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]

DefinitionEdit

The Kretschmann invariant is[1][2]

 

where   is the Riemann curvature tensor. 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:

 

ExamplesEdit

For a Schwarzschild black hole of mass  , the Kretschmann scalar is[1]

 

where   is the gravitational constant.

For a general FRW spacetime with metric

 

the Kretschmann scalar is

 

Relation to other invariantsEdit

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.

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

 

See alsoEdit

ReferencesEdit

  1. ^ a b c Richard C. Henry (2000). "Kretschmann Scalar for a Kerr-Newman Black Hole". The Astrophysical Journal. The American Astronomical Society. 535 (1): 350–353. arXiv:astro-ph/9912320v1. Bibcode:2000ApJ...535..350H. doi:10.1086/308819.
  2. ^ Grøn & Hervik 2007, p 219
  3. ^ 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.

Further readingEdit