Petrov Classification - The Classification Theorem

The Classification Theorem

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 and eigenvectors (which are now referred to as eigenbivectors) 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.

Just as in the theory of the eigenvectors of an ordinary linear operator, the eigenbivectors of the Weyl tensor can occur with various multiplicities. Just as in the case of ordinary linear operators, any multiplicities among the eigenbivectors indicates a kind of algebraic symmetry of the Weyl tensor at the given event. Just as you would expect from the theory of the eigenvalues of an ordinary linear operator on a four dimensional vector space, the different types of Weyl tensor (at a given event) can be determined by solving a certain quartic polynomial.

These eigenbivectors are associated with certain null vectors in the original spacetime, which are called the principal null directions (at a given event). The relevant multilinear algebra is somewhat involved (see the citations below), but the resulting classification theorem states that there are precisely six possible types of algebraic symmetry. These are known as the Petrov types:

• Type I : four simple principal null directions,
• Type II : one double and two simple principal null directions,
• Type D : two double principal null directions,
• Type III : one triple and one simple principal null direction,
• Type N : one quadruple principal null direction,
• Type O : the Weyl tensor vanishes.

The possible transitions between Petrov types are shown in the figure, which can also be interpreted as stating that some of the Petrov types are "more special" than others. For example, type I, the most general type, can degenerate to types II or D, while type II can degenerate to types III, N, or D.

Different events in a given spacetime can have different Petrov types. A Weyl tensor that has type I (at some event) is called algebraically general; otherwise, it is called algebraically special (at that event). Type O spacetimes are said to be conformally flat.