The problem is that of finding a Hertz potential, given a solution (or both solutions)
to the Teukolsky equation. To make this more precise, we look to the expression of the
curvature perturbations, I',, (s = +2) and #'4 (S = -2), in terms of the Hertz potentials.
If we take the potential to satisfy the s = -2 Teukolsky equation then the perturbation
exists in the in going radiation gauge (IR G) and we have that
2,',, = DDDD[lIRG,] and (1 21)
2p-42;: I[4 IRG 12p- ('281 IRG (1221)
where = [de + s cot 8 i cs e 0,] + is sin 08d and D = A-l[(T2 + 2 t r d + a84] define
derivatives in orthogonall) null directions, p = -(r ia cos 8)-1 and WIRG is the potential.
While for a potential satisfying the s = +2 Teukolsky equation, we have a perturbation in
the outgoing radiation gauge (OR G), where
2p-4 4 = 2 dd 2 ORG] and (1 23)
2,II ,, = ORG 12p- ('2i31 ORG] ) (1-24)
where a = ~pp[(r2 + 2)dt rd + a84] and the complex conjugate of the operator
defined above, are also derivatives in null directions (mutually orthogonal to each other
and those defined by the operators in the IR G). These are the equations we would like
to invert for the potentials WIRG and ~ORG. Once this is done, the potential may then he
used to construct the metric perturbation. We now look at several different approaches to
this problem.
1.3.2.1 Ori's construction for Kerr
In principle, with solutions for the Weyl curvature perturbations on all of spacetime,
one could integrate along null directions to undo the derivatives in Equations 1-21 or
1-22 (or their ORG counterparts). Ori [23] has recently performed this task-integrating
Equation 1-21 in order to find the potential WIRG in terms of I',,