Self-Energy and Stability of the Classical Electron

The classical theory of the electron, as proposed by Abraham and Lorentz, is usually presented as beset by the difficulty that the momentum and velocity of its Coulomb field are incorrectly related kinematically: p = 4/3m_sv, where m_s is the electromagnetic mass defined by the electromagnetic self-energy. This problem also persists in the relativistic theory. It is shown here that the difficulty is eliminated from the relativistic theory by treating the integrals over the electromagnetic field in a relativistic fashion, i.e., taking note of their dependence on the motion of the electron. The surface dependence of the integrals representing the electromagnetic momentum and energy of the particle is essential and occurs whenever the matter tensor is not introduced. The nonrelativistic limit of this formulation then also leads to the correct relationship p = m_sv. The corrected Abraham-Lorentz theory still contains the stability problem, but this problem is no longer related to the transformation properties. It can be removed by renormalization.