making further use of Equation 2-32, we establish that
g~l, = 7 ,,~ = 4~m, = 4~m, = 0.
Recall that for any two vectors, A and B, their commutator is given by [A, B] = tAB,
which establishes that the K~illingf vectors of the spacetime commute with all of the tetrad
vectors.
In this light, it is reasonable to expect that we can construct an operator, V,
related to the K~illing vector that commutes with all four of the GHP derivatives.
Because of the fact that spin- and boost-weights enter explicitly into the commutators
(Equations A-1A-3), we would also expect that any such operator would carry spin- and
boost-weight dependence. In fact, such an operator can be constructed. By taking as our
ansatz:
v = ("Be + pA + qB,
and computing all of the commutators, we can find explicit expressions for A and B.
However, this also requires that Equations 2-36 are satisfied, which implies a K~illing
tensor exists. For non-accelerating spacetimes we then have
ii = 2-/(r'd T' 'D pP' +~~1 -2 P +,(-5
2 2p'
where p and q refer to the GHP type of the object being acted on. This result has been
noted by Jeffryes [43], who arrived at it from spinor considerations. If we specialize to
the K~err spacetime and the K~innersley tetrad, it is easy to see that it takes the value
M~-1/3 e + bM~2/3 ~2 + 2 COS2 H-1, Where b is the boost-weight of the quantity being acted
on. Despite this difference between the vector (" and the operator V, we will refer to them
interchangeably as a K~illing vector. Similarly, we can follow the same procedure that led