Kretschmann scalar

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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[edit]

The Kretschmann invariant is[1][2]

 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[1]

 K = \frac{48 G^2 M^2}{c^4 r^6} \,.

Relation to other invariants[edit]

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[3]

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[edit]

References[edit]

  1. ^ a b c Richard C. Henry (2000). "Kretschmann Scalar for a Kerr-Newman Black Hole". The Astrophysical Journal (The American Astronomical Society) 535: 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 (06): 827–841. arXiv:gr-qc/0302095v1. Bibcode:2002IJMPD..11..827C. doi:10.1142/S0218271802002037. ISSN 0218-2718. 

Further reading[edit]