RADIATIVE MULTIPOLE MOMENTS OF INTEGER-SPIN FIELDS IN CURVED SPACETIME

Radiative multipole moments of scalar, electromagnetic, and linearized gravitational fields in Schwarzschild spacetime are computed to third order in nu in a weak-field, slow-motion approximation, where nu is a characteristic velocity associated with the motion of the source. The se moments are defined for all three types of radiation by relations o f the form Psi(t,(x) over right arrow)=r(-1)Sigma(lm)M(lm)(u)Y-lm(thet a,phi), where Psi is the radiation field at infinity and M-lm are the radiative moments, functions of retarded time u=t-r-2M In(r/2M-1); M i s the mass parameter of the Schwarzschild spacetime and (t,(x) over ri ght arrow)=(t,r,theta,phi) are the usual Schwarzschild coordinates. Fo r all three types of radiation the moments share the same mathematical structure: To zeroth order in nu, the radiative moments are given by relations of the form M-lm(u)proportional to(d/du)(l) integral rho(u,( x) over right arrow)r(l) (Y) over bar(lm)(theta,phi)d (x) over right a rrow, where rho is the source of the radiation. A radiative moment of order l is therefore given by the corresponding source moment differen tiated l times with respect to retarded time. To second order in nu, a dditional terms appear inside the spatial integrals, and the radiative moments become M-lm(u)proportional to(d/du)(l) integral[1+O(r(2) part ial derivative(u)(2))+0(M/r)]rho r(l) (Y) over bar(lm)d (x) over right arrow. The term involving r(2) partial derivative(u)(2) can be interp reted as a special-relativistic correction to the wave-generation prob lem. The term involving M/r comes from general relativity. These corre ction terms of order nu(2) are near-zone corrections which depend on t he detailed behavior of the source. Furthermore, the radiative multipo le moments are still local functions of Ik, as they depend on the stat e of the source at retarded time u only. To third order in nu, the rad iative moments become M-lm(u)-->M-lm,(u)+2M integral(-infinity)(u)[ln( u-u')+const]M-lm(u') du', where overdots indicate differentiation with respect to u'. This expression shows that the O(nu(3)) correction ter ms occur outside the spatial integrals, so that they do not depend on the detailed behavior of the source. Furthermore, the radiative multip ole moments now display a nonlocality in time, as they depend on the s tate of the source at all times prior to the retarded time u, with the factor In(u-u') assigning most of the weight to the source's recent p ast. (The term involving the constant is actually local.) The correcti on terms of order nu(3) are wave-propagation corrections which are heu ristically understood as arising from the scattering of the radiation by the spacetime curvature surrounding the source. The radiative multi pole moments are computed explicitly for all three types of radiation by taking advantage of the symmetries of the Schwarzschild metric to s eparate the variables in the wave equations. Our calculations show tha t the truly nonlocal wave-propagation correction-the term involving In (u-u')-takes a universal form which is independent of multipole order and field type. We also show that in general relativity, temporal and spatial curvatures contribute equally to the wave-propagation correcti ons. Finally, we produce an alternative derivation of the radiative mo ments of a scalar field based on the retarded Green's function of DeWi tt and Brehme. This calculation shows that the tail part of the Green' s function is entirely responsible for the wave-propagation correction s in the radiative moments. [S0556-2821(97)03820-4]. PACS number(s): 0 4.25.Nx, 04.30.Db, 04.40.Nr.