Full Poincare-invariant equation of motion for an extended charged particle in an electromagnetic field

Abstract

For the rigid, nonrotating motion of an extended charge in an arbitrary electromagnetic field, an equation of motion is derived by Lorentz-invariantly calculating the 4-Lorentz force = external 4-force + 4-self-force, acting upon the charge. The equation of motion is invariant under any full Poincare transformation. It is a differential-integral equation for the particle's 4-centre in Minkowski space as a function of proper time. The twofold proper time integrated form of the equation of motion is suited to derive the solution iteratively if the initial conditions and the external electromagnetic field are given and the iteration procedere converges. Diverse approximations for the self-force, and with that also for the equation of motion, are derived. The (approximated) equation of motion is solved for two examples: (1) Rectilinear hyperbolic motion in a homogeneous electric field. (2) Spiral (nonrelativistic) motion in a homogeneous magnetic field. In both cases amazingly simple solutions result. The advantages of the equation of motion against other ones, for example the Lorentz-Dirac equation, are discussed. Keywords: Equation of motion, extended charged particle, self-force, effective inertial mass.