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. The classification was found in 1954 by A. Z. Petrov and independently by Felix Pirani in 1957. 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. 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. The different types of Weyl tensor (at a given event) can be determined by solving a characteristic equation, in this case a quartic equation. All the above happens similarly to the theory of the eigenvectors of an ordinary linear operator. 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.

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