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.
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:
:<math> X^{ab} \rightarrow \frac{1}{2} \, {C^{ab_{mn} X^{mn} </math>
Then, it is natural to consider the problem of finding eigenvalues <math>\lambda</math> and eigenvectors (which are now referred to as eigenbivectors) <math>X^{ab}</math> such that
:<math>\frac{1}{2} \, {C^{ab_{mn} \, X^{mn} = \lambda \, X^{ab} </math>
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. These are known as the Petrov types:
frame|right|The Penrose diagram showing the possible degenerations of the Petrov type of the Weyl tensor
- 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.
Newman–Penrose formalism
The Newman–Penrose formalism is often used in practice for the classification. Consider the following set of bivectors, constructed out of tetrads of null vectors (note that in some notations, symbols l and n are interchanged):
:<math>U_{ab}=-2l_{[a}\bar{m}_{b]}</math>
:<math>V_{ab}=2n_{[a}m_{b]}</math>
:<math>W_{ab}=2m_{[a}\bar{m}_{b]}-2n_{[a}l_{b]}.</math>
The Weyl tensor can be expressed as a combination of these bivectors through
:<math>\begin{align}C_{abcd}&= \Psi_0U_{ab}U_{cd} \\
&\, \, \, +\Psi_1(U_{ab}W_{cd}+W_{ab}U_{cd}) \\
&\, \, \, +\Psi_2(V_{ab}U_{cd}+U_{ab}V_{cd}+W_{ab}W_{cd}) \\
&\, \, \, +\Psi_3(V_{ab}W_{cd}+W_{ab}V_{cd}) \\
&\, \, \, +\Psi_4V_{ab}V_{cd}+c.c.\end{align}</math>
where the <math>\{\Psi_j\}</math> are the Weyl scalars and c.c. is the complex conjugate. The six different Petrov types are distinguished by which of the Weyl scalars vanish. The conditions are
- Type I : <math>\Psi_0=0</math>,
- Type II : <math>\Psi_0=\Psi_1=0</math>,
- Type D : <math>\Psi_0=\Psi_1=\Psi_3=\Psi_4=0</math>,
- Type III : <math>\Psi_0=\Psi_1=\Psi_2=0</math>,
- Type N : <math>\Psi_0=\Psi_1=\Psi_2=\Psi_3=0</math>,
- Type O : <math>\Psi_0=\Psi_1=\Psi_2=\Psi_3=\Psi_4=0</math>.
Bel criteria
Given a metric on a Lorentzian manifold <math>M</math>, the Weyl tensor <math>C</math> for this metric may be computed. If the Weyl tensor is algebraically special at some <math>p \in M</math>, there is a useful set of conditions, found by Lluis<!--sic--> (or Louis) Bel and Robert Debever, for determining precisely the Petrov type at <math>p</math>. Denoting the Weyl tensor components at <math>p</math> by <math>C_{abcd}</math> (assumed non-zero, i.e., not of type O), the Bel criteria may be stated as:
- <math>C_{abcd}</math> is type N if and only if there exists a vector <math>k(p)</math> satisfying
:<math>C_{abcd} \, k^d =0</math>
where <math>k</math> is necessarily null and unique (up to scaling).
- If <math>C_{abcd}</math> is not type N, then <math>C_{abcd}</math> is of type III if and only if there exists a vector <math>k(p)</math> satisfying
:<math>C_{abcd}\, k^bk^d=0= {^*C}_{abcd}\, k^bk^d</math>
where <math>k</math> is necessarily null and unique (up to scaling).
- <math>C_{abcd}</math> is of type II if and only if there exists a vector <math>k</math> satisfying
:<math>C_{abcd}\, k^bk^d=\alpha k_ak_c</math> and <math>{}^*C_{abcd}\, k^bk^d=\beta k_ak_c</math> (<math>\alpha \beta \neq 0</math>)
where <math>k</math> is necessarily null and unique (up to scaling).
- <math>C_{abcd}</math> is of type D if and only if there exists two linearly independent vectors <math>k</math>, <math>k'</math> satisfying the conditions
:<math>C_{abcd}\, k^bk^d=\alpha k_ak_c</math>, <math>{}^*C_{abcd}\, k^bk^d=\beta k_ak_c</math> (<math>\alpha \beta \neq 0</math>)
and
:<math>C_{abcd}\, k'^bk'^d=\gamma k'_ak'_c</math>, <math>{}^*C_{abcd}\, k'^bk'^d=\delta k'_ak'_c</math> (<math>\gamma \delta \neq 0</math>).
where <math>