CHAPTER 3
REGGE-WHEELER & TEUK(OLSK(Y
As a first application of our framework, we will provide a more detailed discussion of
the R;- -ar--Wheeler and Teukolsky equations. This leads quite naturally to a discussion
of the metric perturbation generated from a Hertz potential, which will phIi-. a ill l.) .r role
in subsequent chapters. Our starting point is a general discussion of parity that does not
assume either spherical symmetry or angular separation from the outset.
3.1 Parity Decomposition of Spin- and Boost-Weighted Scalars
One crucial feature of the R;- ear--Wheeler analysis is the identification of even and
odd-parity modes. In the context of spherically symmetric backgrounds, where angular
dependence can be separated off using spherical harmonics, it is sufficiently simple to
achieve this decomposition by considering the behavior of the spherical harmonics under
a parity transformation directly. For (scalar, vector or tensor) functions defined on more
general 2-surfaces, this task can be cumbersome, if not outright impossible. Furthermore,
narrowing our focus to angular functions obfuscates the fact that there is something more
fundamental happening. It is the goal of this section to provide a more general description
of the parity decomposition, applicable to more general 2-surfaces without appealing to
separation of variables. We will also see that the GHP formalism is uniquely suited to
this description. The decomposition theorems we make use of are proven by Detweiler and
Whiting [50].
Our first assumption is that our spacetime manifold, M~, admits a spacelike, closed
2-surface, S, topologically a 2-sphere, with positive Gaussian curvature and a positive
definite metric given by
where m, and m, are two members of a null tetrad. For a spherically symmetric
background Fab is proportional to the metric of the (round) 2-sphere and m" and m"
can be directly associated with the background metric. More generally, we allow for the