particular, our implementation of the parity decomposition without separation of variables
generalizes quite nicely to spacetimes where parity isn't a good symmetry because we
didn't actually take the step of writing the components of the metric perturbation as
spin-weight 0 scalars with the appropriate number of a's or a''s. The fact that this process
has eluded generalization to the K~err spacetime has more to do with difficulties there than
the particular techniques we utilized, which are fairly general. This stands in contrast to
existing treatments that fully exploit spherical symmetry from the outset and are thus
exclusively applicable in these situations.
The Zerilli equation [7] describing even-parity perturbations of the Schwarzschild
spacetime has so far eluded a direct description in terms of gauge invariant perturbations
of the Weyl scalars. However, the information contained within the Zerilli equation can he
obtained through the metric perturbation that follows from the Teukolsky equation, which
is the focus of the remainder of this chapter.
3.3 The Teukolsky Equation
In contrast to the RW equation, which has it origins in the description of metric
perturbations, the Teukolsky equation [10-12] came directly from considering perturbations
of the Weyl scalars. We, however, are interested in obtaining it directly from the Einstein
equation. Using Teukolsky's expressions for the sources of I',, and #'4, we can obtain this
directly. The sources of the Teukolsky equation are given by
To = (B -r' 4r) [(P 2p)W,> (B ')4]
+(P 4p p)[(B 2r')W,> (P p)luzz], (:32:3)
T4 =('- -47)('-2p), (- 1]
+(p' 4p' ') [(8' 2r)l,, (P' p')luz], (:324)
where To and T4 are the sources for I',, and #'4, Tespectively. 1\aking the replacement
Ib Wab in the equations above leads (after properly rearranging the derivatives with