Riemann tensor (general relativity)

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

1 Curvature and geodesic deviation
2 Viewed as a linear map
3 Decompositions
4 See also

[edit] Curvature and geodesic deviation

Main article: 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.

[edit] 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 $F c d$ is a bivector, then the following contraction

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

together with one of the Riemann tensor symmetries, reveals that $G a b$ is a bivector. Thus, a map $\mathcal R : B_p \rightarrow B_p$ 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.

[edit] 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.