0 60 10 32 v 1 9 J an 2 00 6 Gauge Invariant Treatment of the Energy Carried by a Gravitational Wave

We present a completely gauge invariant treatment of the energy carried by a gravitational fluctuation in a general curved background. Via a varia-tional principle we construct an energy-momentum tensor for gravitational fluctuations whose covariant conservation condition is gauge invariant. With contraction of this energy-momentum tensor with a Killing vector of the background allowing us to convert the covariant conservation condition into an ordinary one, via spatial integration we are able to relate the time derivative of the total energy to an asymptotic spatial momentum flux, with this integral relation itself also being completely gauge invariant. It is only in making the simplification of setting the asymptotic momentum flux to zero that one actually loses manifest gauge invariance, with only invariance under asymp-totically flat gauge transformations 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 gravita-tional antenna in a completely gauge invariant manner, no matter how badly behaved at infinity the gauge function might be. In standard treatments of the energy carried by a gravitational wave, the use of a non-covariant energy-momentum pseudo-tensor totally obscures the covariance and gauge issues involved, while additionally forcing one to only admit those particular gauge transformations which are are asymptotically flat. However, with the full gauge invariance of general relativity equally holding for asymptotically badly-behaved gauge transformations as well, and with the response of a gravitational antenna to a gravitational wave needing to be invariant under all gauge transformations both well-or badly-behaved if such a response is to be physically meaningful, it is necessary to provide a treatment of gravitational waves which takes the badly-behaved ones into account as well. In this paper we provide such a treatment, using an approach which retains full gauge invariance at every step of the way.