Even though the energy carried by a gravitational wave is not itself gauge
invariant, the interaction with a gravitational antenna of the gravitational
wave which carries that energy is. It therefore has to be possible to make some
statements which involve the energy which are in fact gauge invariant, and it
is the objective of this paper to provide them. In order to develop a gauge
invariant treatment of the issues involved, we construct a specific action for
gravitational fluctuations which is gauge invariant to second perturbative
order. Then, via variation of this action, we obtain an energy-momentum tensor
for perturbative gravitational fluctuations around a general curved background
whose covariant conservation condition is also fully gauge invariant to second
order. Contraction of this energy-momentum tensor with a Killing vector of the
background conveniently allows us to convert this covariant conservation
condition into an ordinary conservation condition which is also gauge invariant
through second order. Then, via spatial integration we are able to obtain a
relation involving the time derivative of the total energy of the fluctuation
and its asymptotic spatial momentum flux which is also completely gauge
invariant through second order. It is only in making the simplification of
setting the asymptotic momentum flux to zero that one would actually lose
manifest gauge invariance, with only invariance under those particular gauge
transformations which leave the asymptotic momentum flux zero then remaining.
However, if one works in an arbitrary gauge where the asymptotic momentum flux
is non-zero, the gravitational wave will then deliver both energy and momentum
to a gravitational antenna in a completely gauge invariant manner, no matter
how badly behaved at infinity the gauge function might be.Comment: 13 pages, revtex4. Final version. To appear in Phys. Rev.
