allows us to use Equation :38 in arbitrary backgrounds. Furthermore in the limit that of
and of become surface-forming (e.g. the a 0 limit of the K~err spacetime), Equation :39
becomes Equation :37. This is one avenue for understanding why parity pt i.is such
an important role in the perturbation theory of spherically symmetric backgrounds.
In the context of null tetrad formalisms we can see the seemingly unmotivated act of
performing parity decomposition, which does not generalize well, as arising from the
quite natural (and perhaps more fundamental) act of separating quantities into their
real and imaginary parts, which is entirely general. In this light, it makes sense that our
attention would be focused on parity because the first perturbative analysis took place
in the spherically symmetric Schwarzschild background in which one cannot differentiate
between the two decompositions but parity has significance there. Regardless, the only use
we make of these results, except for some remarks in ChI Ilpter 5, is below in the case of the
Schwarzschild background where the point is moot.
3.2 Regge-Wheeler
In this section we will provide a perturbative analysis equivalent to that of R< -~-- and
Wheeler for the odd-parity sector of the Schwarzschild spacetime. Though the results are
well known, our methods and language are sufficiently different and original that they shed
some new light on and bring an interesting perspective to the subject. The two keys to
our analysis are essentially the same as those of RW: the parity decomposition and the
RW gauge. Having already discussed the former, we will look now at the latter before
proceeding with the analysis.
3.2.1 The Regge-Wheeler Gauge
R< -~-- and Wheeler describe their gauge choice in terms of the -e-mode decomposition
of a gauge vector. This description is inadequate for our purpose and so our first task is to
translate the RW gauge into mode-independent form. This has been performed by Barack
and Ori [24] who obtained
sin2 Bh a ha~ = 0, (:310)