4.5 Discussion
With our new form of the perturbed Einstein equations, use of NP methods has
allowed us to treat the quite general class of type II spacetimes without either choosing
coordinates or finding a metric. In this context, the Held technique has allowed us to
exploit our form of the equations by enabling partial integration in solving SiI = 0 while
investigating the existence of the IRG. Additionally, the Held technique has allowed
us to completely characterize the residual gauge freedom and use it in the radiation
gauge construction. By explicit demonstration, this work establishes our new form of
the perturbed Einstein equations as a powerful tool within perturbation theory, both in
conjunction with the Held technique and otherwise.
For perturbations with 1 = 0, our characterization of the residual gauge freedom
is sufficiently complete that we can explicitly demonstrate the required gauge choice
to remove any non-zero solution for the trace obtained via SiI = 0. Thus, in type II
spacetimes, radiation gauges can be established by a genuine gauge choice, even if only
after a solution of SiI = 0 is chosen.
There are subtle differences between the general type II case and the more restricted
type D case, as there are also in the construction of Hertz potentials for the two cases.
Stewart [21] writes out the type II case rather fully for an IRG. In this case, the
perturbation in 90 is tetrad and gauge invariant, while the potential satisfies the adjoint
(in the sense detailed by Wald [55]) of the s = +2 Teukolsky equation. Remarkably, in the
type D case, this adjoint is actually the s = -2 Teukolsky equation, also satisfied by the
gauge and tetrad invariant perturbation in 94. In the type II case, the adjoint equation
is the same as in type D, but 94 is HO longer tetrad invariant. Compared to the type D
result, the expression for 94 giVen by Stewart has many extra terms depending on W' and
o-', so presumably it does not satisfy the same equation as the potential. As a consequence,
metric reconstruction would be restricted to being built around the perturbation for Wo
(c.f. the comments at the end of OsI Ilpter 3).