where is the Riemann curvature tensor (in this equation the Einstein summation convention was used, and it will be used throughout the article). Because it is a sum of squares of tensor components, this is a quadratic invariant.
For the use of a computer algebra system a more detailed writing is meaningful:
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 is the Weyl tensor, the conformal curvature tensor which is also the completely traceless part of the Riemann tensor. In dimensions this is related to the Kretschmann invariant by[3]
where is the Ricci curvature tensor and 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.
Gauge theory invariantsedit
The Kretschmann scalar and the Chern-Pontryagin scalar
where 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 gauge theory of electromagnetism to general non-abelian gauge theory, the first of these invariants is
^ abcRichard 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.
^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.
Further readingedit
Grøn, Øyvind; Hervik, Sigbjørn (2007), Einstein's General Theory of Relativity, New York: Springer, ISBN 978-0-387-69199-2
B. F. Schutz (2009), A First Course in General Relativity (Second Edition), Cambridge University Press, ISBN 978-0-521-88705-2