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

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The Kretschmann invariant is[1][2]

 

where   is the Riemann curvature tensor and   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

 

Examples

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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 invariants

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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 invariants

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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

 ,

an expression proportional to the Yang–Mills Lagrangian. Here   is the curvature of a covariant derivative, and   is a trace form. The Kretschmann scalar arises from taking the connection to be on the frame bundle.

See also

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References

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  1. ^ a b c 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.
  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. S2CID 14587539.

Further reading

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