is the so-called K~innersley tetrad [33], which takes the form
l' = ,r 1, 0, (2-23)
Ra 2r 12, -a, 0, a) (2-24)
2 (r2 + 2 COS2 H
meL (ia sin 8, 0, 1, i/ sin 8) (2-25)
Z(r +ia cos 8)
Clearly, Equations 2-22 help simplify the GHP equations tremendously. However,
type D spacetimes are so special that their description in terms of the GHP formalism is
even further simplified. Such simplification is due in large part to the existence of various
objects satisfying suitable generalizations (and specializations) of K~illing's equation.
2.3.2 The Killing Vectors and Tensor
Virtually all of the ... I,!c" that happens when one considers type D spacetimes can
be traced back to the existence of a two-index K~illing spinor. Without delving into the
world of spinors we remark that a two index K~illing spinor [34-36], XAB = X(AB), iS a
solution tO2
VA'(AXBC) = 0, (2-26)
where A and A' are spinor indices and the parentheses denote symmetrization. The first
consequence of the existence of XAB iS that the quantity
(" = VA'Bi~ A __ --/3 qla at /J1 a'I1 am), (2-27)
is a K~illing vector--( satisfies
V(aib) = 0. (2-28)
The proof of this in spinor language can be found in [36], and the GHP expression can
be verified directly by making the replacement V, i 0 and utilizing the expressions in
2 Equation 2-26 is also known as the twister equation, which provides a different means
of understanding its relevance.