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
… excerpt ends here. Continue reading the full article.
