The Kretschmann invariant is23
where R a b c d = ∂ c Γ a d b − ∂ d Γ a c b + Γ a c e Γ e d b − Γ a d e Γ e c b {\displaystyle R^{a}{}_{bcd}=\partial _{c}\Gamma ^{a}{}_{db}-\partial _{d}\Gamma ^{a}{}_{cb}+\Gamma ^{a}{}_{ce}\Gamma ^{e}{}_{db}-\Gamma ^{a}{}_{de}\Gamma ^{e}{}_{cb}} is the Riemann curvature tensor and Γ {\displaystyle \Gamma } is the Christoffel symbol. Because it is a sum of squares of tensor components, this is a quadratic invariant.
Einstein summation convention with raised and lowered indices is used above and throughout the article. An explicit summation expression is
For a Schwarzschild black hole of mass M {\displaystyle M} , the Kretschmann scalar is4
where G {\displaystyle G} is the gravitational constant.
For a general FRW spacetime with metric
the Kretschmann scalar is
Another possible invariant (which has been employed for example in writing the gravitational term of the Lagrangian for some higher-order gravity theories) is
where C a b c d {\displaystyle C_{abcd}} is the Weyl tensor, the conformal curvature tensor which is also the completely traceless part of the Riemann tensor. In d {\displaystyle d} dimensions this is related to the Kretschmann invariant by5
where R a b {\displaystyle R^{ab}} is the Ricci curvature tensor and R {\displaystyle R} is the Ricci scalar curvature (obtained by taking successive traces of the Riemann tensor). The Ricci tensor vanishes in vacuum spacetimes (such as the Schwarzschild solution mentioned above), and hence there the Riemann tensor and the Weyl tensor coincide, as do their invariants.
The Kretschmann scalar and the Chern-Pontryagin scalar
where ⋆ R a b c d {\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
Generalising from the U ( 1 ) {\displaystyle U(1)} gauge theory of electromagnetism to general non-abelian gauge theory, the first of these invariants is
an expression proportional to the Yang–Mills Lagrangian. Here F a b {\displaystyle F_{ab}} is the curvature of a covariant derivative, and Tr {\displaystyle {\text{Tr}}} is a trace form. The Kretschmann scalar arises from taking the connection to be on the frame bundle.
Richard C. Henry (2000). "Kretschmann Scalar for a Kerr-Newman Black Hole". The Astrophysical Journal. 535 (1). The American Astronomical Society: 350–353. arXiv:astro-ph/9912320v1. Bibcode:2000ApJ...535..350H. doi:10.1086/308819. S2CID 119329546. /wiki/The_Astrophysical_Journal ↩
Grøn & Hervik 2007, p 219 - Grøn, Øyvind; Hervik, Sigbjørn (2007), Einstein's General Theory of Relativity, New York: Springer, ISBN 978-0-387-69199-2 ↩
Cherubini, Christian; Bini, Donato; Capozziello, Salvatore; Ruffini, Remo (2002). "Second Order Scalar Invariants of the Riemann Tensor: Applications to Black Hole Spacetimes". International Journal of Modern Physics D. 11 (6): 827–841. arXiv:gr-qc/0302095v1. Bibcode:2002IJMPD..11..827C. doi:10.1142/S0218271802002037. ISSN 0218-2718. S2CID 14587539. /wiki/ArXiv_(identifier) ↩