decomposition that takes into account both spin- and boost- weight. In the next chapter
we will make some more precise statements in this direction.
Recall our expression for the perturbed Einstein equations:
1 1 1
Sab c ~Cc ab ~a b cc ~c(a b)c gab ~c c dd ~c d cd)
2 2 2
By making the replacement V, i 0 and understanding hab aS referring to the tetrad
components of the metric perturbation given in Equation 2-63, we arrive at the perturbed
Einstein equations in GHP form:
1 1 1
Sab c~c ab -OaOb" cc c8O(a b)c gab c"Oc dd Oc d cd), (2-65)
2 2 2
which (right now, at least) don't look all that different! The tetrad components of
Equation 2-65 for an arbitrary algebraically special background spacetime are given in
Appendix B. Aside from the obvious cosmetic differences, there are several key distinctions
between Equation 2-65 and the standard form of metric perturbation theory worth
pointing out. First of all, our form lacks the background Einstein equations present in the
standard treatment. Taking their place are the background GHP equations and Bianchi
identities. Perhaps more importantly is the inherent coordinate independence. Coupled
with the concepts of spin- and boost-weight, this allows for a certain structural intuition
not present in coordinate based techniques. This point of view will be stressed throughout.
Writing Equation 2-65 is one thing, but actually computing it is another question
entirely, which we now turn our attention to.
2.6.2 GHPtools The Details
To perform such a computation for an arbitrary background spacetime is no small
task, even (or rather especially) in the standard tensor language. For this the aid of Maple
was enlisted. Unfortunately, at the time the computation was performed, there were no
Maple packages available for performing all such computations at the level of generality