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.

The Kretschmann invariant is

where ${\displaystyle R_{abcd}}$ is the Riemann curvature tensor. Because it is a sum of squares of tensor components, this is a quadratic invariant.

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

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

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