(ed.). Gravitational waves (IOP, 2001)(422s).

52 Waves and energygeometries consist of a slowly and smoothly changing background metric whichis altered by perturbations of small amplitude and high frequency. If L and λ arethe characteristic lengths over which the background and ‘ripple’ metrics changesignificantly, we assume that the ratio λ/L will be very much smaller than unityand that |h µν | is of the same order of smallness as λ/L. In this way the totalmetric remains slowly changing on a macroscopic scale, while the high-frequencywave, when averaged over several wavelengths, will be the principal source of thecurvature of the background metric. This is the ‘short-wave’ approximation [24].Obviously this is a direct generalization of the treatment in chapter 2.5.2.1 Gauge transformation and invarianceConsider an infinitesimal coordinate transformation generated by a vector fieldξ α ,x α → x α + ξ α . (5.8)In the new coordinate system, neglecting quadratic and higher terms in h αβ ,itisnot hard to show that the general gauge transformation of the metric ish µν → h µν − ξ µ;ν − ξ ν;µ , (5.9)where a semicolon denotes the covariant derivative. We assume that thederivatives of the coordinate displacement field are of the same order as the metricperturbation: |ξ α,β |∼|h αβ |.Isaacson [24] showed that the gauge transformation of the Ricci andRiemann curvature tensors has the property( ) λ 2¯R µν (1) − R(1) µν ≈ (5.10)L( )¯R (1)λ 2αµβν − R(1) αµβν ≈ Lwhere R µν (1) and R (1)αµβνare the first order of Ricci and Riemann tensors (inpowers of perturbation h µν ) and an overbar denotes their values after the gaugetransformation. In our high-frequency limit, therefore, these tensors are gaugeinvariantto linear order, just as in linearized theory.5.2.2 Gravitational-wave actionLet us suppose that we are in vacuum so we have only the metric, no matter fields,but we work in the high-frequency approximation. The full metric is g µν (smoothbackground metric) +h µν (high-frequency perturbation). Our purpose is to showthat the wave field can be treated as a ‘matter’ field, with a Lagrangian and its