Fields of Moving Multipoles

Abstract

IN free space the electromagnetic potential φ^µ of Maxwell's equations is related to the sources of the field j^µ by the equation □φ^µ = j^µ. If the source of the field is an arbitrarily moving charge, the retarded solution for φ^µ may be expressed as an integral over the world line of the charge (z^µ = z^µ (τ)): where D_ret is the four-dimensional retarded Green's function. Concise expressions for the fields F^µν = φ^µ,ν − φ^ν,µ can then be obtained by differentiating under the integral sign in (1). The same processes may be applied to the calculation of the radiation fields F^µν_rad = F^µν_ret − F^µν_adv (by changing the Green's function) and the method affords an effective means of evaluating the radiation field on the world line of the charge¹. The latter fields have the value: