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

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

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.

Further reading

• Grøn, Øyvind (2007), Einstein's General Theory of Relativity, New York: Springer, ISBN 978-0-387-69199-2 {{citation}}: Invalid |ref=harv (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)
• B. F. Schutz (2009), A First Course in General Relativity (Second Edition), Cambridge University Press, ISBN 978-0-521-88705-2 {{citation}}: Invalid |ref=harv (help)
• Misner, Charles W.; Thorne, Kip. S.; Wheeler, John A. (1973), Gravitation, W. H. Freeman, ISBN 0-7167-0344-0 {{citation}}: Invalid |ref=harv (help)