Returning to the Einstein equations with this simplified description of the metric
perturbation leads, after some manipulation, to the R;- ea-~~;-Wheeler-Zerilli equations, which
can be compactly described in a single expression, namely
82 o,e d2 o1e
-m -" +-Vo~e(r)qQ = 0, (1-15)
where the letters 'o' and 'e' stand for odd and even, respectively, r* = r + In(-~ 1) just
pushes the horizon out to infinity and ., is the appropriate master variable. Two aspects
of this result are noteworthy: (1) the two degrees of freedom have completely decoupled
and (2) these equations are separable in the Schwarzschild background. These two features
are desirable for any perturbative description of any background spacetime.
1.2 Perturbations of Kerr Black Hole Spacetimes
Unfortunately, the techniques used by RW to obtain a perturbative description of the
Schwarzschild spacetime are of little use when the background geometry possesses only
axial symmetry. Such is the case for the K~err geometry, which describes a rotating black
hole. In Bci-; r-Lindquist coordinates, its metric takes the form
ds2 __ 2Mr,, 4Mar sin2 8P"2 d
P2 22
where p2 __ r2 2 a"COS2 Ha 2 2M~r + a2, M~ is the mass and a = J/M~ is the angular
momentum per mass of the black hole. The spin coefficient formalism of Geroch, Held and
Penrose [9] developed in the next chapter has proved to be fundamental in virtually every
perturbative description of the K~err spacetime.
The first successful perturbation analysis of the K~err geometry was performed
by Teukolsky in a series of papers beginning in 1973 [10-12]. Teukolsky took as his
starting point the perturbed Bianchi identities in a spin coefficient formalism. Each
quantity is perturbed away from its background value and only first order terms are kept.
Equivalently, though with considerably more effort, Teukolsky's result can also be seen as