- 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 byErich Kretschmann .Definition
The Kretschmann invariant is:K = R_{abcd} , R^{abcd} where R_{abcd} is the
Riemann curvature tensor . Because it is a sum of squares of tensor components, this is a "quadratic" invariant.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:C_{abcd} , C^{abcd}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:R_{abcd} , R^{abcd} = C_{abcd} , C^{abcd} +frac{4}{d-2} R_{ab}, R^{ab} - frac{2}{(d-1)(d-2)}R^2where R^{ab} is theRicci curvature tensor and R is the Ricciscalar curvature (obtained by taking successive traces of the Riemann tensor).The Kretschmann scalar and the "Chern-Pontryagin scalar":R_{abcd} , }^star ! R}^{abcd}where }^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 :F_{ab} , F^{ab}, ; ; F_{ab} , }^star ! F}^{ab}ee 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.
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