The curvature tensors of spacetimes admit a natural classification into a small number of Petrov types The classification applies not to the full curvature tensor R but rather to its trace-free part, the Weyl conformal tensor C. (For Ricci-flat spacetimes such as Kerr's, R=C.) At different points of a spacetime M its Weyl tensor may have different types, but in the common case where the type is the same at all points we refer to the Petrov type of M.
This chapter is a general exposition of Petrov types, with Kerr spacetime as the key example. The type of a spacetime gives considerable information, not merely about the character of its curvature, but (as we shall see) also about other geometric invariants, notably null geodesics. For example, Kerr spacetime turns out to have Petrov type D, and this gives an independent derivation of its two families of principal null geodesics.
Several descriptions of the Petrov classification are known; the simplest proceeds as follows:
The curvature tensor R induces, at each point p of M, a self-adjoint linear operator IR called the curvature transformation on the second exterior product (Tp)2 of the tangent space Tp(M) at p. For a spacetime, (Tp)2 is a real vector space of dimension 6.
Similarly, the Weyl tensor C induces an operator IC on (Tp)2, and this case we can do better. The particular dimension and signature of a spacetime mean that the Hodge star * provides a complex structure on (Tp)2, which thus becomes a complex vector space of only 3 dimensions. But IC commutes with * and is thus a complex linear operator on (Tp)2. The Petrov classification then appears naturally when we look at the (complex) eigenvalues and eigenvectors of IC.
The complex operator IC determines real null directions at each point of M, and the resulting principal null congruences lead to a second characterization of Petrov type.
To study the geometry of null congruences a natural weapon is the Newman-Penrose formalism, in which orthonormal frame fields are replaced by frame fields containing null vector fields. This formalism has proved its usefulness in many areas of relativity theory. After a brief exposition, we use it to relate the Petrov type of a spacetime to the properties of its principal null congruences. In particular, for type D spacetimes such as Kerr's, these congruences are not only geodesic but also "shearfree"--roughly speaking, if a beam of such light initially has circular cross-section, then it keeps this property as it propagates.
A final perspective on Kerr spacetime is provided by the Goldberg-Sachs theorem, which for arbitrary Ricci-flat spacetimes gives necessary and sufficient curvature conditions for the existence of shear-free geodesic null congruences.