functions. This incompatibility does not arise for Schwarzschild, where reconstruction from
solutions of the RW equation can translate into comparable metric reconstruction from the
Weyl scalars, since there is a unique way of representing tensors on the sphere.
The spin weighted spherical (and spheroidal) harmonics fail to be defined for e < |8|
and thus the Teukolsky equation can give us no information about the -E = 0, 1 modes.
This is not a surprise since Iel, and #'4 are comporterts of the curvature tensor, which
carries information about the quadrupole (and higher multiple) generated gravitational
waves. In fact, Wald has shown [16] that for vacuum perturbations each of I',, and
('4 is SUffleient to characterize the perturbation of the spacetime, up to shifts in mass
and angular momentum. In Schwarzschild, these lower multiple moments can he
expressed appropriately in terms of spherical harmonics using the RW formalism, but any
comparable expressions for the K~err case would be incompatible with metric coefficients
constructed from spin weight +2 functions (i.e., they would be expressed in different
bases). Yet, these low--A multiple moments are urgently sought, since they convey
information about the energy and both the axial and non-axial components of the angular
momentum of a particle in orbit around the black hole. 1\oreover, in recent calculations
demonstrating the precise relation of the -E = 0, 1 multipoles in Schwarzschild to shifts in
the mass and angular momentum, Detweiler and Poisson [17] emphatically point out that
such shifts are just as important as the radiating multipoles for describing the motion of a
small black hole orbiting a supermassive black hole. The non-radiated multiple moments
are the subject of C'!s Ilter 6.
Solutions of the Teukolsky equation lead quite naturally to metric perturbations
through the use of Hertz potentials which solve Equation 1-17. We now turn our attention
to this subject.
1.3 Metric Perturbations of Black Hole Spacetimes
The first explicit solutions for metric perturbations given in terms of Hertz potentials
were written down by C'!,l~!!. 1,.---1:! [18] and Cohen and K~egeles [19]. This work was