of the perturbation (6Q) and the background (4~Q). Quantities that satisfy 4~Q = 0
for every (0 are therefore called gauge invariant. It is straightforward to see that the
perturbation of Q is gauge invariant if and only if: (1) Q vanishes in the background,
(2) Q is a constant scalar in the background or (3) Q is a constant linear combination of
K~roenecker deltas. This is a result originally due to Sachs [47]. A direct consequence of
this fact is that the metric perturbation, arguably the most fundamental quantity we deal
with, fails to be gauge invariant. Fortunately, type D spacetimes come equipped with two
gauge invariants, I,, and tb4, Which have simple expressions in terms of the components of
the metric perturbation. As we will see, appropriate use of gauge freedom simplifies our
computations tremendously.
2.6 GHPtools A New Framework for Perturbation Theory
With the basic formalism in place, we are ready to present the tools that form the
basis of the subsequent chapters. The motivation for our framework comes from two
places: (1) the desire to take advantage of gauge freedom in standard metric perturbation
theory and (2) the success of the GHP formalism in perturbation theory. As mentioned
in the previous chapter, gauge freedom proved absolutely crucial for the RW analysis
and that of Cohen & K~egeles [20], C!!. I.1, i.---1:! [18], and Stewart [21], and it will
certainly pll li- a central role in any future description of metric perturbations. The
second ingredient, the GHP formalism comes with several advantages. First of all, the
inherent coordinate independence and notational economy makes calculations in general
spacetimes tractable. Furthermore, by virtue of the Goldberg-Sachs theorem, we can deal
with the entire class of type D spacetimes at once. Additionally, spin- and boost- weights
provide useful bookkeeping and, as we'll see, a useful context for understanding the roles
that various quantities pll li-. Last but not least, the use of a spin coefficient formalism has
proved absolutely crucial for studying perturbations of anything other than spherically
symmetric spacetimes. We will put these ideas together to compute the perturbed Einstein
equations in a mixed tetrad-tensor form. This is the heart of our work.