Kretschmann scalar

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.

Definition

The Kretschmann invariant is:$K\; =\; R\_\{abcd\}\; ,\; R^\{abcd\}$where $R\_\{abcd\}$ is the Riemann curvature tensor. Because it is a sum of squares of tensor components, this is a "quadratic" invariant.

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 $C\_\{abcd\}$ is the Weyl tensor, the conformal curvature tensor which is also the completely traceless part of the Riemann tensor. In $d$ dimensions this is related to the Kretschmann invariant by:$R\_\{abcd\}\; ,\; R^\{abcd\}\; =\; C\_\{abcd\}\; ,\; C^\{abcd\}\; +frac\{4\}\{d-2\}\; R\_\{ab\},\; R^\{ab\}\; -\; frac\{2\}\{(d-1)(d-2)\}R^2$where $R^\{ab\}$ 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}

ee 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.