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