such as one coming from a source. The importance of this consideration can be seen
immediately from Equation B-1 of Appendix B, in which every term would be removed
by Equations 4-5 and 4-6, rendering Equation B-1 inoperable whenever it has a non-zero
source. In the next section we will look to the perturbed Einstein equations to determine
the circumstances under which we can safely impose all five of the conditions that
constitute the IRG.
It is useful to note the similarity between the full IRG, (4-1), and the more commonly
known transverse traceless (TT) gauge defined by
V"hab = 0, gab ab = 0, (4-7)
which, at a glance, also appears to be over-specified. In fact, the TT gauge exists for
any vacuum perturbation of an arbitrary, globally hyperbolic, vacuum solution [61],
because imposing the differential part of the gauge does not exhaust all of the available
gauge freedom. Interestingly enough, Stewart's analysis in terms of Hertz potentials
[21] begins by considering a metric perturbation in the TT gauge. However, in order to
construct the curved space analogue of a Hertz potential, he is compelled to perform a
transformation that destroys Equation 4-7 and instead yields a metric perturbation in
the IRG.2 Furthermore it appears that the restriction to type II spacetimes is essential
for Stewart's analysis. From these observations, we expect radiation gauges to exist under
conditions less general than those required for the existence of the TT gauge. At the
same time, we should not be surprised that the IRG inherits the feature of residual gauge
freedom.
Consider a gauge transformation on the metric perturbation generated by a gauge
vector, go. To create a transformed metric in the 1- h gauge, the gauge conditions in
2 See [21] or the electromagnetic example in C'!s Ilter 1 for a more detailed explanation.