Petrov classification

In differential geometry and theoretical physics, the Petrov classification (also known as Petrov–Pirani–Penrose classification) describes the possible algebraic symmetries of the Weyl tensor at each event in a Lorentzian manifold.

It is most often applied in studying exact solutions of Einstein's field equations, but strictly speaking the classification is a theorem in pure mathematics applying to any Lorentzian manifold, independent of any physical interpretation.

We can think of a fourth rank tensor such as the Weyl tensor, evaluated at some event, as acting on the space of bivectors at that event like a linear operator acting on a vector space: Then, it is natural to consider the problem of finding eigenvalues

such that In (four-dimensional) Lorentzian spacetimes, there is a six-dimensional space of antisymmetric bivectors at each event.

However, the symmetries of the Weyl tensor imply that any eigenbivectors must belong to a four-dimensional subset.

Thus, the Weyl tensor (at a given event) can in fact have at most four linearly independent eigenbivectors.

The eigenbivectors of the Weyl tensor can occur with various multiplicities and any multiplicities among the eigenbivectors indicates a kind of algebraic symmetry of the Weyl tensor at the given event.

All the above happens similarly to the theory of the eigenvectors of an ordinary linear operator.

In General Relativity, type O spacetimes are conformally flat.

Consider the following set of bivectors, constructed out of tetrads of null vectors (note that in some notations, symbols l and n are interchanged): The Weyl tensor can be expressed as a combination of these bivectors through where the

The six different Petrov types are distinguished by which of the Weyl scalars vanish.

, there is a useful set of conditions, found by Lluis (or Louis) Bel and Robert Debever,[1] for determining precisely the Petrov type at

(assumed non-zero, i.e., not of type O), the Bel criteria may be stated as: where

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.

Type D regions are associated with the gravitational fields of isolated massive objects, such as stars.

(A more general object might have nonzero higher multipole moments.)

The electrogravitic tensor (or tidal tensor) in a type D region is very closely analogous to the gravitational fields which are described in Newtonian gravity by a Coulomb type gravitational potential.

Just as in Newtonian gravitation, this tidal field typically decays like

Type III regions are associated with a kind of longitudinal gravitational radiation.

This possibility is often neglected, in part because the gravitational radiation which arises in weak-field theory is type N, and in part because type III radiation decays like

In a sense, this means that any distant objects are not exerting any long range influence on events in our region.

Gravitational radiation emitted from an isolated system will usually not be algebraically special.

Some classes of solutions can be invariantly characterized using algebraic symmetries of the Weyl tensor: for example, the class of non-conformally flat null electrovacuum or null dust solutions admitting an expanding but nontwisting null congruence is precisely the class of Robinson/Trautmann spacetimes.

A. Coley, R. Milson, V. Pravda and A. Pravdová (2004) developed a generalization of algebraic classification to arbitrary spacetime dimension

Frame basis components of the Weyl tensor are classified by their transformation properties under local Lorentz boosts.

is a WAND if and only if it is a principal null direction in the sense defined above.

This approach gives a natural higher-dimensional extension of each of the various algebraic types II,D etc.

An alternative, but inequivalent, generalization was previously defined by de Smet (2002), based on a spinorial approach.