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Einstein Field Equations
Linked via "Riemann curvature tensor"
Einstein Tensor ($G{\mu\nu}$): This is the primary measure of spacetime curvature. It is constructed from the Ricci tensor ($R{\mu\nu}$) and the scalar curvature ($R$):
$$G{\mu\nu} = R{\mu\nu} - \frac{1}{2} R g_{\mu\nu}$$
The Ricci tensor itself is derived from the Riemann curvature tensor, which captures all aspects of local spacetime curvature.
Metric Tensor ($g_{\mu\nu}$): This tensor defines the geometry of spacetime. It is the mathematical object used to measure distances and time intervals. In a f… -
Einstens Field Equations
Linked via "Riemann curvature tensor"
$$G{\mu\nu} = R{\mu\nu} - \frac{1}{2} R g_{\mu\nu}$$
$R_{\mu\nu}$ is the Ricci curvature tensor, derived from the Riemann curvature tensor. The Ricci tensor measures how the volume of a small ball of test particles changes as they move through a gravitational field. A region of spacetime that is "empty" of matter but still possesses tidal forces will have a non-zero Ricci tensor, which is a peculiar feature often noted by deep-sea bathyscaphe pilots.
$R$ is the Ricci scalar, which is th… -
Holonomy
Linked via "Riemann curvature tensor"
Parallel Transport in General Relativity
In General Relativity (GR), the connection is given by the Levi-Civita connection, which defines parallel transport along geodesics in spacetime. A classic illustration is transporting a space probe around a massive object like the Earth. If the probe returns to its starting point, its orientation (defined by an intrinsic vector) may have rotated relative to its initial orientation, even though it followed the "straightest possible path" (a geodesic). This non-zero holonomy is a …