background. Now suppose we've identified some geometric quantity of interest (could be
scalar, vector, tensor, etc., for simplicity we write it with no indices), Q = Q(A), and we
are interested in its first order perturbation, 6Q, towards the physical spacetime, evaluated
in the background. Before we can compute anything we must confront the issue of how
to relate quantities on two different curved manifolds. One can imagine introducing a
(suitably well-behaved) vector field, (", that connects points in the physical spacetime to
points in the background. Then, to compute 6Q, we evaluate Q at some point p + 6p in the
physical spacetime, pull the result back along (" to the background spacetime, subtract
from it the value of Q at a point p in the background, divide by 6p and take the limit as
6p 0 The mathematical apparatus for performing this task is the Lie derivative. Thus,
the first order perturbation, 6Q, to a quantity, Q, evaluated in the background spacetime
is given by
6Q = 4Q(A)~= (o (261)
The important point about this prescription is the fact that (" not only fails to be unique,
but there is, in general, no preferred choice for it. A choice of (" is more commonly known
as a choice of gauge. According to Equation 2-61, the difference between 6Q computed
with (" and rl" is given by
sQg sQ, = 4_~,Q,
and so we define 6Q, the gauge transformation of 6Q by
6Q = 6Q1 4Q3. (2-62)
Note that a gauge transformation in this sense represents a change in the way we identify
points in the physical spacetime with points in the background. This is to be distinguished
from a coordinate transformation, which changes the labeling of coordinates in both the
physical and background spacetimes.
The significance of Equation 2-62 is that unless 4~Q = 0 for every (", there is some
ambiguity in identifying the perturbation-we can't differentiate between the contributions