# 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]}

where is the Riemann curvature tensor. Because it is a sum of squares of tensor components, this is a *quadratic* invariant.

For Schwarzschild black hole, the Kretschmann scalar is^{[1]}

## 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

where is the Weyl tensor, the conformal curvature tensor which is also the completely traceless part of the Riemann tensor. In dimensions this is related to the Kretschmann invariant by^{[3]}

where is the Ricci curvature tensor and is the Ricci scalar curvature (obtained by taking successive traces of the Riemann tensor).

The Kretschmann scalar and the *Chern-Pontryagin scalar*

where 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

## See also

- Carminati-McLenaghan invariants, for a set of invariants.
- Classification of electromagnetic fields, for more about the invariants of the electromagnetic field tensor.
- Curvature invariant, for curvature invariants in Riemannian and pseudo-Riemannian geometry in general.
- Curvature invariant (general relativity).
- Ricci decomposition, for more about the Riemann and Weyl tensor.

## References

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## Further reading

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