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.
Physical Sciences>Physics>Relativity>General Relativity>Equations>Tensor
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Date Modified: 2022.0224