities. Matrix elements are time-dependent complex numbers in infinite Hilbert space. Squared-amplitude absolute values give probabilities of energy-level transitions.
Matrix cells include all direct and cross-channel electronic transitions. Cells (linear-equation terms) with both indices the same are for directly emitted or absorbed photons. Cells (linear-equation terms) with different indices are for cross channels.
Because transition-matrix amplitudes are renormalized, sum of all state probabilities is one. Transition matrices are mathematically equivalent to Schrödinger wave equations, because time-dependent complex numbers represent anharmonic oscillators.
quanta
Matrix cells represent discrete energy changes and so quanta. Matrices are not continuous.
deterministic
Particles move from energy state to energy state deterministically, with probabilities.
space
Transition matrices are not about space. There is no position or trajectory information.
space: no fields
Energy and momentum transfers are quanta. There are no fields.
space: uncertainty
Matrices use non-commutative symbol algebra, not wave-equation Hamiltonian-equation variables. The uncertainty principle depends on wave behavior. Non-commuting operators are certain, so matrix theory does not account for uncertainty.
time
Transition matrices can change over time.
tensor
Quantum-mechanical matrices are similar to general-relativity symmetric tensors. Hermitean-matrix principal-axis transformation is a unitary-Hilbert-space tensor. If transformation is independent of time, tensor is a diagonal matrix. However, quadratic distance form is invariant, so transformations are unitary, not orthogonal as in general relativity.
S-matrix theory additions {Regge calculus} can group hadron mesons and baryons. Hadron masses and angular momenta have groups {Regge hypothesis}. Hadron groups lie on a line {Regge trajectory} plotting angular momenta versus mass squared. Because mesons and baryons have same relation between mass and angular momentum, and both depend on quarks, their internal dynamics must be the same.
simplexes
Flat simplexes joined edge to edge, face to face, and vertex to vertex can approximate continuous space. For two-dimensional spaces, all curvature is at vertexes. For four-dimensional spaces, all curvature is at triangles. Curvature is where masses and particles are.
For hadrons, exchange-transition scattering-amplitude sum equals direct-channel-transition scattering-amplitude sum {dual resonance theory}. Hadrons are zero-rest-mass-string quantum states. String ends move at light speed. Strings can break, rejoin, rotate, and oscillate. String tension is potential energy. Quarks are at string ends, so strings are one-dimensional gauge fields. Dual-resonance theory requires hadrons {pomeron} with no quarks. Dual-resonance theory predicts infinite hadrons, with heavier masses {Regge recurrences}. Dual-resonance theory predicts that maximum temperature is 10^12 K.
Perhaps, rather than calculus of continuous variables, discrete algebra {algebraic physics} can describe physical laws using groups or matrices.
Perhaps, rather than calculus of continuous variables, spins or other quanta can be space, time, energy, and/or mass units, making discrete-number physics {combinatorial physics}.
Abstract Euclidean or non-Euclidean space {configuration space} {phase space, quantum mechanics} can have any number of dimensions and discrete or continuous points, with vectors from origin to points.
physical space and classical configuration space
Particles have center-of-gravity positions and momenta. In three-dimensional physical space, particle positions have three coordinates. Positions are real numbers, over an infinite range. In three-dimensional physical space, particle momenta have three coordinates. Momenta are real numbers, over an infinite range. Classical configuration space has six dimensions for each particle. In three-dimensional physical space, one particle has six-dimension configuration space: three dimensions for space coordinates and three dimensions for momentum coordinates. Two particles have twelve-dimension configuration space. For an N-particle system, classical configuration space has 6*N dimensions. Systems must have a finite number of particles, because universe is not infinitely big. Classical configuration space has Euclidean topology.
Phase space represents particle positions and momenta. For one particle, particle physical-space position coordinates can be the same as particle configuration-space position coordinates. For more than one particle, particle physical-space position coordinates are put on different configuration-space dimensions. For one particle, particle physical-space momentum coordinates are the same as measured in physical space at that position. For more than one particle, particle physical-space momentum coordinates are put on different configuration-space dimensions. In general, configuration space includes physical space for only one particle.
Particle positions and momenta are independent dimensions, because particles are independent. In classical physical space, a particle has a real-number density function, and particles have independent real-number density functions that add to make system density function.
To simplify, assume one particle and that the y-axis and z-axis positions and momenta are zero, so configuration space has x-axis perpendicular to x-momentum-axis. Assume that one particle moves in the positive direction along the x-axis. For no external forces and so constant momentum, configuration space has a straight-line trajectory parallel to the x-axis. For constant external force in the positive direction along the x-axis and so increasing momentum, configuration space has a straight-line trajectory with positive slope to the x-axis. For two particles under the same conditions, configuration space has four independent dimensions and two independent straight-line trajectories.
To account for rotations and angular momenta, configuration space can have three more dimensions for each particle.
quantum mechanics
In quantum mechanics, particle positions and momenta have three complex-number coordinates. Configuration space has six dimensions for each particle, but each dimension has two dependent components: real and imaginary. If particles interact, particle dimensions are not independent. For example, when processes create two photons, photon spins entangle.
In quantum-mechanics configuration space, the system density function is not the sum of particle complex-number wave functions. Quantum-mechanical configuration space has non-Euclidean topology.
states
Configuration-space points represent all possible physical-system states. Assume one particle and that y-axis and z-axis positions and momenta are zero, so configuration space has x-axis perpendicular to x-momentum-axis. Assume that one particle moves in the positive direction along x-axis. For no external forces and so constant momentum, quantum-mechanical configuration space has evenly-spaced points along a straight-line trajectory parallel to x-axis. For constant external force in the positive direction along x-axis and so increasing momentum, quantum-mechanical configuration space has unevenly-spaced points along a straight-line trajectory with positive slope to x-axis. Assume that particle is inside a box, and particle has elastic collisions with box walls, then particle has higher probability of being in the box than outside.
Number of possible states is infinite, because matter waves are infinitely long, because configuration-space dimensions are infinite. Particle positions are anywhere along dimension, because matter waves are infinitely long. Particle momenta are anywhere along dimension, because mass can increase indefinitely.
states: lattice
In continuous physical space, number of positions is infinite. Using a lattice of points, separated by a fixed distance, makes number of positions over an interval finite, for computer calculation.
time
Over time, system coordinates stay orthogonal, and states that are orthogonal stay orthogonal. Scalar products stay constant {unitary evolution, spaces}. Relations between vectors do not change.
time: steps
Over continuous time, number of times is infinite. Using time steps, separated by a fixed interval, can make number of times over an interval finite.
momentum or energy levels
Over continuous momentum or energy, number of levels is infinite. Using quanta, separated by a fixed interval, can make number of levels over an interval finite.
spin angular momentum levels
Spin angular momenta can be 0, +1/2, -1/2, 1, -1, +3/2, -3/2, and so on. For particle systems, total spin angular-momentum levels can be 0 (0, +1/2, -1/2, 1, -1, +3/2, or -3/2, and so on), 2 (+1/2 or -1/2), 3 (+1, 0, or -1), 4 (+3/2, +1/2, -1/2, or -3/2), and so on.
waves
Classical configuration space has no matter waves, because it has only real numbers and so no real-number/imaginary-number interactions. Quantum-mechanical configuration space has complex numbers and resonating matter waves. Complex-number wavefunctions represent all possible particle positions and momenta, or energies and times, and their probabilities. Matter waves cause space, time, energy, and momentum quanta and the uncertainty principle. Possible configuration-space points are possible particle states (state vector), because they are wavefunction solutions. Matter waves only relate to electromagnetic waves for a system with one photon. Matter waves are not in physical space, do not travel, and have no energy.
Mathematical spaces {complex vector space} {Hilbert space, quantum mechanics} can have complex-number vectors that originate at origin.
dimensions
Mathematical spaces can have from zero to infinite number of dimensions (coordinates), all of same type. Mathematical-space points have values for all coordinates.
vectors
Complex vectors are not lines, like real vectors, but are planes because they have two components, real and imaginary. Complex vectors can vary over time and so are waves with phase and amplitude. Phase goes from 0 to 2 * pi. Vector length is wave amplitude.
Hilbert-space vectors represent same state no matter what length, because only space direction is a physical property.
vectors: normalization
Because only direction matters, normalized vectors can all have amplitude one (unit vector), making square equal one.
vectors: scalar product
Vectors have scalar products with themselves {Hermitean scalar product}, to make squared length. Scalar products commute, so relations are symmetrical. If two coordinate vectors have scalar product zero, they are orthogonal and independent. Two vectors typically are not orthogonal, but spin states of spin-1/2 particles are orthogonal, as are integer multiples of spin 1/2.
transformations
If coordinate relations are linear, coordinate systems can transform, using translation, rotation, and reflection.
Hilbert-space states can have different coordinates {transformation theory}.
Relativity is important at high speed or gravity. Quantum mechanics is important at small distances and energies. Theories {quantum relativity} try to unite relativity and quantum mechanics.
Space-time time and quantum-mechanics time are not compatible. By uncertainty principle and complementarity principle, relativistic space-time and quantum-mechanics space-time are not compatible. In quantum mechanics, space-time is one history in superspace, with all possible histories inside, which all interact to give actual space-time. Space-time geometry has probability and phase and cannot be at any location. In relativity, physics is local, and space-time is relativistic.
fluid
From far away, fluids and crystals are continuous as in relativity, but from nearby they are discrete as in quantum mechanics. Fluids can model space-time curvature. Sound propagating in turbulently flowing fluid has similarities to light propagating in curved space-time.
fluid: black hole
Black-hole-radiation Hawking effect occurs at continuous event horizon at vacuum ground-state energy. Sound waves must have wavelength longer than distance between molecules. Hawking-effect photons start with wavelength less than black-hole diameter. Gravity pulls emitted photons, so wavelength becomes longer.
fluid: low temperature
At near-zero temperature, sounds can have phonon quanta. Flow changes are slow compared to molecular changes, so phonons have ground-state energy. In non-accelerating fluid, wavelength, frequency, and speed stay constant. In accelerating fluid, wavelength and speed increase. As wavelength approaches distance between molecules, molecular interactions cause speed in different fluids to differ. If speed stays constant {Type I behavior}, quantum effects do not matter. If speed decreases {Type II behavior}, phonons just outside event horizon can go below horizon speed and first fall in but then go out. If speed increases {Type III behavior}, phonons just inside event horizon can exceed horizon speed and escape.
fluid: surface waves
Surface waves on deeper and shallower flowing water can model event-horizon behavior.
fluid: inertia
Fluids have inertia, which affects motions. Electromagnetism has self-energy. Perhaps, inertia and self-energy relate.
unification by harmonic oscillators
In quantum mechanics, continuous fields are virtual-particle streams. Fields can carry waves. Infinite-length virtual-particle streams can be harmonic oscillators. Perhaps, quantum-mechanical waves are virtual-particle harmonic oscillators.
General relativity uses tensors to represent continuous fields. Tensors can represent harmonic oscillators. Perhaps, general-relativity tensors are harmonic oscillators.
Perhaps, harmonic oscillators unify general relativity and quantum mechanics by combining waves and quanta.
Crystals are lattices. Quantized space-times can be lattices {crystals, general relativity} {general relativity, crystals}.
Crystal defects are disinclinations or dislocations. Dislocations are disinclination and anti-disinclination pairs. Disinclinations are dislocation series.
Zero-curvature space-time lattices have no crystal defects. Curved space-time lattices have disinclinations. Space-time lattice torsions have dislocations (line defects such as edge and screw dislocations).
In crystals, dislocations are disinclination and anti-disinclination pairs, and disinclinations are dislocation series, so crystal curvature and torsion are interchangeable. Perhaps, force fields are series of units, and units have disinclinations. However, general relativity does not allow torsion, only curvature.
Perhaps, gravity is weak because it involves shorter unit distances than electromagnetism.
At 10^16 GeV, all forces except gravitation are equal in strength. At 10^18 GeV, all forces are equal in strength. Why is this unifying energy so high {hierarchy problem}?
Nozzles {Laval nozzle}, such as rocket nozzles, can have narrowing, in which fluid exceeds sound speed but makes no shock wave. Narrowing pushes sound going upstream back. Original sound wavelength is distance between molecules. Above boundary, pushing back increases wavelength. Below boundary, pushing back makes sound faster than it can travel. At boundary, at near-zero temperature, sound emits thermal-phonon pairs. One pair member can go up flow, and one down flow. At near-zero temperature, narrow region acts like black-hole event horizon.
At Planck length, space-time is energetic and discontinuous and has nodes, loops (quantum loop), kinks, knots, intersections, and links {quantum foam}, depending on spins.
Weak force and electromagnetic force can unite with special relativity {quantum electroweak theory}. Field has photons and has Z and W particles, not force lines. Field can change from photons and Z and W particles to particles and back. Weak force has symmetry.
Perhaps, space-time averages all possible 4-simplex matter-wave superpositions {Euclidean quantum gravity}. If space and time are equivalent dimensions, time has no direction, and physics has no causality. If space and time are not equivalent dimensions, time has direction, and physics has causality, so simplexes connect {causal dynamical triangulations}.
Quantum mechanics can unify with general relativity {quantum gravity}|. Quantum gravity is unitary.
gravity
Gravity curves space-time, and space-time determines mass motions {Wheeler-DeWitt equation}. Gravitons and interactions among gravitons determine curvature, but interactions are small if curvature is much larger than Planck length. Interactions take all possible paths, because no information is available about interaction.
gravity: metrics
For cosmology, measurements must be from within and so local. Metrics can have no singularities. Euclidean metrics can be local and can have two types, connected and disconnected.
Connected metrics are broad bounded space-time regions, with a local measurement region. Connected metrics have a boundary, and so boundary conditions. Connected metrics have few paths.
Disconnected metrics are compact unbounded space-time regions, with all local measurements. Disconnected metrics have no boundary, and so no boundary conditions. Disconnected metrics have almost all paths.
wavefunction
Universe wavefunction determines particle positions and depends on three spatial-dimension metrics and on particle. It does not depend on time, because compact metric has no preferred time. It does not depend on coordinate choice, becasue coordinates are equivalent.
Observers can see only part of space, so universe has mixed quantum state, which implies decoherence and classical physics. Superpositions do not happen, because gravitational effects cancel superpositions.
Fermions have Fermi-Dirac statistics, and bosons have Bose-Einstein statistics, and there are no other particle types {spin statistics theorem}, because quantum field theory functionals either commute or anti-commute.
To relate fermions to bosons, theories {supergravity}| can use three spatial dimensions, one time dimension, and seven more spatial dimensions to form high-curvature and high-energy-density seven-spheres. Supergravity is supersymmetry using curved spatial dimensions, seven curled-up dimensions, and gravity.
To describe phenomena that involve massive objects at short distances, such as black holes and Big Bang, theories {theory of everything}| {final theory} must unite general relativity and quantum mechanics. String theory derives from quantum mechanics.
Quantum mechanics can combine with general relativity to make quantum field theory {relativistic quantum mechanics}| {quantum field theory}. Relativistic quantum mechanics accounts for all force types, allows particle creation and destruction, is invariant under Lorentz transformations, requires negative energy levels, and predicts antiparticles. Quantum-field theories modify relativity with quantum mechanics and include quantum electrodynamics, quantum chromodynamics, and grand unified theories.
Non-relativistic quantum mechanics does not require particle spin and does not require Hilbert space. By relativity, observed values cannot affect each other faster than light. Relativistic quantum mechanics requires Hilbert space. In (relativistic) quantum field theory, functionals of quantum fields either commute or anti-commute, because otherwise they would interact faster than light. Relativistic quantum mechanics requires particle spin, to allow commutation and anti-commutation. Fermions anti-commute, and bosons commute. In (relativistic) quantum field theory, these are the only allowed particle types. Other non-commutative relations allow faster than light affects, because of their other components. Relativistic quantum-mechanics operator commutation properties determine Pauli exclusion principle. (Non-relativistic quantum-mechanics operator commutation properties determine Heisenberg uncertainty principle.)
Electromagnetic waves are vector waves, but non-relativistic quantum-mechanics wavefunctions are scalar waves. Scalar waves have no polarization, so non-relativistic quantum-mechanics wavefunctions cannot represent spin. Relativistic quantum-mechanics wavefunctions are scalar waves with spinors and so are vector waves. Vector waves have polarization and can be plane-polarized or circularly polarized, and spin applies to circular polarization. Relativistic quantum-mechanics wavefunctions can represent particle spin. Circular-polarization rate represents particle spin.
Theories {unified field theory}| try to unite all forces and particles. Strong, weak, and electromagnetic forces unify at 10^28 K at distances of 10^-31 meters, when universe was 10^-39 second old, if supersymmetry is true and superpartners exist. Weak and electromagnetic forces unify at 10^15 K.
Theories {grand unified theories}| {Grand Unification} (GUTS) use a new gauge boson that affects both quarks and leptons and so unifies strong and electromagnetic forces.
requirements
Complete unified theory must have perfect symmetry at high temperature, high energy, and short distances and have different and lower symmetry for current universe. Theory must relate three quark and lepton generations {horizontal symmetry}. Maintaining symmetry to preserve conservation laws requires forces.
First symmetry loss creates the twelve hyperweak-force bosons. Next symmetry loss creates the eight strong-force gluons. Next symmetry loss creates the three weak-force intermediate vector bosons. These symmetry losses give bosons their masses.
unity
Particles can have inner electric field surrounded by region with particle creations and annihilations that decrease field. Inner electric field is stronger than electromagnetism and decreases by less than radius squared.
Particles can have inner strong or weak force field surrounded by region with particle creations and annihilations that increase field. Inner field is weaker than strong or weak force and decreases by more than radius.
Decrease of strong nuclear forces and increases of electric and weak forces can meet to unify all forces.
weak and strong forces
Rotation between weak and strong forces became constant when symmetry broke at an angle {Cabibbo angle}.
weak force and electromagnetism
Weinberg-angle coupling constant for isospin and electroweak hypercharge has value close to that predicted by grand unified theory.
Strong nuclear force can unite with special relativity {quantum chromodynamics}| (QCD).
color
Long-range color force causes short-range strong nuclear force. Like electric charge, color conserves.
electric charge
Particles with integral electric charge have no color, because their colors add to white or black. Particles with fractional electric charge have color, because their colors do not add to white or black. For example, pions have up quark and down antiquark, so charge is -1 (-2/3 + -1/3), and color and complementary color add to white. Protons have two up quarks and one down quark, so charge adds to +1 (+2/3 + +2/3 + -1/3), and colors red, green, and blue add to white. In particles, two up quarks must have different colors, because same colors repel.
strength
Close quarks interact weakly, because net color is zero. Farther quarks interact more strongly, because net color is more.
free quarks
Fractional-charge colorful particles cannot exist by themselves, because they cannot break free of strong force. For high energy and temperature, distances are short, and quarks and gluons do not strongly interact {asymptotic freedom}.
vectors
Quantum chromodynamics uses three complex gauge-field vectors, for red, green, and blue, and so is non-Abelian. Cyan, magenta, and yellow are vectors in opposite directions. Colors add by vector addition, so vectors make a color wheel in complex plane.
gauge
Quantum chromodynamics is a hadron gauge theory and uses the SU(3) symmetry group. Strong force has symmetry, because quark color does not matter, only net color.
strong-force exchange particle
Strong-force field has gluons, not force lines, and can change from gluons to particles and back.
lattice
Three-dimensional lattices can approximate continuous space as discontinuous nodes. Nodes represent possible quark locations. Paths between nodes represent quark interactions, and lattice lines are forces connecting quarks. Because strong force is constant with distance after short distance, number of lines between two quarks is constant.
string theory
Strings in five-dimensional dynamic space, and particles in four-dimensional boundary of QCD-force space, have equivalent mathematics. When QCD forces are strong, strings interact weakly. In string theory, QCD viscosity is like black-hole gravity-wave absorption.
Electromagnetism can unite with special relativity {quantum electrodynamics}| (QED) {relativistic quantum field theory}. From electron charge and mass, quantum electrodynamics can predict all charged-particle interactions. Quantum electrodynamics describes electromagnetic photon-electron/proton/ion interactions using quantum mechanics. Possible paths have amplitudes and probabilities. Path number is infinite, but some cancel and some end (sum over histories). Feynman diagrams illustrate paths.
field
Electric field has photons, not force lines. Electromagnetic force has symmetry.
photons
Photons are electric-field excitations. Sources emit photons, and sinks absorb photons. Field can change from photons to particles and back.
quasiparticle
Electrons {quasiparticle, electron} move through material with higher or lower mass than rest mass, because they interact more or less with material electric fields. Electrons moving at relativistic speed tunnel through barriers {Klein paradox}. Electrons {Dirac quasiparticle} moving at relativistic speeds have low effective mass, because they have accompanying virtual antiparticles, which subtract mass, that materialize from vacuum. In vacuum, time is short, so frequency and energy are high enough to make particle-antiparticle pairs. Antiparticles attract to fields that repel particles, so Dirac quasiparticles tunnel.
string theory
String theory derives from quantum-electrodynamics approximation methods {perturbation theory}.
special relativity
Quantum mechanics can combine with special relativity, for use in flat space-time or in time-independent space-time. Time can include imaginary time, which rotates time axis {Wick rotation} and transforms Minkowski into Euclidean space. Gravitons have features that are not gravitational-field excitations.
At energy levels that are low compared to interacting-particle mass, forces are negligible {effective field theory}. Gravitation has negligible force.
Quantum electrodynamics, quantum chromodynamics, and quantum electroweak theory form unified theory {particle physics standard model} {standard model of particle physics} {standard theory}.
particles
Quarks, leptons, and intermediate vector bosons are wave bundles in fields. Top quark has 175 GeV. Proton has 1 GeV.
Why are there three particle generations, rather than just one? The first generation makes consistent theory with need for higher-mass particles.
Particle masses, charges, and spins relate by the Yang-Mills gauge group in the particle Standard Model. That gauge group is the direct product of the Special Unitary group for three gluons, Special Unitary group for two intermediate vector bosons, and Unitary group for one photon: SU(3) x SU(2) x U(1). Therefore, the Yang-Mills gauge group has SU(3), SU(2), and U(1) as subgroups. SU(3) is for strong-force quark and gluon color, is non-Abelian, and has no invariant subgroups, so its matrix is traceless. SU(2) is for weak-force pion and W-and-Z boson strangeness, is non-Abelian, and has no invariant subgroups, so its matrix is traceless. U(1) is for electromagnetic electron and positron electric charge and is Abelian and normal. Unitary groups have unitary square matrices, as generators. Special groups have square-matrix determinants = 1.
field
Standard theory is renormalizable quantum-field theory. Quantum-field theory is for energies that are high compared to particle mass, so it is not about gravitation.
gauge symmetry
Only quantum differences are important, not absolute values.
gauge symmetry: renormalization
Redefining 18 physical constants {renormalizable} can remove infinite quantities.
other forces: mass
Gravitation is about mass. Standard Model does not predict quark and lepton masses, unless it adds a scalar field. Scalar field probably has quanta and so Higgs particles, with masses of 100 to 300 GeV.
other forces: supersymmetry
Perhaps, a new force allows protons to be unstable with half-life 10^31 to 10^34 years. Perhaps, new force gives mass 10^-11 GeV to neutrinos.
In quantum-field theories, matter positive frequencies can go forward in time, and antimatter negative frequencies can go backward in time {twistor, quantum mechanics}| (Penrose). In Minkowski space, twistors are spinors and complex-conjugate spinors.
Riemann sphere
Complex numbers graph to planes. Plane can be at Riemann sphere equator. Pole point can be at infinity. Line from pole through plane can intersect Riemann sphere. Real numbers are on equator. Positive frequencies are in upper hemisphere. Riemann sphere is twistor space. Twistor space has two plane dimensions and three space-time-point dimensions. Adding spin makes six real dimensions {projective twistor space}.
space-time and quantum mechanics
Perhaps, general relativity and quantum mechanics unify using twistors. Space-time relates to quantum-mechanics complex amplitudes through Riemann spheres. Riemann-sphere space-time points have light-ray sets. Space-time events are Riemann-sphere directions, showing which past events can affect future event. In twistor space, light rays are points, so twistor space is not local. Photons have right or left circular polarization {helicity}. Half-spin particles have up and down spin superpositions, as observer sees Riemann sphere. Riemann spheres can have inscribed icosahedrons, which define 20 sphere points. Points join three edges, which can be like three space dimensions. Points combine two independent entangled fermion spins, with spin +1/2 or -1/2. Riemann tensor has 20 components in flat space-time. Perhaps, complex numbers can relate general relativistic space-time to spin quantum mechanics [Penrose, 2004]. At different velocities, transformation groups {Möbius transformation, twistor} can find curvature.
Quantum mechanics can combine with special relativity {gauge theory}.
boson
Forces have force fields and exchange bosons. Bosons are quanta. Field quanta are bosons. Gauge transformations are boson exchanges. Boson exchange carries energy and momentum quanta between fermions. Field is for relativity, and quanta are for quantum mechanics.
Higgs particles are bosons that generate masses for particles. Hadrons are bosons in multiplets for charge and isotopic spin.
groups
Conservation laws determine symmetries and gauge transformations, which form mathematical groups. Quantum electrodynamics is lepton gauge theory and uses symmetry group U(1). Quantum chromodynamics is hadron gauge theory and uses symmetry group SU(3). Electroweak theory [1973] is gauge theory for weak interactions and electromagnetism and uses symmetry group SU(2) x U(1).
Symmetry {gauge symmetry}| requires that only quantum differences are important, not absolute values.
Continuous point sets are manifolds {base space}. Manifold points can have internal spaces {fiber space}, with internal dimensions {fiber, mathematics}. Fiber spaces are manifolds. Fibers do not intersect. Fibers project to points {canonical projection}.
fiber bundles
Combined base and fiber space {fiber bundle}| {bundle} has dimension number equal to sum of fiber-space and base-space dimensions. Base space can be curve. Curve points have line tangents to curve. Tangents are fiber spaces.
Curved-surface points have planes tangent to surface. Tangent planes are fiber spaces.
vector bundle
Fiber spaces can be vector spaces {vector bundle}.
twisting
If fiber spaces are the same for all base-space points, base space and fiber space can make product space {untwisted bundle}. If fiber spaces are not all the same, base space and fiber space can make a symmetrical locally untwisted product space {twisted bundle} with a mathematical group. For example, particle spins can be fiber bundles. Base-space spins go to fiber-space phase relations.
curvature
Curvature can be connections between fibers in fiber bundles, with rule {path-lifting rule} for getting to fiber-space point from base-space point.
gauge fields
Gauge fields can be connections between fiber-bundle fibers. Bundles can have locally constant values {bundle connection}, which are like gauge connections. Connections represent field phase shifts {path lift}.
tangent bundle
Base spaces can have tangent vectors as fiber spaces {tangent bundle} or covectors as fiber spaces {cotangent bundle}. Base spaces can be two-dimensional spheres. Fiber spaces can be circles. Bundles {Hopf fibration} {Clifford bundle} can be three-dimensional spheres.
Quantum mechanics can combine with general relativity by gauge-theory extension {relativistic gauge theory}. Base field or space represents physical space-time events. Total field or space represents quantum wavefunctions or symmetry transformations. Base-space points project to total-space points to make fibers.
Perhaps, fermions and bosons can interchange using a new force {technicolor theory}| {supersymmetry} {Supersymmetric Standard Model} (SSM). Fermions and bosons have quarks, which are fermions. Supersymmetry unites half-integer-spin fermions and integer-spin bosons.
fermion
Particles with odd number of quarks are fermions, which have half-integer spins. Fermions have negative ground-state energy.
boson
Particles with even number of quarks are bosons, which have integer spins. Bosons have positive ground-state energy.
stability
Fermion-boson interaction can cancel ground-state energies, leaving small stable energies.
force
Fermions and bosons can have a new force. The new exchange particles have 1000-GeV energies, with range from 10^2 GeV to 10^16 GeV. Because force strength depends on particle energy, the new force is the strongest force.
spin: superpartner
Particles pair with massive superpartners with spin 1/2 more or less than particle spin. Fermions have boson superpartners, such as squark, sneutrino, and selectron. Bosons have fermion superpartners, such as gravitino, higgsino, photino, gluino, wino, and zino.
spin: change and symmetry
Perhaps, besides space, time, and orientation symmetries, angular-momentum components {spin symmetry} can unite all forces and particles.
spin: space dimensions
Supersymmetry spin change requires extra spatial dimensions {Grassmann dimension, spin}.
spin: symmetry
Supersymmetry uses graded Lie algebra {superalgebra}.
detection
Instruments have not yet detected superpartners or fermion decay to bosons. Perhaps, universe origin had supersymmetry but universe now has broken symmetry.
hierarchy problem
At 10^16 GeV, all forces except gravitation are equal in strength. At 10^18 GeV, all forces are equal in strength. Why is this unifying energy so high (hierarchy problem)? Supersymmetry uses high energies and can resolve this problem.
supergravity
Supersymmetry applies to flat space-time Yang-Mills-field strong and weak nuclear forces and to electromagnetic fields, but can extend to gravity.
Standard Model
Supersymmetry can add to Standard Model. Standard-Model particles have superpartners {Minimal Supersymmetric Standard Model}.
In a supersymmetry model {interacting boson model}| [Arima and Iachello, 1975], atomic nuclei can have nucleon pairs. Even numbers of protons and neutrons, as in platinum, can have three dynamical-symmetry classes. Even numbers of protons and odd numbers of neutrons, and vice versa, and odd numbers of protons and numbers, relate to even-even case. Interacting bosons make nuclei behavior independent of particles and of special relativity, except for mass. If boson and fermion numbers are constant, supersymmetry can predict odd-odd case for heavy atoms, such as gold 196 with 79 p and 117 n, which has doublet ground state.
Particles have massive paired particles {superpartner}, with spin 1/2 more or less than particle spin.
Supersymmetry requires extra dimensions {Grassmann dimension, supersymmetry}.
At Planck scale, space-time is quantum foam and has nodes, loops {quantum loop}, kinks, knots, intersections, and links, depending on spins. A loop theory {quantum loop theory}| {loop quantum gravity} can represent quantum-foam loops and their kinks (but not intersections, links, and knots).
Quantum-loop lengths are multiples of Planck length. Quantum-loop areas are multiples of Planck area. Quantum-loop volumes are multiples of Planck volume.
Because spins can transfer, quantum loops can interact. Force fields have interacting quantum loops.
Quantum loops define space dimensions (background independent). Perhaps, space is intertwined quantum loops.
Interacting quantum loops can make fractals. Perhaps, space-time has fractal structure at Planck lengths. Fractals can remove the infinities that appear in relativity and quantum mechanics.
Quantum loop theory derives from general relativity and adds quantum mechanics. Quantum loop theory uses ideas of supergravity, twistor theory, string theory, and non-commutative geometries.
At quantum distances, quantum loops have repulsion.
photons
Loop quantum gravity predicts that high-energy photons, such as gamma rays, have faster speeds than low-energy photons, such as infrared rays. A test of this hypothesis is to measure if the microwave background radiation scatters photons. If photons all have same light speed, they scatter, but if some have faster speed, they do not scatter.
Quantum loop theory does not assume space-time existence {background independence, space}. Quantum loops determine matter and energy, which cause space-time geometry. In contrast, string theory assumes space-time (background dependence).
Behavior can be the same in any coordinate system {diffeomorphism invariance}.
In networks {spin network}|, nodes are space states, and edges connecting nodes are state transitions. Particle motions are translations from node to node, with possible momentum and angular-momentum changes. Transition series travel along edges {path, graph} {graph, path}. Therefore, spin networks are extensions of quantum mechanics and describe space-time quantum states and their transitions. They can approximate all space-time geometries, quantum states, and motions. Spin networks are about a single time.
nodes
Different node types represent different particles and their positions and angular momenta. Simple nodes are volume quanta. Networks have total volume equal to total angular momentum.
edges
Different edge types represent different fields, energies, or forces between masses. Simple lines between nodes are area quanta. Edge lines indicate momentum.
quantum loops
Spin networks can represent linked quantum loops and their possible kinks. Therefore, spin networks can approximate quantum foam. However, spin networks cannot represent quantum-foam intersections, links, or knots.
Spin networks can evolve in time {spin foam}|. Time is in multiples of Planck time. Spin-foam lines are nodes plus one quantized time dimension, and represent space-time states. Spin-foam surfaces are edges plus one quantized time dimension, and represent space-time state transitions. Spin-foam nodes are where spin-foam lines intersect and represent quanta. Spin foams are extensions of quantum mechanics and can represent electronic transitions and particle interactions. However, spin foams cannot represent quantum-foam intersections, links, or knots.
One-dimensional vibrating loops or line segments {string, physics} account for matter, force, and energy. Strings vibrate at specific frequencies, amplitudes, phases, and modes {superstring theory}| {string theory} to represent particles.
length and mass
String lengths are multiples of Planck length. Uncertainty principle requires strings to be longer than Planck length.
particles
Strings are fundamental particles and have no sub-structure or sub-particles. String vibration modes make elementary particles. String theory requires an infinite series of elementary particles with increasing masses. High-mass elementary particles are unstable. Particle interactions merge two strings into one string or split one string into two strings.
particles: dilaton
Force strengths depend on string 11th-space-time-dimension length (dilaton). Short dilatons represent weak nuclear forces. Long dilatons represent strong nuclear forces. Dilaton lengths represent electromagnetism, and dilaton length variations change electromagnetic fields.
Before universe origin, dilatons are long, and forces are strong. At universe origin, dilatons are short, and forces are weak. Observing intergalactic magnetic-field changes is a test for dilatons and so can indicate universe-origin conditions.
mass
String mass is proportional to string length. Shorter strings have higher vibration frequencies and so higher masses.
waves and resonance
Strings vibrate at light speed as high-frequency waves. Shorter strings have higher vibration frequencies. String waves dampen Planck-scale quantum fluctuations. Waves have Planck lengths and so are not observable. String-wave resonances make particles and forces.
space
Strings have dimension and vibrate, so strings occupy space. Strings require background space.
energy
Strings have high energy and mass, because they are stiff vibrators, with 10^39 tons tension {Planck tension}. However, quantum fluctuations decrease this energy.
In zero-rest-mass particles, string rest masses cancel, leaving low-energy vibrations. Zero-rest-mass strings have relativistic mass and so can have momentum and angular momentum.
In elementary particles with mass, almost all string rest masses cancel.
spatial dimensions
Uncertainty principle causes strings to have a smallest length, equal to Planck length. Strings occupy space because they vibrate.
String waves vibrate in three extended spatial dimensions, seven curled-up Planck-size spatial dimensions (Joyce manifolds), and one time dimension. Physical scalar and vector fields determine the number and properties of infinite and curled-up dimensions. String waves can vibrate in dimensions, wrap around dimensions, and travel around dimensions. Winding around large dimensions takes more energy, because string stretches. Moving around large dimensions takes less energy, because frequency is less. The effects balance each other.
spatial dimensions: winding
In a curled-up dimension, string vibrations occur along the dimension (dimensional vibration) and go around the dimension {winding mode}. They can go around more than once. Curled-up dimensions cannot be smaller and their vibrations have a minimum, so waves have quantum size and energy.
spatial dimensions: size
Curled-up dimensions have two radii, one across curl and one around dimension. Because vibrations increase as quantum fluctuations increase, if one dimension becomes smaller, other dimension becomes larger. Winding-mode vibrations depend directly on dimension radius. Dimensional vibrations depend inversely on radius. Because winding mode vibrations and dimensional vibrations are reciprocal, curled-up dimensions have same physics as dimensions with exchanged winding-mode and dimensional-vibration radius.
spatial dimensions: physical forces
Electromagnetic waves, matter waves, gluons, and W and Z particles travel only in three-dimensional space, not curled-up dimensions. Gravity waves can travel in all dimensions.
interactions
Strings can split into two strings, or two strings can merge into one string. Splitting and merging probabilities depend on a positive-number constant {string coupling constant}. Constant is less than one for weak coupling or greater than one for strong coupling.
finite
String theory has finite quantities and so removes field-theory infinity problems and quantum-mechanics renormalization problems. (However, eleven-dimensional quantum-field theory can have finite particles and quantities.)
strings
Strings are one-dimensional Planck-length-multiple line segments or circles, and M-theory p-branes are multi-dimensional Planck-length-multiple areas. Particles are Planck-length-multiple strings that harmonically vibrate in dimensions.
strings: general relativity
At greater-than-Planck-length distances and larger-than-energy-quantum energies, string theory and general relativity have same form.
strings: quantum mechanics
String-vibration wave equations have harmonic-frequency wavefunction solutions. Discrete wave frequencies represent energy quanta, which account for particle masses. String wavefunctions are essentially the same as particle quantum-mechanical wavefunctions.
strings: dimensions
Space has three infinite dimensions and some number of compactified dimensions. Compactified dimensions have relations.
strings: vibration components
As fundamental units, strings have no internal parts, structure, or forces. Because longitudinal vibrations require internal parts, structure, or forces, strings have no longitudinal vibrations.
Strings vibrate transversely across all dimensions, so vibrations have many components. For curled-up dimensions, vibration frequency varies directly with radius inverse, so shorter strings have wave higher frequencies. For curled-up dimensions, strings can also vibrate transversely around the dimension (wind), and winding-vibration frequency varies directly with radius, so longer strings have higher winding-wave frequencies. Therefore, winding vibrations are duals to the other transverse vibrations.
In each dimension, strings vibrate at resonant frequencies, determined by string length and tension. Vibration components have complex-number frequencies, which determine real and imaginary wave-energy components, which determine real and imaginary particle-mass components. Because complex-number operations can result in positive or negative values, superpositions of wave components can make positive or negative wave energies and positive or negative particle masses. Quantum energy fluctuations can also be positive or negative, so virtual particles can have positive or negative masses.
Strings cannot have zero length, because then they are not strings, which must have tension between endpoints. Strings cannot have no vibrations, because they must have tension and endpoints and so fundamental frequency.
strings: particles
String theory accounts for all Standard-Model particles.
Electric charge is at open-string endpoints or spread around closed string. Zero-rest-mass oriented open strings are U(n) bosons. Photons are open strings, are vectors with spin 1, and have wave amplitude zero. For local interactions, photons have clockwise or counterclockwise transverse-wave amplitude-vector rotation around oriented-string long axis to account for integer spin. Photons have no mass, so strings have zero-point lowest-energy state. Photons have photons as antiparticles. For global interactions, electric charge can be positive or negative and that corresponds to orientation, requiring otherwise-same opposite oriented open strings or otherwise-same clockwise and counterclockwise motions around closed strings. Electric charge does not have anti-charge, only exactly opposite positive and negative charge. Electrons and positrons have clockwise or counterclockwise transverse-wave amplitude-vector rotation around long axis to account for half-integer spin.
Strangeness is at open-string endpoints or spread around closed string. Vector bosons are open strings and are vectors with spin 1. Strangeness can be zero or one, corresponding to absence or presence, requiring non-opposite oriented open strings or non-opposite clockwise and counterclockwise motions around closed strings (violating parity). Strangeness does not have anti-strangeness. For local interactions, Z intermediate vector bosons have clockwise or counterclockwise transverse-wave amplitude-vector rotation around oriented-string long axis to account for integer spin. W intermediate vector bosons have both clockwise and counterclockwise transverse-wave amplitude-vector rotation around oriented-string long axis to account for zero spin. Intermediate vector bosons have mass, so strings have intermediate-energy state. Intermediate vector bosons have intermediate vector bosons as antiparticles. For global interactions, pions have clockwise or counterclockwise transverse-wave amplitude-vector rotation around oriented-string long axis to account for half-integer spin.
Color charge is at open-string endpoints or spread around closed string. Gluons are open strings and are vectors with spin 1. Color has three vectors that add to zero, corresponding to an equiangular triangle, requiring complex-number oriented open strings or complex-number clockwise and counterclockwise motions around closed strings. Colors have anti-colors. For local interactions, gluons have clockwise or counterclockwise transverse-wave amplitude-vector rotation around long axis to account for integer spin. Gluons have mass, so strings have high-energy state. Gluons have gluons as antiparticles. For global interactions, quarks have clockwise or counterclockwise transverse-wave amplitude-vector rotation around long axis to account for half-integer spin.
Gravitons are closed strings, are symmetric tensors with spin 2, and have wave amplitude zero. Mass-energy is positive and scalar. Mass has negative scalar anti-mass. For local interactions, gravitons have two clockwise or counterclockwise transverse-wave amplitude-vector spins, one around each tensor axis. Gravitons have no mass, so strings have zero-point energy. Gravitons have gravitons as antiparticles. Fermion and boson waves together make tensor that has one symmetry, which makes tensor gauge and so closed string with spin 2.
Dilatons are closed strings and are scalars with spin 0. Axions are closed strings and are antisymmetric tensors with spin 0. Zero-rest-mass unoriented open strings are SO(n) or Sp(n) bosons.
strings: virtual particles
At small distances, string-theory quantum mechanics allows virtual particles. Strings always change to strings, never to no strings, because, by uncertainty principle, zero-length strings have infinite energy. The no-string (vacuum) state cannot exist.
One zero-point-energy string can become two virtual-particle strings. Two virtual-particle strings can become one zero-point-energy string.
Strings preserve all symmetries and conservation laws.
Real-particle strings have longer lengths, lower energies, and longer lifetimes. Virtual-particle strings have short lengths, high energies, and short lifetimes.
strings: particle properties
String (Planck-multiple) lengths and (high) tensions determine transverse vibration modes and account for particle energies, masses, rotations, and other properties.
String endpoints rotate around center, or closed strings rotate, so strings account for particle spin. String orientations that differ only in direction can represent clockwise and counterclockwise spin. String orientations that differ in direction and other properties can represent parity or no parity. Closed strings can account for zero-rest-mass spin-2 particles (graviton).
matrix theory
Perhaps, space is intertwined strings or zero-branes {matrix theory} and so is not background-independent.
quantum loop
Perhaps, quantum loops are the background for strings. Larger loops can be strings. Perhaps, strings are waves in spin networks.
comparison to points
Strings have one dimension, are Planck length or higher, and have waves. Points have zero dimension, are smaller than Planck scale, and have no waves.
Point particles {point particle} are zero-dimensional points with quanta. Point particles have quantum-mechanical waves. Point particles preserve symmetries and conservation laws.
virtual particles
Point-particle quantum mechanics allows virtual particles. Two virtual point particles can appear (particle creation) from space-vacuum energy (negative-energy-particle field) fluctuations. Two virtual point particles can become space-vacuum photon energy (particle annihilation).
energy
Because point particles have only one point, and space has no compactified dimensions, point particles have only positive real-number energies and masses. Point particles have no size and do not change size, so they cannot have infinite energy.
particle properties
Point particles have no mechanism for particle spin, orientation, or other properties and no mechanism to make spin-2 bosons (gravitons).
Point particles can have any rest mass and so can have zero rest mass.
space dimensions
Point particles have one point of zero dimension and so do not require unobserved dimensions. Point particles have no mechanism to specify number of space-time dimensions.
Strings can be closed loops {closed string} or have ends that freely move {open string}. Open-string ends have boundary conditions {Dirichlet boundary conditions}, typically different for different dimensions. For example, electron strings move in the three infinite dimensions but do not move in the seven curled-up dimensions.
High-speed strings appear to have discrete line segments {string bit}. String bits have quantized length, energy, momentum, and angular momentum. String-bit minimum length is Planck length. Planck length string bit has minimum momentum. Longer string bits have multiples of Planck length and multiples of minimum momentum.
Expanded string quantum-field theory {M-theory}| adds vibrating disks, blobs, toruses, and higher-dimension branes, to explain elementary particles, forces, and energies. M-theory branes create space-time (background independence). M-theory has ten space dimensions (three infinite and seven curled up) and one time dimension. People do not yet know M-theory physical principles.
Spaces can be lattices {Wilson's loops} {Wilson loops}. Particles are at nodes. Gravity, electric, and strong-force field lines are on lattice lines between nodes. For electric fields, field lines diverge, decrease with distance, and approximate continuity. For color-charge fields, field lines stay apart and have constant force.
String theory assumes space-time {background dependence}, in which strings move. M-theory branes create space-time, so M-theory has background independence.
Space-time points have zero dimension and do not vibrate, so real-number coordinates represent them. Strings and quantum loops have one dimension and vibrate. Strings and quantum loops have uncertain positions and motions, by the uncertainty principle. Complex-number matrices can represent string (and quantum-loop) positions and motions. Complex-number matrix operations do not commute {non-commutative geometry}. At large scales, diagonal matrices can approximate complex-number matrices, and diagonal matrices have commutative operations (commutative geometry).
Force strengths depend on no-spin closed-string 11th-space-time-dimension scalar length {dilaton}. Short dilatons represent weak nuclear forces. Long dilatons represent strong nuclear forces. Dilaton lengths represent electromagnetism, and dilaton length variations change electromagnetic fields.
Before universe origin, dilatons are long, and forces are strong. At universe origin, dilatons are short, and forces are weak. Observing intergalactic magnetic-field changes is a test for dilatons and so can indicate universe-origin conditions.
Magnetic-field photons can make dilaton-related antisymmetric-tensor no-spin closed-strings {axion} that have less than one millionth electron mass, no charge, and zero average quantum field. Magnetic-field axions can make photons. Therefore, axions allow strong nuclear forces to maintain charge-parity (CP) symmetry between antiparticles and particles.
Cosmic-microwave-background temperature fluctuations are small, have Gaussian distribution, and have same amplitude for large space regions. Cosmic-microwave-background temperature fluctuations arise mostly from density differences and partly from gravity waves. However, string theories without axions allow no density differences. Axions determine large-scale universe temperature fluctuations [Adams, 2002].
Strings, disks, blobs, toruses, and higher-dimension objects {brane} can vibrate.
Rather than strings, fundamental elements can be membranes {supermembrane} {2-brane} {3-brane} {p-brane}|, of dimension 2, 3, or any natural number p. Minimum length is Planck length. Branes vibrate and have high-energy waves. Branes require ten space dimensions (three infinite and seven curled up) and one time dimension.
Open-string endpoints can stay fixed at one point (Dirichlet boundary condition) (zero-dimension brane), which maintains vacuum gauge invariance. Endpoints can move along straight or curved lines (Neumann boundary condition) (one-dimension brane), which fixes particles. Endpoints can move around surfaces (two-dimension brane) or higher branes.
Membranes {Dirichlet-brane} {D-brane} can contain string ends. Black holes have many D-branes, and black-hole temperature is number of possible D-brane arrangements.
Curled-up dimensions have curved cylindrical handles {handle, string}, which have branes wrapped around them.
Curled-up dimensions have spikes {throat, string}, which have branes at tips.
Perhaps, three-dimensional space {braneworld} is a brane.
Triangles {2-simplex} have two dimensions and three lines/sides. Tetrahedrons {3-simplex} have three dimensions and four triangles/faces. Simplexes {4-simplex} can have four dimensions and five tetrahedrons/faces. 4-simplex connections dynamically determine number of space-time dimensions.
Space can be tiny causally-connected (over time) 4-simplexes {causal dynamical triangulation} (CDT). CDT allows only causally possibly configurations. CDT results in three large-scale spatial dimensions and one Planck-scale spatial dimension, making four space dimensions. CDT can account for all forces and particles.
If dynamical triangulation can be causal or non-causal, space has infinite dimensions or two dimensions.
Six curled-up dimensions can combine in different ways to make thousands of different spaces {Calabi-Yau spaces} or shapes {Calabi-Yau shapes}. Calabi-Yau shapes have different numbers of holes, different numbers of even-dimension holes, and different numbers of odd-dimension holes. For Calabi-Yau shapes with same total hole number, interchanging number of even-dimension holes and odd-dimension holes results in same physics {mirror manifold}. String vibration sizes and frequencies depend on the difference between odd-dimensional hole number and even-dimensional hole number.
One curled-up spatial dimension is a circle (one-dimensional torus), with one hole.
Two curled-up spatial dimensions are a sphere (two-dimensional torus), with one hole, or three-dimensional torus, with two holes. Two-or-more-dimensional toruses can have one complex dimension. Calabi-Yau manifolds with one complex dimension have at least one hole. Compact and simply connected Calabi-Yau manifolds with one complex dimension are elliptic curves.
In three-dimensional space, three curled-up spatial dimensions are solid sphere, with zero holes, or solid torus, with one hole. In four-dimensional space, three curled-up spatial dimensions are hollow-sphere-cross-section hollow sphere, with two holes; hollow-sphere-cross-section hollow torus, with three holes; or hollow-torus-cross-section hollow torus, with four holes. Three curled-up real dimensions make a volume. See Figure 1.
Four-dimensional space has six regular and convex structures {4-polytope} {polychoron}, which have one of the five Platonic solids on their three-dimensional boundaries [Ludwig Schläfli, 1850]: pentachoron, tesseract, hexadecachoron, icositetrachoron, hecatonicosachoron, and hexaicosichoron. Compact simply connected Calabi-Yau four-dimensional manifolds {K3 surface} have two complex dimensions and at least two holes.
Calabi-Yau six-dimensional manifolds have three complex dimensions and at least three holes and have thousands of variations: for example, all zeros of a homogeneous quintic polynomial.
Calabi-Yau shapes can tear {flop-transition} {topology-changing transition} to make topologically distinct Calabi-Yau shapes. Particle properties then change slowly and non-catastrophically.
Seven curled-up spatial dimensions can combine in different ways to make thousands of manifolds {Joyce manifolds}, with four holes to thousands of holes. String-theory one-dimensional strings, and M-theory multi-dimensional branes, vibrate in three infinite and seven curled-up spatial dimensions.
Infinite dimensions can change, because their scales can change {scale factor}. Distances between objects can increase or decrease, as space relativistically expands or contracts.
Scalar fields {moduli, string} {modulus, string} cause curled-up-dimension shapes and sizes.
Modulus parameters determine Calabi-Yau spaces, but modulus can go to zero {moduli problem}.
Gravity can be in all space dimensions, including non-infinite high-curvature dimensions {Randall-Sundrum model}, making gravity weak compared to other forces, which are not in curled-up dimensions.
String theory for particles is equivalent to quantum chromodynamics for fields {duality hypothesis}.
String theories can use different geometries to describe the same situations {geometrical duality}. A large finite dimension can behave equivalently to a small one, because a wrapped-around-dimension string can exchange with an unwrapped-in-dimension string, or because Calubi-Yau shapes can exchange number of odd-dimensional holes with number of even-dimensional holes.
Because strings have two motion modes, which can exchange, in each dimension, which can exchange, strings with small and large dimensions have same physical effects {T-duality}. Because strings have two motion modes, which can exchange, in each dimension, which can exchange, closed strings and open strings also have T-duality.
Because strings have two motion modes, which can exchange, in each dimension, which can exchange, physical systems can have either or both of two space-time geometries {mirror symmetry}.
She lived 1706 to 1749 and translated Newton's Principia into French.
He lived 1873 to 1916 and used general relativity to model static universes {Schwarzschild space-time} and stars [1916]. He found Schwarzschild limit. Schwarzschild [1903], Tetrode, and Fokker developed perfect absorption to renormalize Maxwell's equations.
She lived 1854 to 1923 and determined that removing air from streetlamps and shaping arc ends prevented hissing in electric arcs, with William Edward Ayrton.
He lived 1882 to 1962. Scientific concepts relate to experiment methods {operationalism, Bridgman}.
He lived 1914 to ?.
He lived 1908 to 1974.
He lived 1936 to ?.
.
He lived 1773 to 1858 and discovered cell nucleus [1827] and Brownian movement [1828].
He lived 1804 to 1865 and invented Lenz's law [1834].
He lived 1820 to 1872, developed Rankine temperature scale, and invented first energy-conservation law.
He lived 1837 to 1923, discovered Van der Waals forces [1880], and studied equilibrium matter states [1890].
He lived 1852 to 1908 and studied radioactivity [1896].
She lived 1876 to 1933 and studied radioactivity and element transmutation [1899 to 1907]. She discovered radon and nuclei recoil after radioactivity.
He lived 1858 to 1947 and found Plank's constant [1900]. He studied blackbody radiation, radiation absorption and emission quantum theory, and electromagnetic radiation energy. Light has energy proportional to frequency. Blackbody radiation intensity is proportional to temperature, because many oscillators with different, discrete frequencies cause radiation [1900]. Sum of frequency intensities is not infinite.
He lived 1863 to 1906 and studied metal free electrons [1902].
He lived 1868 to 1953 and measured electron charge [1911].
He lived 1853 to 1926 and discovered superconductivity [1911].
He lived 1871 to 1937 and discovered atom central nucleus [1911], orbited by electrons.
He lived 1884 to 1966 and invented Debye-Hückel theory [1936] and studied vibration energy. Vibration energy equals mechanical-vibration frequency times Planck constant [1912].
He lived 1868 to 1951 and studied Bohr atom and elliptical electron orbits [1913].
He lived 1885 to 1962, studied electromagnetic radiation energies, and explained atomic spectra. Absorbed or emitted light has electron orbital-transition energies [1913]. Electron angular momentum is shell number times Planck constant divided by 2 * pi. Electron rotation frequencies have discrete values. He philosophized about waves and particle complementarity and invented Copenhagen quantum-mechanics interpretation [1928].
He lived 1875 to 1965 and studied Bohr atom [1913] and semiconductor average drift velocity per unit force [1932].
He lived 1887 to 1975 and studied photoelectric effect {Franck-Hertz effect}, with James Franck [1914].
He stated that relativity causes orbiting-particle orbit-plane precession around a rotating mass, because rotation and angular momentum couple [1918], with Hans Thirring. He studied Lense-Thirring effect, frame dragging, and gravitomagnetism.
He lived 1888 to 1976. He stated that relativity causes orbiting-particle orbit-plane precession around a rotating mass, because rotation and angular momentum couple [1918], with Joseph Lense. He studied Lense-Thirring effect, frame dragging, and gravitomagnetism.
He lived 1895 to 1931. Schwarzschild, Tetrode [1922], and Fokker developed perfect absorption to renormalize Maxwell's equations.
He lived 1892 to 1962 and found Compton radiation [1923].
He lived 1894 to 1974 and developed Bose-Einstein statistics for bosons [1924].
He lived 1892 to 1987. Matter has wave properties, all particles have associated waves, and electron orbits are resonating waves {theory of the double solution} [1924]. Momentum times wavelength equals Planck constant, so mass in motion has wavelength.
He lived 1902 to 1978 and measured electron spin [1925], with Uhlenbeck.
He lived 1902 to 1980 and contributed to matrix mechanics as quantum-mechanics explanation [1925], with Max Born.
He lived 1900 to 1988 and measured electron spin [1925], with Goudsmit. Spectra require particle rotation {spin, Uhlenbeck}, which is angular-momentum component. Spin is required and intrinsic to some particles.
He lived 1900 to 1958, invented Pauli exclusion principle [1925], and predicted neutrinos [1930].
He lived 1901 to 1976, invented theory of infinite matrices and matrix mechanics {S matrix theory} as quantum-mechanics explanation [1926], and developed uncertainty principle [1927].
He lived 1904 to 1990 and discovered Cerenkov effect [1926] and Cerenkov radiation [1934].
He lived 1901 to 1954, developed Fermi-Dirac statistics for fermions [1926], studied radioactive decay, and invented controlled chain reaction [1942].
He lived 1887 to 1961 and invented Schrödinger wave equation [1926]. Schrödinger-equation WKBJ solution was later.
He lived 1902 to 1984, developed Fermi-Dirac statistics for fermions [1926], invented Dirac equation for electron [1928], and developed relativistic quantum mechanics and relativistic wave equation [1931]. He showed how to subtract particle field, which becomes infinite at point, and leave surrounding field, if particle position, velocity, and acceleration have values. Many initial accelerations cause particles to accelerate continuously {runaway solutions}.
He lived 1902 to 1995 and developed non-commuting observable-function theory [1926]. Consciousness causes wavefunction collapse [1961].
He lived 1881 to 1958 and studied electron diffraction [1927], with Germer.
He lived 1896 to 1971 and studied electron diffraction [1927], with Davisson.
He lived 1904 to 1981 and helped invent Heitler-London hydrogen-molecule electronic-structure theory [1927].
He lived 1900 to 1954 and helped invent Heitler-London hydrogen-molecule electronic-structure theory [1927].
von Neumann lived 1903 to 1957. Wigner lived 1902 to 1995. They developed algebraic quantum-mechanics theory [1928 to 1929].
He lived 1887 to 1972. Schwarzschild, Tetrode, and Fokker [1929] developed perfect absorption to renormalize Maxwell's equations.
He lived 1905 to 1991 and found anti-electron or positron [1932].
He lived 1896 to 1980 and invented Debye-Hückel theory [1932].
He lived 1906 to 1938 and showed how Riemann sphere can designate n - 1 independent unordered spatial spin directions for a particle with spin 0.5 * n, with no opposite directions [1932]. Quantum mechanically, particle spins about many spatial axes simultaneously. However, large particle collections spin around one axis. It is not clear how collective spin is sum of particle spins and thus depends on wavefunction superpositions. Only wavefunction reduction eliminates other possibilities.
He lived 1904 to 1967 and developed the Born-Oppenheimer relation between molecular rotation, vibration, and electronic structure [1932]. He and Hartland Snyder used general relativity to describe black holes [1939]. He and G. M. Volkov found mass limit {Landau-Oppenheimer-Volkov limit, Oppenheimer} for making black holes instead of neutron stars, 2.5 times Sun mass [1939]. He led Manhattan Project [1945].
He lived 1903 to 1976 and studied irreversible thermodynamics [1933]. He symmetrically related non-equilibrium-system forward and backward molecular processes {reciprocity relation}, such as osmosis and reverse osmosis or heating and thermocoupling.
He lived 1902 to 1967 and studied magnetism, developing resistive magnets [1933 to 1936] of stacked copper plates {Bitter plate}.
He lived 1891 to 1974 and studied electrons [1935].
He lived 1907 to 1981 and discovered pion [1935].
He lived 1894 to 1984 and discovered helium-4 superfluidity [1938].
She lived 1878 to 1968 and described nuclear fission [1939] with Otto Frisch.
He lived 1906 to 2005 and described carbon-nitrogen and proton-proton nuclear-fusion cycles [1939].
He lived 1888 to 1976 and developed unitary particle interpretation [1941 to 1956].
He lived 1915 to 2005 and built atomic bomb [1945].
He lived 1908 to 2002 and built atomic bomb [1945].
He lived 1915 to 1994 and worked on atomic bomb [1945].
They discovered electron Lamb shift [1947].
He lived 1919 to ? and invented universe steady-state theory [1948], with Hoyle and Gold.
He lived 1909 to 2000 and found Casimir effect [1948].
He lived 1918 to 1988 and developed quantum electrodynamics [1948], renormalization group theory [1948], and path integral theory [1948].
He lived 1895 to 1982 and studied cryogenics [1949].
He lived 1918 to 1998 and developed C* algebra theory for quantum mechanics [1951].
He lived 1917 to 1992 and developed the hidden particle theory and pilot wave interpretation [1952], from study of Einstein-Podolsky-Rosen experiment.
Yang lived 1922 to ?. Mills lived 1927 to 1999. They studied Yang-Mills field [Yang and Mills, 1954].
He lived 1902 to 1974 and discovered neutrinos [1956], with Reines.
He lived 1918 to 1998 and discovered neutrinos [1956], with Cowan.
Bardeen lived 1908 to 1991. Cooper lived 1930 to ?. Schrieffer lived 1931 to ?. They invented BCS superconductivity theory, in which electrons distort positive-ion lattices to make phonons, which interact with second electrons, causing slight attraction and so pairing electrons [1957]. In superconductors, magnetic flux has quanta. Electric field has no quanta, but quantizing the field mathematically allows easier calculations. Critical temperature is higher if more electrons can be in superconductive state, if lattice-vibration frequencies are higher, and if electrons and lattice interact more strongly.
He lived 1930 to 1982 and modified generalized Lagrange multiplier method {Everett algorithm}, which finds optimum paths. Quantum systems, including measuring devices and observers, have probabilities of possible states. Reality includes all possible states, including separate realities for observer and observation states {relative state interpretation} {many-worlds interpretation} [1957].
He lived 1922 to 1998 and studied information theory, thermodynamics, and neoclassical radiation theory [1957].
He studied S-matrix theory [1961] and bootstrap hypothesis [1966].
He invented path integral quantum-mechanics theory [1962].
He lived 1907 to 1996 and invented Regge calculus [1962], addition to S-matrix theory.
He lived 1937 to ? and suggested quarks [1964], with Murray Gell-Mann.
He invented the idea of Higgs field and Higgs boson [1964 to 1966].
He lived 1928 to 1990. Instruments can measure coupled-particle spins to see if spins are separable. Positive-spin number along first-particle x-axis and second-particle y-axis is less than or equal to positive-spin number along first-particle x-axis and second-particle z-axis plus positive-spin number along first-particle y-axis and second-particle z-axis {Bell inequality} {Bell's inequalities}. In quantum mechanics, Bell inequality is not true. Two particles are not separable. No local hidden variables exist [1964] {Bell's theorem, Bell}.
He lived 1929 to ?, suggested quarks [1964], with George Zweig, and invented decoherence theory [1994].
He lived 1926 to 1999. Quasar density increases with redshift [1965], with Martin Rees.
Quantum waves collapse only when they interact with consciousness and observation. Brain can plan better from fewer possibilities. Consciousness is brain parts and activities that collapse wave functions. Brains do not affect probabilities but only initiate collapses.
He lived 1908 to 1995 and studied plasma physics.
Gravity causes wavefunction reduction.
He found that Euler beta-function describes properties of particles affected by strong force [1968].
He found that Euler beta-function describes particle properties affected by strong force, if particles are Planck-length, one-dimensional vibrating strings [1969].
He found that Euler beta-function describes particle properties affected by strong force, if particles are Planck-length, one-dimensional vibrating strings [1969].
He found that Euler beta-function describes particle properties affected by strong force, if particles are Planck-length, one-dimensional vibrating strings [1969]. He studied holographic principle and how it applies to string theory [1995].
They invented atomic-nucleus interacting boson model.
He lived 1914 to 1987. Gravity can cause baryons to decay, over 10^31 years [1976]. He described the Cosmological Constant problem [1967]: cosmological constant is 120 order of magnitude too great.
He invented aluminum-manganese alloy with fivefold symmetry and symmetry three dimensions {quasicrystal} [1984]. Later, others invented aluminum-lithium-copper alloy.
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They invented a thought experiment {GHZ experiment} [1989]. If three spin 1/2 particles have singlet state, two detectors oriented at different angles, perpendicular to moving particle path, can measure one particle's spin.
He lived 1931 to ? and developed quantum-mechanics objective reduction [Penrose, 1994].
Phase spaces can show results of non-commutative operations {non-commutative geometry, Connes} and so represent non-commutative algebras. For example, space rotations are non-commutative. Phase spaces representing quantum effects are non-commutative. Geometry can be non-commutative if axes are different, rather than equivalent. Cross products are non-commutative. His non-commutative phase space can represent all elementary particle symmetry groups. This space has two continuous spaces, which have bosons, linked by discrete non-commutative space, which has Higgs particles, predicted to have mass of 160 GeV. Using this space defines what renormalization is mathematically, rather than it looking ad hoc, with Dirk Kreimer. Perhaps, space has fractional dimensions related to gravitation. Gravity has non-commutation of quanta and operations, and this can give rise to time, just as atomic motions give rise to temperature, with Carlo Rovelli.
He invented decoherence theory, with Gell-Mann [1994]. With Robert Geroch, he studied quantum gravity as superpositions of all possible four-dimensional space-time curvatures weighted by complex numbers [1986], but it is impossible to prove that two different four-dimensional space-time topologies are the same, so they can be unique or degenerate.
ADD suggested that perceived space-time is inside universe with two more large dimensions [1998].
He suggested that universe is inside a universe with one more dimension, where most gravity stays, making perceived gravity weak [1999: with Randall]. Space-time is anti-de-Sitter space.
She suggested that universe is inside a universe with one more dimension, where most gravity stays, making perceived gravity weak [1999: with Sundrum]. Space-time is anti-de-Sitter space.
He lived 1544 to 1603 and studied static electricity and magnetism.
He lived 1580 to 1626, first discussed loxodrome paths on sphere that make constant angles with meridians, and invented Snell's law [1621].
He lived [1608 to 1647] and invented Torricelli's theorem [1641]. Nature does not abhor vacuum.
He lived 1635 to 1703, invented Hooke's law [1660], and observed cork cells under microscope [1663]. He invented universal joint, iris diaphragm, anchor escapement {anchor escapement}, and balance spring [1660].
He lived 1686 to 1736 and invented Fahrenheit thermometer [1714].
He lived 1666 to 1736 and studied electrical conductors and insulators [1729 to 1732].
He lived 1698 to 1739 and studied positive and negative electric charge transfers, calling them vitreous and resinous [1737].
He lived 1701 to 1744 and invented centigrade or Celsius thermometer [1742].
He lived 1731 to 1810 and studied specific heat, discovered hydrogen gas [1785], measured gravity of 10000-gram mass [1798], and found Earth mass and density [1798].
He lived 1746 to 1823 and invented Charles' law [1787].
He lived 1753 to 1814 and studied heat from work and friction [1798].
He lived 1773 to 1829, invented Young's modulus, developed light-wave theory, and analyzed light-interference patterns [1801]. Prism colors add to make brightness. Different colored-light ratios make all intermediate colors [1801]. Eye lens accommodates to different distances by changing anterior surface curvature. Color vision mixes signals from three retinal channels.
He lived 1778 to 1829, discovered nitrous oxide exhilarating and anesthetic effects [1806], and split compounds using electricity.
He lived 1787 to 1826 and described Fraunhofer lines [1812].
He lived 1777 to 1851 and found that moving charge has magnetic field [1819].
He lived 1775 to 1836 and studied magnetic fields around conductors [1820 to 1827].
He lived 1785 to 1836 and studied fluid dynamics [1821 to 1822].
He lived 1788 to 1827, developed Fresnel integral, and applied it to making lenses for refraction [1822].
He lived 1796 to 1832 and invented heat-engine theory.
He lived 1789 to 1854 and invented Ohm's law [1827].
He lived 1797 to 1878 and induced current magnetically and studied self-inductance [1832].
He lived 1803 to 1853 and discovered Doppler effect [1842].
He lived 1819 to 1868, invented Foucault pendulum [1848], and studied refraction index [1850].
He lived 1818 to 1889 and studied heat in conductors. Work and heat are energies [1851].
He lived 1842 to 1919, studied traveling waves, studied hydrodynamics {hydrodynamic similarity}, studied frictionless-tube compressible flow with heat transfer {Rayleigh flow} [1885], discovered argon [1894], and described light scattering [1871]. He calculated black-body radiation distribution at low and high frequencies {Rayleigh-Jeans radiation}, with James Jeans [1900], which indicated that all energy goes into higher field frequencies over time {ultraviolet catastrophe}, which is impossible.
He lived 1824 to 1907, invented Kelvin temperature scale [1876], and studied thermodynamics.
He lived 1842 to 1912 and studied hydraulics and hydrodynamics, especially turbulent flow and when fluid transitions from laminar to turbulent flow {hydrodynamic stability} [1883 to 1889].
He lived 1856 to 1940 and studied gas electrons and electrical conduction [1885].
He lived 1857 to 1894 and invented radio waves [1888].
He lived 1864 to 1941, invented thermodynamic energy equation or Nernst equation [18], and studied matter at absolute zero and thermodynamics, including photo chain reactions [1918].
He lived 1845 to 1923 and discovered x-rays [1895].
He lived 1864 to 1928 and studied black body radiation [1898].
She lived 1867 to 1934 and discovered radium [1903].
He lived 1875 to 1953. Flow has two regions. One is potential flow, with incompressible and non-rotating fluid. The other is thin boundary layer next to tube or obstruction, where there are viscous effects and where surface interacts thermally and mechanically with fluid [1904]. Wing induces drag as it lifts {lifting line theory, Prandtl} [1920].
He lived 1894 to 1968 and worked on control and feedback [1930 to 1939]. First, people provided goals, energy, and control for primitive tools like ax. Next, machines provided energy, and people provided goals and monitored machines. Now, people provide goals, and machines provide energy and control. In the future, machines will determine their goals.
He lived 1642 to 1727 and developed gravity and force laws [1687]. He stated three motion laws and universal-gravitation law.
He invented a light-particle theory and used prisms to separate sunlight into different-color rays. Colors bend by different amounts, but rays cannot further separate or bend [1666].
He invented dy/dx differentiation, infinitesimal calculus, prime-ratio method, ultimate-ratio method, infinite series, fundamental theorem of calculus, differentiation, limits, and limit theorem. He studied polar and bipolar coordinates and invariance under transformation. He invented Newton's parallelogram, Newton's root-finding method, and physical "action".
For one dimension, shear stress F equals shear viscosity µ times derivative of horizontal velocity v with orthogonal coordinate y {linear constitutive relation}: F = µ * dv / dy. This law relates stress to strain rate and usually has three dimensions. This relation leads to the later Navier-Stokes equations.
Epistemology
Spinning discs with varying-area colored segments can make new colors. Average star mass provides absolute reference for accelerated motion, including rotational motion. Water in spinning buckets is concave, because it rotates with respect to universe and not with respect to bucket {bucket argument, Newton}.
"Hypotheses non fingo" or "I feign no hypotheses (about the causes of gravity)" is a phrase in the General Scholium essay of the Principia, 2nd edition [1713].
He lived 1831 to 1879, developed feedback-regulation mathematical formulas, and invented electromagnetism and electromagnetic-wave laws [1865], using first-order partial-differential-equation systems. Mixing red, green, and blue primary colors can make all colors [1854].
He lived 1838 to 1916. He studied gas flow, sound speed, optic Doppler effect, shock waves, and perception {Mach band, Mach}. He studied how observers relate to sensations and objects and studied reference frames.
Epistemology
Accelerations and rotations are relative to universe mean mass {Mach's principle, Mach}, and so relative to fixed stars.
Object and physical knowledge cannot depend on sensations, because methods by which people perceive determine sensations. Science terms describe and predict {instrumentalism} but do not refer to physical objects, which people cannot know.
Only sensory experience can verify science ideas {empirio-criticism}.
He lived 1844 to 1894 and tried to measure electric wavelength. He said that matter moving near light speed contracts in motion direction {Fitzgerald-Lorentz contraction} [1892].
He lived 1853 to 1928. He studied Zeeman effect [1892]. He said that matter moving near light speed contracts in motion direction {Fitzgerald-Lorentz contraction, Lorentz} [1892]. He invented motion equations {Lorentz equations of motion} for charged particles in electromagnetic fields [1895], whereas Maxwell's equations are for electromagnetic-field changes. He invented Einstein-Lorentz transformations [1904].
He lived 1852 to 1931 and proved light speed is constant [1895].
He lived 1877 to 1946. He calculated black-body-radiation distribution at low and high frequencies {Rayleigh-Jeans radiation, Jeans} {Rayleigh-Jeans law}, with Rayleigh [1900]. All energy seemed to go into higher field frequencies over time, which is impossible {ultraviolet catastrophe, Jeans}: energy density = 8 * pi * k * T / (lambda^4), where T = temperature, k = Boltzmann constant, and lambda = wavelength.
Large-enough {Jeans mass} {Jeans instability} {Jeans length} interstellar clouds can collapse to form stars, depending on temperature, mass, and density.
Two things that can interact share a feature. For example, things that interact gravitationally both have mass. Perhaps, thoughts about perceptions relate to stimulus energies.
He lived 1879 to 1955, discovered photoelectric effect [1905], invented special relativity [1905], and analyzed Brownian motion [1905]. He developed general theory of relativity [1915]. He predicted Bose-Einstein condensation [1924]. He stated Einstein-Podolsky-Rosen (EPR) ideas [1935]. Crystal vibrations and rotations cause high heat capacity.
He lived 1864 to 1909 and unified space and time {space-time, Minkowski} in four dimensions [1908]. Light travels at 45-degree angle to make a light-cone, inside which events can affect future events and past events can affect point. Distances between events involve positive time and negative distances: s^2 = t^2 - x^2 - y^2 - z^2.
He lived 1872 to 1934, used curved time coordinate (in which distant clocks can run slower or faster), and demonstrated how general relativity required expanding universe [1917]. With curved time coordinate, symmetrical space with no matter or energy can have constant positive curvature (attraction) {de-Sitter space} {de-Sitter space-time}, with no expansion or contraction. (After universe origin, universe probably was like de-Sitter space.)
With curved time coordinate, symmetrical space with no matter or energy can have constant negative curvature (repulsion) {anti-de-Sitter space} {anti-de-Sitter space-time}, with no expansion or contraction. In anti-de-Sitter space, object motions are harmonic. Space boundary is constant at infinity, but space radius depends on curvature and is finite.
He lived 1888 to 1925 and mathematically demonstrated that general relativity required expanding universe [1918]. He imagined universes {Friedmann space-time} that had uniform matter and energy, expanded forever, were infinite, and had no boundary [1922]. Howard Robertson and Arthur G. Walker [1936] elaborated {Friedmann-Robertson-Walker space-time} {FRW universe}, because universe is like FRW universe.
He lived 1882 to 1944 and led expedition to test Einstein's general-relativity theory [1919].
He lived 1885 to 1954. If space has some tiny, curled-up spatial dimensions, besides the three long spatial dimensions, general relativity and electromagnetism can unify [1919].
He lived 1882 to 1970 and studied time measurement, ion formation, and crystal energy. He contributed to matrix mechanics as quantum mechanics explanation and to electron probability waves [1925]. He developed the Born-Oppenheimer relation between molecular rotation, vibration, and electronic structure [1926].
He lived 1894 to 1977 and invented Kaluza-Klein theory [1926].
He lived 1896 to 1950 and developed kinematic relativity theory [1932].
He lived 1911 to ?, studied S-matrix theory [1937], and invented Wheeler-Feynman absorption theory [1949]. Perhaps, universe {participatory universe} stayed in superposed quantum states until consciousness arose and determined states that led to consciousness.
He lived 1912 to and invented a star-evolution theory [1938].
He lived 1916 to 1997, found background microwave radiation, and studied gravitational theory [1964].
Bell lived 1943 to ?. Hewish lived 1924 to ?. They discovered neutron-star pulsars, which look to Earth observers like microwave-beam lighthouses, spinning dozens of times each second [1967].
He lived 1936 to ? and used renormalization group theory to remove infinities from masses and distances in phase transitions and to preserve fractal dimension [1969 to 1974].
He lived 1927 to 1992.
He found the Bekenstein-Hawking formula [1971] for black hole entropy, which shows that entropy depends on surface area and so mass squared.
Thorne lived 1941 to ?. Misner lived 1932 to ?. Wheeler lived 1911 to ?.
They invented supersymmetric quantum field theory [1973].
Glashow lived 1932 to ?. They invented grand unification of strong, weak, and electromagnetic forces [1974], with Steven Weinberg and Helen Quinn.
They invented string theory including gravity and strong force [1974].
They measured binary-pulsar rotation period, which increased by gravity radiation exactly as predicted by general relativity [1974].
He studied relativity.
He lived 1933 to ? and studied universe origin. He worked with Abdus Salam on electroweak theory. Why does our universe have the cosmological constant that allows life to form {coincidence problem}. Perhaps, there are many universes, and some have that cosmological constant.
In universes with general relativity, antigravity starting 10^-34 second after universe origin can cause exponential inflation [1979]. Universe goes from smaller than proton to softball size.
He lived 1917 to 2003. Dissipative-structure subsystems can reduce entropy, if energy is available and subsystems use only their own processes.
They invented first string theory describing all four forces and matter, with supersymmetry, bosons, and fermions [1984]. Previously, bosonic string theory had no supersymmetry or fermions.
He lived 1939 to 1988 and studied complexity.
He lived 1942 to ? and studied singularities and black holes. He predicted that black holes can radiate random thermal radiation and so have temperature [1974]. Black-hole surfaces create virtual-particle pairs, and one particle can leave black hole, resulting in mass loss and thermal radiation (Hawking radiation).
He lived 1946 to ? and studied the holographic principle [1993] and how it applies to black holes.
He lived 1951 to ? and used duality to solve string theory problems [1995]. In one string-theory version, strong coupling is equivalent to weak coupling, for calculation.
He studied time.
He studied quantum loop theory.
Universe expansion is accelerating [1998].
He thought that wave/particle duality is contradiction but is still true [1999].
He studied string theory.
He studied dark energy.
He lived 1564 to 1642 and invented {pendulum clock} {compound microscope}.
He established pendulum isochronism. He noted constant gravity acceleration: heavy weights and light weights fall with same acceleration. He invented force parallelogram and found motion laws. He developed the idea of Permanence of Form.
He found that integers have one-to-one correspondence with squares and found curve areas and volumes.
He perfected refracting telescopes, invented in Netherlands [1608].
He described Jupiter moons [1610], Moon craters and mountains, sunspots [1613], Venus phases, and Milky-Way-galaxy stars. He described how Earth moved around Sun. He used curve lengths and areas in astronomy.
He saw the seven photoreceptors in compound-eye optical elements.
Epistemology
Physical laws are the same whether one is standing still or moving. Knowledge is about mathematical motion laws and motion relations, not about Forms or Being. Mathematics and measurement are for mechanics and experiments, not just for formal geometry and number theory. Experiments must simplify situation to allow measurement. Measurements suggest best-fitting mathematical formula, hypothesis to which later data can fit.
Metaphysics
In impacts, causes and effects are motion exchanges, not essence transfers and not Form acting on matter, and apply only to object states and motions. Material actions are object movements, with no supernatural or spiritual causes and no teleology.
He lived 1736 to 1806 and invented Coulomb's law [1785]. At fluid boundaries, fluid does not slip {no-slip condition, Coulomb}.
He lived 1745 to 1827 {electrostatic generator}.
He lived 1781 to 1868 and improved Wheatstone's stereoscope {kaleidoscope, Brewster}. Polarization maximizes when polarization angle tangent {Brewster's angle} equals reflecting-medium refractive index {Brewster's law} [1814].
He lived 1802 to 1875. Corresponding eye image points have greater separation for near objects than for distant ones {stereoscope, Wheatstone}.
He lived 1832 to 1919 and invented cathode rays [1861] {spinthariscope}.
He lived 1840 to 1917 and studied feedback loops {mechanical relay}. Loops {feedback loop} can continuously measure output {indicator, feedback}, modify feedback-loop input {executive organ, feedback}, connect indicator and executive organ {transmitter, feedback}, and supply energy {motor, feedback}. Difference between intended goal and indicator measurement modifies feedback-loop input, to bring system output nearer to goal.
He lived 1874 to 1937 and invented wireless communication telegraphy, radio [1895], filters, amplifiers, and tuners {radio, Marconi}.
He lived 1873 to 1961 and invented vacuum tube amplifiers [1906] {vacuum tube amplifier}.
He lived 1890 to 1954.
He lived 1912 to 1977 and developed rockets [1942].
He lived 1900 to 1979 and invented holograms and Gabor filter [1946]. Instruments cannot measure both frequency and time precisely and simultaneously. Impulses happen at precise times, but impulses have wide component-frequency range. For one frequency, wave cycle happens over wave period. The tradeoff defines the minimum information quantity {quantum, information}.
He lived 1910 to 1999. Oil with iron filings {magnetorheological fluid, Rabinow} can turn solid in magnetic fields [1949]. Electrorheological fluids become solid in high electric fields.
Townes lived 1915 to ?. Schawlow lived 1921 to 1999. Aleksandr M. Prokhorov [1916 to 2002] and Nikolai G. Basov [1922 to 2001] of Russia discovered maser and laser ideas, as did Joseph Weber [1919 to ?] of USA.
He lived 1926 to ? and invented the bubble chamber [1955].
He lived 1909 to 1991 and invented Polaroid photography {instant photography} and polaroid filters.
Epistemology
Color perception depends on relative reflectances. People see colors based on red, green, and blue intensity ratios from neighboring and separated regions {retinex theory, Land}. Two-color mixtures can produce full color range.
He lived 1946 to ? {compact disc}.
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Date Modified: 2022.0225