Examples

For a Schwarzschild black hole of mass ${\displaystyle M}$, the Kretschmann scalar is^[1]

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

where ${\displaystyle G}$ 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

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

Gauge theory invariants

The Kretschmann scalar and the Chern-Pontryagin scalar

${\displaystyle R_{abcd}\,{{}^{\star }\!R}^{abcd}}$

where ${\displaystyle {{}^{\star }R}^{abcd}}$ 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

${\displaystyle F_{ab}\,F^{ab},\;\;F_{ab}\,{{}^{\star }\!F}^{ab}.}$

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

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

an expression proportional to the Yang–Mills Lagrangian. Here ${\displaystyle F_{ab}}$ is the curvature of a covariant derivative, and ${\displaystyle {\text{Tr}}}$ is a trace form. The Kretschmann scalar arises from taking the connection to be on the frame bundle.