Maxwell stress dyadic in differential-form formalism

Abstract

The classical Maxwell stress tensor (or stress-energy-momentum tensor) is revisited by introducing the dyadic formalism to that of differential forms. Dyadics, as originally introduced by Gibbs to vector analysis, appear suitable companions to differential forms because of their co-ordinate-free character. Basic properties of dyadics together with some useful identities are first derived. It is shown that, in terms of the identities, the Maxwell stress tensor can be given a particularly simple dyadic form. This requires that the Lorentz force density be first expressed as a dyadic quantity mapping trivectors to vectors and, in four-dimensional representation, the Lorentz force-power density as a dyadic mapping from quadrivectors to vectors. Finally, it is shown that to be able to define the force density in terms of a stress dyadic, the macroscopic electromagnetic medium (assumed linear, homogeneous and time-independent) must satisfy a certain symmetry condition which turns out to equal the Lorentz reciprocity condition for time-harmonic fields.