The first step is to introduce new derivative operators P/ and 8I = 8 such that they
commute with P when acting on quantities that P annihilates,3
[9, 1 P]xo = [P, 8]xo 0, [P, 8 ]xo 0, (4-11)
where [a, b] denotes the commutator between a and b. The explicit form of the operators
is given in Appendix C. The next step, the heart of Held's method, is to exploit the
GHP equation Pp = p2, and its complex conjugate, Pp = p2, to express everything as
a polynomial in terms of p and p, with coefficients that are annihilated by P. Held's
method is then brought to completion by choosing four independent quantities to
use as coordinates [56, 62]. In this work, we will not take this extra step. For type II
spacetimes (and the accelerating C-metrics), this step has not been carried out, while for
all remaining type D spacetimes, it has been carried through to completion [45, 46].
In a spacetime more general than type II, there is no possibility of having a repeated
PND. When a repeated PND exists, we can appeal to the Goldberg-Sachs theorem [32]
and set is = o- = Wo = ~1 = 0 in Equations B-1-B-7. Following Held's partial integration
of Petrov type II backgrounds [56], we also perform a null rotation (keeping la fixed, but
changing n") to set -r = 0. As a consequence, it follows from the GHP equations that
-r' = 0. Now we are in a position to address the question of when the full IRG can be
imposed. First we apply the 1- & gauge conditions in Equations 4-5 to Equations B-1-B-7.
While most of the perturbed Einstein equations depend on several components of the
metric perturbation, after imposing Equations 4-5, the expression for SiI depends only on
hmm and the ll-component of the perturbed Einstein tensor simply becomes
{(pD p p) + 2pp~hmm {((9 2p)(P + p p)}hmm = 8xri, (4-12)
3 Such quantities are denoted with the degree mark, o, as in Pxo = g