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

17.5 Post-Newtonian wave generation17.5.1 Field equationsPost-Newtonian wave generation 349We consider a general compact-support stress–energy tensor T µν describing theisolated source, and we look for the solutions, in the form of a (formal) post-Newtonian expansion, of the Einstein field equations,R µν − 1 2 gµν R = 8πGc 4 T µν , (17.54)and thus also of their consequence, the equations of motion ∇ ν T µν = 0 of thesource. We impose the condition of harmonic coordinates, i.e. the gauge condition∂ ν h µν = 0; h µν = √ −gg µν − η µν , (17.55)where g and g µν denote the determinant and inverse of the covariant metric g µν ,and where η µν is a Minkowski metric: η µν = diag(−1, 1, 1, 1). Then the Einsteinfield equations (17.54) can be replaced by the so-called relaxed equations, whichtake the form of simple wave equations,£h µν = 16πGc 4 τ µν , (17.56)where the box operator is the flat d’Alembertian £ = η µν ∂ µ ∂ ν , and where thesource term τ µν can be viewed as the stress–energy pseudotensor of the matterand gravitational fields in harmonic coordinates. It is given byτ µν =|g|T µν +c416πG µν . (17.57)τ µν is not a generally-covariant tensor, but only a Lorentz tensor relative to theMinkowski metric η µν . As a consequence of the gauge condition (17.55), τ µν isconserved in the usual sense,∂ ν τ µν = 0 (17.58)(this is equivalent to ∇ ν T µν = 0). The gravitational source term µν is a quitecomplicated, highly nonlinear (quadratic at least) functional of h µν and its firstandsecond-spacetime derivatives.We supplement the resolution of the field equations (17.55) and (17.56) bythe requirement that the source does not receive any radiation from other sourceslocated very far away. Such a requirement of ‘no-incoming radiation’ is to beimposed at Minkowskian past null infinity (taking advantage of the presence ofthe Minkowski metric η µν ); this corresponds to the limit r =|x| →+∞witht + r/c = constant. (Please do not confuse this r with the same r denoting theseparation between the two bodies in section 17.4.) The precise formulation ofthe no-incoming radiation condition is[ ∂limr→+∞ ∂r (rhµν ) + ∂ ]c∂t (rhµν ) (x, t) = 0. (17.59)t+ r c =constant