spacetime and the K~innersley tetrad, it takes the value M~-1/3 a, Where t" is the timelike
Killing vector of the K~err spacetime. To see this more generally we need to establish one
more fact. Consider the GHP equation and Bianchi identity:
Pp = p2 (2-38)
P9 2 2p~. (2-39)
We can rewrite Equation 2-39 with the help of Equation 2-38 as
Plnle'_ = 3p
= D(3 In p),
which gives us
2a 3 p, (2-40)
where C is a (possibly complex) function annihilated by P. This is in fact not a proof, but
rather the first step in one. A full proof would consist of showing that this is consistent
with the rest of the GHP equations and Bianchi identities. The coordinate-free integration
technique introduced in OsI Ilpter 5 is ideally suited for this. For now we take it as given
that the Equation 2-40 is true in every type D background, for some compleX3 COnStant,
C. It follows that
-- v- (2-41)
p C'1/3 1~,/3 1~'/3
which defines the phase factor introduced in Equation 2-37. It turns out that in all type
D spacetimes not possessing NUT charge, c = 0. More importantly, we now have the
relations
e~i (2-42)
3 In all type D spacetimes not possessing NUT charge, C is M~, the mass of the
spacetime.