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.
5-Physics-Relativity-General Relativity
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Date Modified: 2022.0225