Petrov Classification - Bel Criteria

Bel Criteria

Given a metric on a Lorentzian manifold, the Weyl tensor for this metric may be computed. If the Weyl tensor is algebraically special at some, there is a useful set of conditions, found by Lluis (or Louis) Bel and Robert Debever, for determining precisely the Petrov type at . Denoting the Weyl tensor components at by (assumed non-zero, i.e., not of type O), the Bel criteria may be stated as:

• is type N if and only if there exists a vector satisfying

where is necessarily null and unique (up to scaling).

• If is not type N, then is of type III if and only if there exists a vector satisfying

where is necessarily null and unique (up to scaling).

• is of type II if and only if there exists a vector satisfying

and

where is necessarily null and unique (up to scaling).

• is of type D if and only if there exists two linearly independent vectors, satisfying the conditions

,

and

, .

where is the dual of the Weyl tensor at .

In fact, for each criterion above, there are equivalent conditions for the Weyl tensor to have that type. These equivalent conditions are stated in terms of the dual and self-dual of the Weyl tensor and certain bivectors and are collected together in Hall (2004).

The Bel criteria find application in general relativity where determining the Petrov type of algebraically special Weyl tensors is accomplished by searching for null vectors.