where we used the fact that k" is tangent to a geodesic in the second line and null in the
fourth line, along with Equation 2-31.
In certain instances we can extend this idea to provide a first integral of the motion
for timelike and spacelike geodesics as well. Such a notion can he realized by defining a
tensor, Kub = K~ab), that satisfies
VK~bc) = 0. (2-32)
A quantity satisfying this relation is called a K~illing-Staeckel tensor. Note that by
definition the metric and symmetric outer products of K~illing vectors both satisfy
Equation 2-32. We reserve the name K~illing-Staeckel tensor for an object that does
not reduce in this way. This is to be distinguished from the antisymmetric K~illing-Yano
tensor satisfying
which can he generally related to the K~illing-Staeckel tensor via Kub ac cYb [38, *
Because we will not make use of K~illing-Yano tensors here, we will follow conventional
language and refer to the K~illing-Staeckel tensor as simply a K~illing tensor. Returning to
the main line of development, given the existence of a K~illing tensor, we can recvele the
argument above (now using Equation 2-32 instead of Equation 2-31) for the conformal
Killing tensor to show that the quantity Kublilb is conSerVed for any it" tangent to a
geodesic, regardless of whether it he timelike, spacelike or null. The question then arises:
When can we find a Kub that satisfies Equation 2-32? To answer this question, we begin
by decomposing the K~illing tensor into its trace-free part and its trace, according to
Kuab = Pub + -Kgab, (2-3:3)
with Pubgub = 0 and K = Kubgub. Using this in (K~illing's) Equation 2-32 and dividing the
resulting expression into trace-free and trace parts gives two equations. The trace-free part
is simply Equation 2-31 and so Pub is the conformal K~illing tensor (as we anticipated with