Enigmatic factor of 4/3 in electromagnetic momentum of a moving spherical capacitor

Abstract

The electromagnetic energy-momentum of a moving charged spherical capacitor may be calculated by a 4-vector Lorentz transformation from the energy in the rest frame. However, energy-momentum of the moving system computed directly from electromagnetic fields yields extra terms; in particular a factor of 4/3 in momentum appears, similar to that encountered in the classical electron model, where this enigmatic factor has been a source of scourge for more than a century. There have been many attempts to eliminate this `unwanted' factor, noteworthy among them is a modification in electromagnetic field energy-momentum definition that has entered even standard textbooks. Here it is shown that in a moving charged spherical capacitor, some additional contributions to the electromagnetic energy-momentum arise naturally from electromagnetic forces in system or equivalently from terms in the Maxwell stress tensor; contributions that do not otherwise show up in 4-vector transformations. Furthermore, a similar factor of 4/3 in the momentum of a perfect fluid comprising a randomly moving ultra-relativistic gas molecules or an isotropic photon gas, filling an {\em uncharged} spherical capacitor in motion, appears owing to the contribution of pressure. Thus, genesis of the "enigmatic" factor of 4/3 can be traced to pressure or stress whose presence in the system may be of non-electromagnetic origin and where the proposed modifications in energy-momentum definition do not even come into picture. No modifications in the definitions of energy-momentum of moving fluids have ever been required; physics should be the same in electromagnetic case as well, implying there is nothing amiss in the standard definition of electromagnetic energy-momentum.