4/3 problem, Poynting theorem, and electromagnetic energy-momentum tensor

Abstract

We show that the familiar 4/3 problem originates from the incorrect determination of the momentum of the electromagnetic field generated by an isolated charged particle, which results from an incorrect application of the Poynting theorem to the field. To demonstrate this, we consider a macroscopic problem: splitting a charged oil droplet into two identical smaller droplets in the rest frame K of the original droplet, and in another inertial frame K' moving with respect to K with some constant velocity, and arrive at a physically senseless solution in the framework of the standard approach. To solve this problem in a correct way, we suggest a modified Poynting theorem for an isolated charge and recalculate the momentum of the electromagnetic field of this charge, which yields the usual relativistic relationship between the field energy and momentum. Our approach is not at odds with the idea about "Poincare stresses"; however, in contrast to the standard method, it provides a usual relationship between the "Poincare stress-energy" and the associated momentum stress component. Finally, the continuity equation in four-dimensional form for the case of isolated moving charge is also proposed, which completely resolves the 4/3 problem.