which also defines Ro, a quantity annihilated by P. Then we can rewrite Equation 4-16 as
hm =1ao+aoP P a + bo] (p + p). (4-21)
2 pp 2
In a similar fashion, we rewrite Equation 4-18 as
(4-22)
(8~'d + aa' "b" p"o pro a
in which each coefficient in big square brackets is purely real. Now, suppose we have
a particular solution for SiI = 0 (i.e., ao, ao and be are fixed) and our task is to solve
for the components of the gauge vector which removes this solution. By comparing
Equations 4-21 and 4-22 we see that, for any given (mo and (mo, we can fix (to (up to a
solution of D (to = 0) via
p1 0 o (a + ao -I(mo + igo~), (4-23)
2 2
and we can fix (no by setting
1 1, 1~~" -a~0 1
6 o= (o a.)n + be g' g o plol" -t /m m o (4-24)
2 2 2
to completely eliminate the nonzero hmm, thus imposing the full IRG while still leaving
two completely unconstrained degrees of gauge freedom, (mo and (mo. Once in the IRG,
Equations 4-23 and 4-24, with ao, ao and be set to zero and (mo and (mo arbitrary,
give the remaining components of a gauge vector preserving the IRG. It is currently
unclear how to take advantage of this remaining gauge freedom to simplify the analysis of
perturbations in the full IRG.
4.4 Imposing the IRG in type D
Type D background metrics are of considerable theoretical and observational interest
since they include both the Schwarzschild and K~err black hole spacetimes. K~innersley first
obtained all type D metrics by integrating the N. ein-! lIs-Penrose equations [33]. While the