Gauge Invariant Treatment of the Energy Carried by a Gravitational Wave
Abstract
We present a completely gauge invariant treatment of the energy carried by a gravitational fluctuation in a general curved background. Via a variational 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 asymptotically 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 gravitational antenna in a completely gauge invariant manner, no matter how badly behaved at infinity the gauge function might be.Keywords
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