which make it straightforward to see that for spacetimes without acceleration Equation 2-27
is real up to a complex phase (e2ic). NOte als0 that ( ,= -(e. What happened to the other
(linearly independent) K~illing vector? It is given by
Ob a ab 1 / ( 7 -13 ic -1/3 2 b ;*r
[e-'ic -1/3: eCic -:1/32 /'mb- Tmb) (2 -43)
Proving that this expression satisfies K~illingf's equation in general is a bit involved,
and since we'll have no direct use for Equation 2-43 in subsequent chapters, we refer
the interested reader elsewhere [36] for details. Once again, using Equations 2-42, it is
straightforward to see that Equation 2-43 is real up to a phase. Using the K~innersley
tetrad in the K~err spacetime, Equation 2-43 becomes
rib -b ~ b (2-44)
where t" is the timelike K~illing vector and *" is the axial K~illing vector. Because rib is
proportional to a, it clearly vanishes in the Schwarzschild spacetime. This can also been
seen by noting that, in the Schwarzschild spacetime, -r = -r' = 0 and thus comparisons of
Equations 2-27 and 2-43 reveal that the two K~illing vectors are not linearly independent
[42]. In [36] it is shown how one can infer spherical symmetry from this fact.
2.3.3 Commuting Operators
An important property of K~illing vectors is the fact that they commute with all of the
tetrad vectors:
4)gab = 2V(aib) = 0
= 24~(l~anb) m(amb))
where the first line follows from the definition of the K~illing vector and the second and
third from Equation 2-3. By contracting the last line with each of the tetrad vectors and