40 by the addition of the term V" b b and therefore, in the Lorentz gauge, contributes
nothing to the fields. In practice, Equation 1-20 is used to reduce the Hertz hivector
potential to a single complex (or two real) scalar potential(s). Herein lies the power
of the method. However, moving to curved-space naturally complicates things. While
the wave equations are modified to include curvature pieces, the transformation in
Equation 1-20 is retained (see Cohen and K~egeles [20] and Stewart [21]). As a result,
the field equations are still satisfied and the six components of Hab are still reduced to
two, but the transformation in Equation 1-20 explicitly breaks the Lorentz gauge because
derivatives no longer commute. In this way a new gauge is introduced that brings with it
complications for the inclusion of sources. The necessary and sufficient conditions for the
existence of this gauge are the subject of C'!s Ilter 4.
1.3.2 The Inversion Problem for Gravity
The formulation of the gravitational Hertz potential proceeds analogously to that of
its (flat space) electromagnetic counterpart, with a few differences. For one, the result is a
metric perturbation in one of two complimentary gauges. Additionally, the potential itself
is a solution to the Teukolsky equation for s = +2 (or s = -2; the choice of the sign of
s determines which gauge the metric perturbation is in), though it is not the curvature
perturbation of the metric perturbation it generates. In analogy to the electromagnetic
example above, the components of the metric perturbation are given by two derivatives of
the potential. The natural language in which to express the metric perturbation arising
from the Hertz potential is again the spin coefficient formalism of Newman and Penrose
[22], or its modification due to Geroch, Held and Penrose [9]. Thus we postpone the
formal development of the subject until ('I Ilpter 3, when the necessary formalism is in
place, and instead offer an overview of the general process and documented research on the
topic of reconstructing the metric perturbation from solutions to the Teukolsky equation
(assuming the form of metric perturbation is prescribed), which we will refer to as the
inversion problem.