Relativistic time-dependent multipole analysis for scalar, electromagnetic, and gravitational fields

That one can, in principle, calculate multipole fields for rapidly varying bounded sources, that is, in situations where the long-wavelength approximation is not valid, is well known. However, the usual procedure of doing these calculations using the time-Fourier-transformed fields masks the fact that the multipole moments are indeed true spatial moments of time-dependent source functions. A time-dependent multipole analysis is presented here for scalar, electromagnetic, and linear gravitational fields which shows that multipole moments can always be defined in the form r^'lY₋₌^* (^', ^') {}₋ ({r}^', t - rc) d^3r^', where {}₋ is a suitable l-dependent weighted time average of appropriate source functions. The analysis is done first for the scalar field where new expansions for the retarded Green's function are discussed. Debye potentials are then used to extend the analysis to electromagnetism and linear gravitation.