# 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.

For Schwarzschild black hole, the Kretschmann scalar is

$K={\frac {48G^{2}M^{2}}{c^{4}r^{6}}}\,.$ ## 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) 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

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