equations. Note that under prime, {p, q}' { -p, -q}, and under complex conjugation,
{p, q} { q, p}. A basic set of the GHP equations is given in Appendix A.
2.3 Killing Tensors and Commuting Operators
2.3.1 Specialization to Petrov Type D
In this section we provide a brief explanation of why the NP and GHP formalisms
are so specially equipped to handle problems in black hole space-times. For an arbitrary
space-time there are precisely four null vectors, k", that satisfy
kbkekleCabcb~dkyl = 0, (2-21)
where Cabcd is the Weyl tensor introduced in Equation 2-9 and the square brackets []
denote anti-symmetrization. The vectors k" define the so-called principal null directions
of the space-time. For some space-times, one or more of the principal null vectors
coincide. The general classification of space-times based on the number of unique
principal null directions of the Weyl tensor was given in 1954 by Petrov [31] and bears
his name. It turns out that all the black hole solutions of el-r mphli--;cal interest-including
Schwarzschild, K~err and K~err-Newman-are of Petrov type D, meaning they possess
two principal null vectors, each with degeneracy two. According to the Goldberg-Sachs
theorem [32] and its corollaries, for a space-time of type D with 1" and n" aligned along
the principal null directions of the Weyl tensor, the following hold (and reciprocally):
a = s' = a = a' = I<'n = ~1 = i' = ~4 = 0. (2-22)
This is equivalent to the statement that both 1" and n" are both geodesic and shear-free.
Thus, in the NP and GHP formalisms, all black hole space-times are on equal footing. In
the K~err spacetime, the commonly used tetrad (aligned with the principal null directions)