The resulting proposal for a generalized RW gaugfe is
hmm = 0,
hmm = 0,
(3-14)
(B + atr + b-r')hlm + (8' + atr + b-r')hlm = 0,
(8' + b-r + a-r')hm + (B + b-r + a-r')hm = 0,
where a and b are (generally complex) constants that must be determined by some
other means. Note that the form of Equations 3-14 is restricted by requiring the gauge
restrictions to be invariant under both prime and complex conjugation. The full utility of
the generalized RW gauge remains to be explored, but it is clear that any simplification it
brings will apply uniformly to all type D spacetimes.
3.2.2 The Regge-Wheeler Equation
With the pieces in place, we turn our attention to the odd-parity perturbations of the
Schwarzschild spacetime. Starting with the description of the background, we have
p = p, p' = p', and '_= ,(3-15)
with all other background quantities vanishing, so the situation is immediately simplified.
Next we proceed with the parity decomposition by writing the components of the metric
perturbation as, for example, him = h +Ib, him h""" ihgg, etc. Note the relative
minus signs between the odd-parity bits and their complex conjugates. From here on
we will specialize to odd-parity and thus drop the "odd" labels and factors of i since no
confusion can arise. With this specialization, our gauge conditions now read:
hmm 0
kmm = 0
(3-16)
a'hlm c7le.. = 0
B'hm cll, = 0.