Resolution of the Abraham-Minkowski controversy

The momentum of light inside ponderable media has an electromagnetic part and a mechanical part. The local and instantaneous density of the electromagnetic part of the momentum is given by the Poynting vector divided by the square of the speed of light in vacuum, irrespective of the nature of the electromagnetic fields or the local or global properties of the material media. The mechanical part of the momentum is associated with the action of the electromagnetic field on the atomic constituents of the media, as specified by the Lorentz law of force. Proper interpretation and application of the Maxwell-Lorentz equations within the material bodies as well as at their surfaces and interfaces is all that is needed to obtain a complete picture of the momentum of light, including detailed numerical values at each and every point in space and time. That the Abraham-Minkowski controversy surrounding the momentum of light inside material media has persisted for nearly a century is due perhaps to an insufficient appreciation for the completeness and consistency of the macroscopic Maxwell-Lorentz theory, inadequate treatment of the electromagnetic force and torque at the material boundaries, and an undue emphasis on the necessity of coupling the equations of electrodynamics to those of the theory of elasticity for proper treatment of mechanical momentum. The present paper reports the resolution of the Abraham-Minkowski controversy within the framework of the classical theory of electrodynamics, without resort to such complicating and ultimately unnecessary factors as pseudo-momentum, special surface forces, alternative energy-momentum tensors, and hidden momenta, that have caused so much confusion for such a long period of time. 1. Introduction. The Abraham-Minkowski controversy dates back to the early years of the 20 th Century, when Hermann Minkowski's electromagnetic theory indicated that a light pulse of energy E pulse within a dielectric medium of refractive index n would have a momentum of nE pulse /c (c being the speed of light in vacuum) whereas Max Abraham's theory gave a value of E pulse /(nc) for the same momentum (1, 2). Over the years, many theoretical arguments have been advanced (3-27) and experimental evidence provided (28-38) in support of one theory over the other, but the controversy persists. As it turns out, the Abraham-Minkowski conundrum can be fully resolved if the Maxwell-Lorentz equations of classical electrodynamics are properly interpreted and correctly applied. The solution to the problem of optical momentum in ponderable media is fully consistent with the conservation laws and does not require any new physics or complicated hypotheses. The solution applies to any type of material medium, whether or not the medium is homogeneous, transparent, dispersive, birefringent, magnetic, non-linear, etc.; in particular, it is applicable to negative-index media, where the dielectric permittivity  and magnetic permeability  are both negative-valued. It is not our intention here to describe in mathematical detail the most general solution to the problem of optical momentum in ponderable media. Rather, we shall describe the main features of the solution and give reference to the literature for rigorous derivations (1821, 39 53). We also provide qualitative descriptions of the mechanical momentum imparted to material media in their interactions with the electromagnetic field. (It is our opinion that a clear, step-by-step picture of the mechanism by which electromagnetic and mechanical momenta are exchanged is crucial to an understanding of the role played by mechanical momentum in resolving the Abraham-Minkowski controversy.) The main features of the momentum of light as revealed by a comprehensive analysis of the Maxwell-Lorentz equations without resort to unwarranted approximations or unsubstantiated hypotheses may be summarized as follows: 1) In the simple case of a light pulse propagating inside a transparent, dispersionless dielectric of refractive index n, the total momentum associated with the light pulse is ½ (n+ n 1 )E pulse /c. This momentum consists of two parts, an electromagnetic part that is given by the Abraham formula p EM =E pulse /(nc), and a mechanical part that is given by pmech =½( n n 1 )E pulse /c.