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

Variational principles and the energy in gravitational waves 51=− 1 ∫G µν √h µν −g d 4 x + O(2) (5.3)16πwhere ‘O(2)’ denotes terms quadratic and higher in h µν . All the divergencesobtained in the intermediate steps of this calculation integrate to zero since h µνis of compact support. This variational principle therefore yields the vacuumEinstein equations: G µν = 0.Let us consider how this changes if we include matter. This will help usto see how we can treat gravitational waves as a new kind of ‘matter’ field onspacetime.Suppose we have a matter field, described by a variable (which mayrepresent a vector, a tensor or a set of tensors). It will have a Lagrangian densityL m = L m (, ,α ,...,g µν ) that depends on the field and also on the metric.Normally derivatives of the metric tensor do not appear in L m , since by theequivalence principle, matter fields should behave locally as if they were in flatspacetime, where of course there are no metric derivatives. Variations of L m withrespect to will produce the field equation(s) for the matter system, but here weare more interested in variations with respect to g µν , which is how we will findthe matter field contribution to the gravitational field equations. The total actionhas the form:∫I = (R + 16π L m ) √ −g d 4 x, (5.4)whose variation is∫ √ ∫δ(R −g)δI =h µν d 4 x + 16π ∂(L √m −g)h µν d 4 x. (5.5)δg µν ∂g µνThis variation must yield full Einstein equations, so we must have the followingresult for the stress-energy tensor of matter:T µν√ −g = 2 ∂ L √m −g, (5.6)∂g µνleading toG µν = 8πT µν . (5.7)This way of deriving the stress-energy tensor of the matter field has deepconnections to the conservation laws of general relativity, to the way ofconstructing conserved quantities when the metric has symmetries and to the socalledpseudotensorial definitions of gravitational wave energy (see Landau andLifshitz 1962) [23]. We shall use it in the latter sense.5.2 Variational principles and the energy in gravitationalwavesBefore we introduce the mathematics of gravitational waves, it is important tounderstand which geometries we are going to examine. We have said that these