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-Equations
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Date Modified: 2022.0225