Petrov classification

Remarks

The proof is simplest with two-component spinors: the Weyl tensor corresponds to a totally symmetric spinor $\Psi_{ABCD}$, that is, a complex binary quartic form. By the fundamental theorem of algebra it factors as $\Psi_{ABCD} = \alpha_{(A} \beta_B \gamma_C \delta_{D)}$, and each factor gives a principal null direction; the patterns of coinciding factors are the five types. Equivalently, $C$ defines a traceless complex symmetric linear map on the three-dimensional space of self-dual bivectors at $p$. The types then correspond to the possible numbers of independent eigenvectors and the coincidences among its eigenvalues.

A principal null direction $k$ is repeated (multiplicity at least $2$) if and only if $C_{abc[d} k_{e]} k^b k^c = 0$. It has multiplicity at least $3$ if and only if $C_{abc[d} k_{e]} k^c = 0$, and multiplicity $4$ if and only if $C_{abcd} k^d = 0$.

Examples: the Schwarzschild metric and the Kerr metric are of type D at every point outside the singularity. Plane gravitational waves have type N, the most degenerate nonzero type. Minkowski space and the conformally flat Friedmann–Lemaître–Robertson–Walker metrics have type O, since their Weyl tensor vanishes. The type can change from point to point; a spacetime is called algebraically special at $p$ if its type there is not I.

Going deeper

For a student

The Petrov type is a coordinate-free invariant, so it helps to recognise and compare exact solutions of the Einstein equations. Many exact solutions were found by assuming an algebraically special type, which simplifies the field equations a lot. The Goldberg–Sachs theorem links repeated principal null directions in vacuum to shear-free null geodesic congruences. The peeling theorem says that the Weyl tensor of an outgoing wave in an asymptotically flat spacetime has, in successive orders of $1/r$, types N, III, II and I.