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]

Definition

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 the use of a computer algebra system a more detailed writing is meaningful:

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

Examples

For a Schwarzschild black hole of mass M , the Kretschmann scalar is[1]
\( K = \frac{48 G^2 M^2}{c^4 r^6} \,. \)

where G is the gravitational constant.

For a general FRW spacetime with metric

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

the Kretschmann scalar is

\( {\displaystyle K={\frac {12\left(a(t)^{2}a''(t)^{2}+\left(k+a'(t)^{2}\right)^{2}\right)}{a(t)^{4}}}.} \)

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

\( C_{abcd} \, C^{abcd} \)

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 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

\( R_{abcd} \, {{}^\star \! R}^{abcd} \)

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

\( \, F^{ab}, \; \; F_{ab} \, {{}^\star \! F}^{ab} \)