arising from a wave equation for the perturbed Riemann tensor, using standard methods
[13]. In either case, the result, written here in Bci-;r-Lindquist coordinates, is Teukolsky's
master equation (written here in accord with [14])
d 8 1 8 8 2
d (T2+ 2) +ta_ s(r-M)) -4s(r+iacos0)
8 8 1 8 8
+n" sin2 8 (Sin2 H iS COS 8
8 cos 8 8 cos 8 si2
x As/2 i' __ s~a/2C, s17
where s: = +2 corresponld to the W~eyl scalars ,',, anld I#2-4/3 4, TSpectively. Th'le Weyl
scalars are perturbations of the extremal spin components of the curvature tensor.
The significance of the Weyl scalar ~4 is that far away from the source of gravitational
radiation
~4 N h+ ix, 18
where h+ and hx are the two polarizations of outgoing gravitational radiation in the
transverse traceless gauge. Similar results hold for I',, and incoming radiation. For other
values of s, solutions correspond to fields of other spin: s = 0 is the massless scalar
wave equation, a = +1/2 the Weyl neutrino, a = +1 the Maxwell field, a = +3/2 the
Rarita-Schwinger field, and so on. Note that angular separation necessarily involves time
separation for a / 0.
Separated solutions to Equation 1-17 are of the form ~, = e-iwqeim*,R(r),S(aw,1 8)
(omitting the e, m and w subscripts). The angular functions, sS(aw, 8), are generally
referred to as "spin weighted spheroidal harmonics". In the limit that aw = 0,, s,,em(
reduce to the standard spin weighted spherical harmonics (cf. Appendix D), which are
interrelated by the spin raising and lowering operators, a and 8' [15], developed in the
following chapter. For aw / 0, solutions correspond to functions of different spin weight,
but the ,S(aw, 8) no longer share common eigenvalues. Thus a metric reconstruction
based on spin weight +2 functions would be incompatible with one based on spin weight 0