Equivalence between electromagnetic self-energy and self-mass

Abstract

A cornerstone of physics, Maxwell's theory of electromagnetism, apparently contains a fatal flaw. The standard expressions for the electromagnetic field energy and self-mass of an electron of finite extension do not obey Einstein's famous equation, $E=mc^2$, but instead fulfill this relation with a factor 4/3 on the left-hand side. Many famous physicists have contributed to the debate of this so-called 4/3-problem but without arriving at a complete solution. Here, a comprehensive solution is presented. The problem is caused by an incorrect treatment of rigid-body dynamics. Relativistic effects are important even at low velocities and equivalence between electromagnetic field energy and self-mass of the electron is restored when these effects are included properly. In a description of the translational motion of a rigid body by point-particle dynamics, its mechanical energy and momentum must be defined as a sum of the energies and momenta of its parts for fixed time, not in the laboratory as in the standard expressions but in the rest frame of the body, and for consistency of the description, the energy and momentum of the associated field must be defined in the same way.