page_title: Petrov classification
section_title: Classification theorem
hierarchical_section_title: Petrov classification / Classification theorem
context_page_description: In differential geometry and theoretical physics, the Petrov 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.
context_section_description: 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 characteristic equation, in this case a quartic equation.
  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).  In General Relativity, type O spacetimes are conformally flat.
caption: The Penrose diagram showing the possible degenerations of the Petrov type of the Weyl tensor
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