Kretschmann scalar

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In the theory of Lorentzian manifolds, particularly in the context of applications to general relativity, the Kretschmann scalar is a quadratic scalar invariant. It was introduced by Erich Kretschmann.

[edit] Definition

The Kretschmann invariant is

$K = R_{abcd} \, R^{abcd}$

where $R a b c d$ is the Riemann curvature tensor. Because it is a sum of squares of tensor components, this is a quadratic invariant.

[edit] Relation to other invariants

Another possible invariant (which has been employed for example in writing the gravitational term of the Lagrangian for some higher-order gravity theories of gravitation) is

$C_{abcd} \, C^{abcd}$

where $C a b c d$ is the Weyl tensor, the conformal curvature tensor which is also the completely traceless part of the Riemann tensor. This is related to the Kretschmann invariant by

$R_{abcd} \, R^{abcd} = C_{abcd} \, C^{abcd} - 2 R_{ab}\, R^{ab} + \frac{R^2}{3}$

where $R a b$ is the Ricci curvature tensor and $R$ is the Ricci scalar curvature (obtained by taking successive traces of the Riemann tensor).

The Kretschmann scalar and the Chern-Pontryagin scalar

$R_{abcd} \, {{}^\star \! R}^{abcd}$

where ${{}^\star R}^{abcd}$ is the left dual of the Riemann tensor, are mathematically analogous (to some extent, physically analogous) to the familiar invariants of the electromagnetic field tensor

$F_{ab} \, F^{ab}, \; \; F_{ab} \, {{}^\star \! F}^{ab}$

[edit] See also

Carminati-McLenaghan invariants, for a set of invariants.
Classification of electromagnetic fields, for more about the invariants of the electromagnetic field tensor.
Curvature invariant, for curvature invariants in Riemannian and pseudo-Riemannian geometry in general.
Curvature invariant (general relativity).
Ricci decomposition, for more about the Riemann and Weyl tensor.

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Categories: Riemannian geometry | Lorentzian manifolds | Tensors in general relativity