is a task that was first performed for the NP equations by K~innersley [33] and later by
Held [45] for the GHP equations. In ChI Ilpter 5, we will discuss the latter of these methods
in more detail.
2.5 Issues of Gauge in Perturbation Theory
One of the most important subtleties associated with perturbation theory in general
relativity is the concept of gauge invariance. The principal of general covariance tells
us that the interesting questions to ask are those that have answers that every observer
agrees upon. In the context of full (non-perturbative, nonlinear) relativity, this is ensured
by focusing on quantities that remain unchanged under coordinate transformations. In
perturbation theory, however, there is a new twist to the problem---there are now two
spacetimes of interest: the unperturbed background spacetime consisting of a manifold,
At with metric gAb (henceforth denoted by (Af, gab)) and the physical spacetime, (Af', 9ub '
that includes both the background and the perturbation. The question of how to relate
perturbations of quantities on (Af', 9u~b) to quantities on (Af, gab) in an unambiguous way
is fundamental for a well defined perturbation theory in general relativity. A complete
analysis, in the context of the GHP formalism, of this question was performed by Stewart
and Walker [46], whose basic results will be developed here. Before we address the
relativistic problem, we very briefly review first-order perturbation theory in a flat
spacetime. In that instance, we think of the quantity of interest, q = q(A), as being
parameterized by some A, so that q(0) corresponds to the unperturbed quantity and q(1)
is the fully perturbed quantity whose first-order perturbations we would like to consider.
It follows from writing q(A) as a Taylor series in A that the first-order perturbation, 6q, is
given by
To adapt this idea to our curved space problem, it helps to think of both the background
and physical spacetimes as members of a one-parameter family of spacetimes, (MAx, 9ub(X)
with A = 1 corresponding to the physical spacetime and A = 0 corresponding to the