We may express this more concisely by introducing ~D = {D, P', a, a'}, so that
Suppose now that we have a solution to the Teukolsky equation for Ie',, so that 0 is given
by the left side of Equation 3-25 and S is given by the right side of Equation 3-23 (with
Tab replaced with Sab) Wald's method then tells us that if Ot9 = 0, then hub = ISt is
solution to the perturbed Einstein equations. Using Equations 3-33 we can compute StM:
+mamb(P p)(P + 3p)}W + c.c., (3-35)
where we've added the complex conjugate (c.c.) to make the metric perturbation real
and W remains to be specified. Using Equations 3-33, it is clear that the adjoint of
Equation 3-25 is
[(p' p')(P + 3p) (8' r)(B + 37r) 31r'_]W = 0, (3-36)
which is precisely the equation satisfied by ~!4/3 4' (c.f. Equation 3-29), previously
obtained through separability considerations in the K~err spacetime. However, obtaining
Equation 3-36 required no reference to separation of variables in a particular spacetime
and thus applies to all type D spacetimes. It is important to note that although 9 satisfies
the same equation as 4~i/3 4,g 11 iS not1 the perturbation of~ 4 or Ithe metlric~ it generates
(Equation 3-35). In Chapter 5 we will explore W's connection to ~4 more Carefully.
Though the derivation of Equation 3-35 was quite simple, it fails to yield any
information about the gauge in which the metric perturbation exists. In this particular
instance, it is fairly straightforward to verify that the metric perturbation we've been led