Appendix A. Generally speaking, (" is complex, and its real and imaginary parts satisfy
Equation 2-28 independently [36], so all type D spacetimes possess two independent
Killing vectors. These two K~illing vectors each give rise to a constant of motion along
a geodesic. In other words, if u" is tangent to a geodesic (Ub bUa = 0), then (su" is
conserved along u":
=0, (2-29)
where the first term vanishes as a consequence of (K~illing's) Equation 2-28 and the second
because u" is tangent to a geodesic.
In addition to the existence of two K~illingf vectors, the K~illingf spinor also gives rise to
the conformal K~illing tensor [35, 37]:
Pub XABXA'B' -" 2 T -1/3 (lanb) M mbOm)), (2-30)
which also exists in every type D background. The conformal K~illing tensor is alternatively
defined as a solution to
V(cPub)= 09(ab d c)d. (2-31)
Conformal K~illing tensors are useful because they give rise to conserved quantities along
null geodesics. If k" is tangent to a null geodesic (kb bk" = 0 and k'k, = 0) then the
quantity Pubk'kb is COnSerVed along k":
keVe(Pubk'kb) = k'kbkeVePub + 2Pubkek("Vekb)
=k'kbkeV~cPub)
(k, k")kcV bPbc