carried out at a time when there was not a strong urge to obtain solutions related to
very specific sources, and so it gave a successful way of creating metric perturbations in
vacuum. Recent interest in EMRIs as a source for gravitational waves has developed a
need for metric perturbations related to known sources, for which curvature perturbations
may be obtained by solving Teukolsky's equation. For this class of problem, the source
is highly localized, and most of the perturbed spacetime can still be treated as vacuum.
We give first a description of solutions to the inversion problem in vacuum, pI lingf special
attention to limitations of each approach. Before we proceed, it will be helpful to give a
brief overview of Hertz potentials in the more immediately familiar context of Maxwell's
equations in flat space. Note that the methods presented here rely crucially on spin
coefficient methods, though we have attempted to keep reference to such methods minimal
for now.
1.3.1 Hertz Potentials in Flat space
To illustrate the essentials of Hertz potential methods we consider the source-free
Maxwell equations in flat spacetime, in essentially the form Cohen and K~egeles attempted
to generalize to curved spacetime [20]:
VaFab = 0 and Eabcd aFed = 0. (1-19)
As usual a vector potential, A,, is introduced and the Lorentz gauge,Vas = 0, is imposed
so that the Maxwell equations lead directly to OA, = 0.
Then a Hertz potential Hab iS introduced via A" = VbHab, Where Hab = -Hba, SO
that the Maxwell field, Fab, iS obtainable by two derivatives of Hab. However, Hab iS Only
defined up to a transformation of the type
Hab Hab + cl~cb + a b ~b a, (1-20)
where Mca"b iS completely antisymmetric and oC" = 0. It is easy to see that in flat
spacetime, where derivatives commute, the transformation Equation 1-20 only changes