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