arXiv:1006.1843v1 [physics.optics] 9 Jun 2010 Energy–Momentum Tensor for the Electromagnetic Field in a Dielectric Michael E. Crenshaw and Thomas B. Bahder RDMR-WSS, Aviation and Missile RDEC, US Army RDECOM, Redstone Arsenal, AL 35898, USA (Dated: June 10, 2010) The total momentum of a thermodynamically closed system is unique, as is the total energy. Nevertheless, there is continuing confusion concerning the correct form of the momentum and the energy–momentum tensor for an electromagnetic field interacting with a linear dielectric medium. Here we investigate the energy and momentum in a closed system composed of a propagating elec- tromagnetic field and a negligibly reflecting dielectric. The Gordon momentum is easily identified as the total momentum by the fact that it is, by virtue of being invariant in time, conserved. We construct continuity equations for the energy and the Gordon momentum and use the continu- ity equations to construct an array that has the properties of a traceless, diagonally symmetric energy–momentum tensor. Then the century-old Abraham–Minkowski momentum controversy can be viewed as a consequence of attempting to construct an energy–momentum tensor from continuity equations that contain densities that correspond to nonconserved quantities. I. INTRODUCTION The energy–momentum tensor is a concise way to rep- resent the conservation properties of the flow field. For most types of simple flows, the energy–momentum tensor is well-defined, with the notable exception of the elec- tromagnetic field in a linear dielectric material. The Abraham–Minkoswski controversy [1–9] for the momen- tum of electromagnetic fields in a dielectric began with the derivation of the energy–momentum four-tensor by Minkowski [10]. Noting that the Minkowski tensor is not diagonally symmetric and therefore contains problems with conservation of angular momentum, Abraham [11] proposed an energy–momentum tensor that was symmet- ric, but at the expense of a new phenomenological force. In order to address this constraint and additional issues, Einstein and Laub [7], Nelson [8], and others proposed variants of the energy–momentum tensor. The crux of the Abraham–Minkowski controversy is whether the electromagnetic momentum density in a di- electric is of the Minkowski form g M = 1 c (D × B) (1.1) or the Abraham form g A = 1 c (E × H). (1.2) Experimental efforts to resolve the theoretical impasse in one direction or the other have not been definitive. While some experiments favor the Abraham formula, other ex- periments support Minkowski’s version. Brevick’s [12] analysis of experiments performed by Jones and Richards [13], Ashkin and Dziedzic [14], and others showed that the allocation of momentum between the field and ma- terial was the determining factor in whether a particular experimental result was described by the Abraham or Minkowski form of electromagnetic momentum. Follow- ing Brevick [12], the formula for the field momentum has been shown repeatedly to be essentially arbitrary such that any of the formulas for the field momentum can be combined with an appropriate momentum for the mate- rial to produce the same total momentum [15]. In 1973, Gordon [1] constructed the total momentum from a microscopic model in which the electromagnetic field component of the total momentum is said to be the Abraham momentum and the dielectric is treated as a di- lute collection of electric dipoles with center-of-mass mo- tion in the direction of propagation of the field. Gordon [1] discusses the empirical and experimental validation of the total momentum density g G = n c (E × B) (1.3) and shows that the density g G , integrated over a volume containing the entire field, is invariant in time. Mikura [16] subsequently obtained the total momentum using a more general material model that included polarizabil- ity, magnetizability, electrostriction, and other material considerations. Since then, the total momentum has been constructed using a variety of field and material momenta [15]. The historical development of the electromagnetic energy–momentum tensor for the field in a dielectric has been intertwined with efforts to derive the correct elec- tromagnetic momentum density. The original Minkowski and Abraham tensors, like the Minkowski and Abraham momenta, are now regarded as being associated with the field, alone, requiring a material component to com- plete the total energy–momentum tensor [15]. Although a number of composite energy–momentum tensors have been constructed [15], the tensor forms that have been constructed from field and material constituents fail to satisfy the constraints imposed on the symmetry, trace, or four-divergence of a total energy–momentum tensor. The basic tenant of classical continuum electrodynam- ics is that the electrodynamic properties of a linear ma- terial can be characterized by macroscopic parameters. In this long wavelength limit, the material is modeled as a simple linear dielectric with a linear refractive index n. In reflection, the electromagnetic field exerts radia- tion pressure on the dielectric. Neither the motion of an