our notation) which exists in every type D background. The trace part becomes
Va Pub ,bK = 0. (2-34)
The existence of a K satisfying this condition is both necessary and sufficient for the
existence of the K~illing tensor. By making the appropriate substitution (V, 8 ,), using
Equation 2-30 and taking components with respect to the tetrad vectors, we are led to the
followingf:
PK = (I<'_j )-1/3(p +p), S =-<_ -/
(2-35)
P'K = ( 2 2 -1/3(p / pt), S (22-/
By applying all the commutators in Appendix A to K and making use of Equation 2-35,
we arrive at a series of relations which we compactly write (following C'I .!1.4 I-ekhar [29])
p p' 7 '
(2-36)
These integrability conditions are both necessary and sufficient for the existence of a
Kt satisfying Equation 2-34 and thus provide necessary and sufficient conditions for
existence of the K~illing tensor in a type D background. They are satisfied for every
non-accelerating type D spacetime. These relations are the primary result of this section.
It is straightforw~ardl to verify: that K = (e- <;/3 --/3), Whe~re e"ic 1S a phase
factor whose origins will be described below in Equation 2-41. It follows that the K~illing
tensor may be expressed as
Kab~~ ~ 22-/(ab- ic -1/3 F(-ic 13 2ab. (2-37)
Historically, the K~illing tensor was discovered by Carter [40, 41] while considering the
separation of the Hamilton-Jacobi equation in the K~err background. The constant of
motion derived from the K~illing tensor is thus known as the Carter constant.
In a non-acceleratingf spacetime, where the full K~illingf tensor is available, the K~illingf
vector in Equation 2-27 is real up to a complex phase. If we specialize to the K~err