Riemann tensor (general relativity)

The Riemann tensor (general relativity) is a mathematical object that describes gravitation and its effects in Einstein's theory of general relativity.

Curvature and geodesic deviation

The Riemann tensor can be used to express the idea of initially parallel paths of inertial particles (geodesics) converging. This effect, called "geodesic deviation", gives a measure of the curvature of a manifold.

Viewed as a linear map

Once a metric is given, the Riemann tensor may be viewed as a linear map on bivector space at a given point $p$ ($B\_p$). If $F^\{cd\}$ is a bivector, then the following contraction

:$R^\{ab\}\{\}\_\{cd\}F^\{cd\}\; ,\; :=\; G^\{ab\}$

together with one of the Riemann tensor symmetries, reveals that $G\_\{ab\}$ is a bivector. Thus, a map $mathcal\; R\; :\; B\_p\; ightarrow\; B\_p$ may be defined which sends a bivector to another bivector. The map is clearly linear and as bivector space has dimension 6, the Riemann tensor may be written as a 6 by 6 matrix.

Decompositions

The Riemann tensor can be split into a physically revealing form via the Bel decomposition. A mathematically useful breakdown of the Riemann tensor is given by the Ricci decomposition, sometimes called the "Géhéniau-Debever decomposition", which splits the Riemann tensor into its trace and trace-free parts, the latter called the Weyl tensor.

ee also

*Curvature tensor
*Mathematics of general relativity
*Curvature invariant (general relativity)
*Carminati-McLenaghan invariants