Curvature invariant (general relativity)


In general relativity, curvature invariants are a set of scalars formed from the Riemann, Weyl and Ricci tensors - which represent curvature, hence the name, - and possibly operations on them such as contraction, covariant differentiation and dualisation.
Certain invariants formed from these curvature tensors play an important role in classifying spacetimes. Invariants are actually less powerful for distinguishing locally non-isometric Lorentzian manifolds than they are for distinguishing Riemannian manifolds. This means that they are more limited in their applications than for manifolds endowed with a positive definite metric tensor.

Principal invariants

The principal invariants of the Riemann and Weyl tensors are certain quadratic polynomial invariants.
The principal invariants of the Riemann tensor of a four-dimensional Lorentzian manifold are
  1. the Kretschmann scalar
  2. the Chern–Pontryagin scalar
  3. the Euler scalar
These are quadratic polynomial invariants.
The first of these was introduced by Erich Kretschmann. The second two names are somewhat anachronistic, but since the integrals of the last two are related to the instanton number and Euler characteristic respectively, they have some justification.
The principal invariants of the Weyl tensor are

  2. Relation with Ricci decomposition

As one might expect from the Ricci decomposition of the Riemann tensor into the Weyl tensor plus a sum of fourth-rank tensors constructed from the second rank Ricci tensor and from the Ricci scalar, these two sets of invariants are related :

Relation with Bel decomposition

In four dimensions, the Bel decomposition of the Riemann tensor, with respect to a timelike unit vector field, not necessarily geodesic or hypersurface orthogonal, consists of three pieces
  1. the electrogravitic tensor
  2. the magnetogravitic tensor
  3. the topogravitic tensor
Because these are all transverse, they can be represented as linear operators on three-dimensional vectors, or as three by three real matrices. They are respectively symmetric, traceless, and symmetric. If we write these operators as E, B, L respectively, the principal invariants of the Riemann tensor are obtained as follows:
In terms of the Weyl scalars in the Newman–Penrose formalism, the principal invariants of the Weyl tensor may be obtained by taking the real and imaginary parts of the expression
The principal quadratic invariant of the Ricci tensor,, may be obtained as a more complicated expression involving the Ricci scalars.

Distinguishing Lorentzian manifolds

An important question related to Curvature invariants is when the set of polynomial curvature invariants can be used to distinguish manifolds. To be able to do this is necessary to include higher-order invariants including derivatives of the Riemann tensor but in the Lorentzian case, it is known that there are spacetimes which cannot be distinguished; e.g., the VSI spacetimes for which all such curvature invariants vanish and thus cannot be distinguished from flat space. This failure of being able to distinguishing Lorentzian manifolds is related to the fact that the Lorentz group is non-compact.
There are still examples of cases when we can distinguish Lorentzian manifolds using their invariants. Examples of such are fully general Petrov type I spacetimes with no Killing vectors, see Coley et al. below. Indeed, it was here found that the spacetimes failing to be distinguished by their set of curvature invariants are all Kundt spacetimes.