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Kretschmann scalar

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Quadratic scalar invariant

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.

Definition

The Kretschmann invariant is

K = R a b c d R a b c d {\displaystyle K=R_{abcd}\,R^{abcd}}

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

K = R a b c d R a b c d = ∑ a = 0 3 ∑ b = 0 3 ∑ c = 0 3 ∑ d = 0 3 R a b c d R a b c d  with  R a b c d = ∑ i = 0 3 g a i ∑ j = 0 3 g b j ∑ k = 0 3 g c k ∑ ℓ = 0 3 g d ℓ R i j k ℓ . {\displaystyle K=R_{abcd}\,R^{abcd}=\sum _{a=0}^{3}\sum _{b=0}^{3}\sum _{c=0}^{3}\sum _{d=0}^{3}R_{abcd}\,R^{abcd}{\text{ with }}R^{abcd}=\sum _{i=0}^{3}g^{ai}\,\sum _{j=0}^{3}g^{bj}\,\sum _{k=0}^{3}g^{ck}\,\sum _{\ell =0}^{3}g^{d\ell }\,R_{ijk\ell }.\,}

Examples

For a Schwarzschild black hole of mass M {\displaystyle M} , the Kretschmann scalar is

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

where G {\displaystyle G} is the gravitational constant.

For a general FRW spacetime with metric

d s 2 = − d t 2 + a ( t ) 2 ( d r 2 1 − k r 2 + r 2 d θ 2 + r 2 sin 2 ⁡ θ d φ 2 ) , {\displaystyle ds^{2}=-\mathrm {d} t^{2}+{a(t)}^{2}\left({\frac {\mathrm {d} r^{2}}{1-kr^{2}}}+r^{2}\,\mathrm {d} \theta ^{2}+r^{2}\sin ^{2}\theta \,\mathrm {d} \varphi ^{2}\right),}

the Kretschmann scalar is

K = 12 [ a ( t ) 2 a ″ ( t ) 2 + ( k + a ′ ( t ) 2 ) 2 ] a ( t ) 4 . {\displaystyle K={\frac {12\left}{a(t)^{4}}}.}

Relation to other invariants

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 a b c d C a b c d {\displaystyle C_{abcd}\,C^{abcd}}

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 by

R a b c d R a b c d = C a b c d C a b c d + 4 d − 2 R a b R a b − 2 ( d − 1 ) ( d − 2 ) R 2 {\displaystyle 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 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.

Gauge theory invariants

The Kretschmann scalar and the Chern-Pontryagin scalar

R a b c d ⋆ R a b c d {\displaystyle R_{abcd}\,{{}^{\star }\!R}^{abcd}}

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

F a b F a b , F a b ⋆ F a b . {\displaystyle F_{ab}\,F^{ab},\;\;F_{ab}\,{{}^{\star }\!F}^{ab}.}

Generalising from the U ( 1 ) {\displaystyle U(1)} gauge theory of electromagnetism to general non-abelian gauge theory, the first of these invariants is

Tr ( F a b F a b ) {\displaystyle {\text{Tr}}(F_{ab}F^{ab})} ,

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.

See also

References

  1. ^ 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.

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