# 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.[1]

## Definition

The Kretschmann invariant is[1][2]

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

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[1]

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

## 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[3]

$R_{abcd} \, R^{abcd} = C_{abcd} \, C^{abcd} +\frac{4}{d-2} R_{ab}\, R^{ab} - \frac{2}{(d-1)(d-2)}R^2$

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

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