The Electrodynamics of Inhomogeneous Rotating Media and the Abraham and Minkowski Tensors I: General Theory

This is paper I of a series of two papers, offering a self-contained analysis of the role of electromagnetic stress-energy-momentum tensors in the classical description of continuous polarizable perfectly insulating media. While acknowledging the primary role played by the total stress-energy-momentum tensor we argue that it is meaningful and useful in the context of covariant constitutive theory to assign preferred status to particular parts of this total tensor, when defined with respect to a particular splitting. The relevance of tensors, associated with the electromagnetic fields that appear in Maxwell's equations for polarizable media, to the forces and torques that they induce has been a debated since Minkowski, Einstein and Abraham considered these issues over a century ago. Consistency with the laws of Newtonian continuum mechanics demands that the total force density on any element of a medium be proportional to the local linear acceleration field of that element in an inertial frame and must also arise as part of the divergence of the total stress-energy-momentum tensor. The fact that, unlike the tensor proposed by Minkowski, the divergence of the Abraham tensor depends explicitly on the local acceleration field of the medium as well as the electromagnetic field, sets it apart from many other terms in the total stress-energy-momentum tensor for a medium. We explore how electromagnetic forces or torques on moving media can be defined in terms of a particular 3-form on spacetimes that exhibit particular Killing symmetries. It is shown how the forms associated with translational Killing vector fields lead to explicit expressions for the electromagnetic force densities in stationary media subject to the Minkowski constitutive relations. These are compared with other models involving polarizable media in electromagnetic fields that have been considered in the recent literature.
Comment: 32 pages, 1 figure. Accepted for publication by Proc. Roy. Soc. A