Kretschmann scalar

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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

References

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Further reading

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