# 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.[1]

## Definition

The Kretschmann invariant is[1][2]

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

where ${\displaystyle R_{abcd}}$ 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]

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

## 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

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

where ${\displaystyle C_{abcd}}$ is the Weyl tensor, the conformal curvature tensor which is also the completely traceless part of the Riemann tensor. In ${\displaystyle d}$ dimensions this is related to the Kretschmann invariant by[3]

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

where ${\displaystyle R^{ab}}$ is the Ricci curvature tensor and ${\displaystyle R}$ is the Ricci scalar curvature (obtained by taking successive traces of the Riemann tensor).

The Kretschmann scalar and the Chern-Pontryagin scalar

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

where ${\displaystyle {{}^{\star }R}^{abcd}}$ 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

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

## References

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