Relativity, scaling, and electromagnetic radiation equilibrium for circular orbits

Abstract

The radiation emitted by a charged particle moving in a circular orbit requires that the orbital speed of the particle is less than the speed of light in vacuum. This crucial relativistic restriction is lost in any treatment which combines nonrelativistic mechanics with classical electrodynamics through the nonrelativistic Larmor radiation formula or the dipole approximation, which approximations correspond to taking only the lowest power of velocity. A notable example of the resulting failure involves the derivation of the blackbody radiation spectrum within classical physics. Nature contains a smallest electric charge e and a largest speed c. Both these fundamental constants should appear in a classical theory of nature. We connect the assumptions regarding fundamental constants to the scaling aspects of classical theories. Nonrelativistic mechanics exhibits scaling which is entirely different from that found in relativistic classical electrodynamics where only Coulomb potentials are allowed and the constants e and c both appear. The scaling aspects are reflected in the radiation spectra which different theories predict for thermal radiation equilibrium. The Rayleigh–Jeans spectrum reflects the scaling aspects of nonrelativistic classical mechanics whereas the classical electromagnetic zero-point spectrum and the Planck spectrum share the scaling aspects of relativistic classical electrodynamics which includes both e and c.