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.

[edit] Definition

The Kretschmann invariant is

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

[edit] 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 of gravitation) is

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

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 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

[edit] 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.