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 v in a weak-field, slow-motion approximation, where v is a characteristic velocity associated with the motion of the source. These moments are defined for all three types of radiation by relations of the form Ψ(t, ~ x) = r ∑ lm Mlm(u)Ylm(θ, φ), where Ψ is the radiation field at infinity andMlm are the radiative moments, functions of retarded time u = t − r − 2M ln(r/2M − 1); M is the mass parameter of the Schwarzschild spacetime and (t, ~x) = (t, r, θ, φ) are the usual Schwarzschild coordinates. For all three types of radiation the moments share the same mathematical structure: To zeroth order in v, the radiative moments are given by relations of the form Mlm(u) ∝ (d/du) ∫ ρ(u, ~x) r Ȳlm(θ, φ) d~x, where ρ is the source of the radiation. A radiative moment of order l is therefore given by the corresponding source moment differentiated l times with respect to retarded time. To second order in v, additional terms appear inside the spatial integrals, and the radiative moments become Mlm(u) ∝ (d/du) l ∫ [1 + O(r∂ u) + O(M/r)] ρ r l Ȳlm d~x. The term involving r∂ u can be interpreted as a special-relativistic correction to the wave-generation problem. The term involving M/r comes from general relativity. These correction terms of order v are nearzone corrections which depend on the detailed behavior of the source. Furthermore, the radiative multipole moments are still local functions of u, as they depend on the state of the source at retarded time u only. To third order in v, the radiative moments become Mlm(u) →Mlm(u) + 2M ∫ u −∞ [ln(u− u) + const]M̈lm(u ) du, where dots indicate differentiation with respect to u. This expression shows that the O(v) correction terms occur outside the spatial integrals, so that they do not depend on the detailed behavior of the source. Furthermore, the radiative multipole moments now display a nonlocality in time, as they depend on the state of the source at all times prior to the retarded time u, with the factor ln(u − u) assigning most of the weight to the source’s recent past. (The term involving the constant is actually local.) The correction terms of order v are wave-propagation corrections which are heuristically understood as arising from the scattering of the radiation by the spacetime curvature surrounding the source. The radiative multipole moments are computed explicitly for all three types of radiation by taking advantage of the symmetries of the Schwarzschild metric to separate the variables in the wave equations. Our calculations show that the truly nonlocal wave-propagation correction — the term involving ln(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 corrections. Finally, we produce an alternative derivation of the radiative moments of a scalar field based on the retarded Green’s function of DeWitt and Brehme. This calculation shows that the tail part of the Green’s function is entirely responsible for the wavepropagation corrections in the radiative moments.