Covariant jump conditions in electromagnetism

Abstract

A generally covariant four-dimensional representation of Maxwell's electrodynamics in a generic material medium can be achieved straightforwardly in the metric-free formulation of electromagnetism. In this setup, the electromagnetic phenomena described by two tensor fields, which satisfy Maxwell's equations. A generic tensorial constitutive relation between these fields is an independent ingredient of the theory. By use of different constitutive relations (local and non-local, linear and non-linear, etc.), a wide area of applications can be covered. In the current paper, we present the jump conditions for the fields and for the energy-momentum tensor on an arbitrarily moving surface between two media. From the differential and integral Maxwell equations, we derive the covariant boundary conditions, which are independent of any metric and connection. These conditions include the covariantly defined surface current and are applicable to an arbitrarily moving smooth curved boundary surface. As an application of the presented jump formulas, we derive a Lorentzian type metric as a condition for existence the wave front in isotropic media. This result holds for the ordinary materials as well as for the metamaterials with the negative material constants.

Skewon modification of the light cone structure

“…Once more we have two optic axes. 4) In the last case, 42) there is only one eigenvalue m = 0 of multiplicity four. It corresponds to two eigenvectors q = (β, 0, 0, ±α), and consequently there are again two optic axes.…”

“…The covariant boundary surface problem was studied in [31] and [42]. In particular, when the space-time decomposition of the fields F and H is applied and the fields are assumed to be independent of the time coordinate, Eqs.…”

“…In order to derive the wave-front conditions, we apply the covariant jump conditions, [42], that are a consequence of the integral Maxwell equations (2.1). Consider an arbitrary smooth non-degenerate hypersurface Σ in space-time.…”

“…Our aim is to understand the possible new physical phenomena coming from the additional electromagnetic parameters. Recently there has been considerable progress in this area, [27], [28], [29], [30], [32], [33], [34], [35], [37], [38], [41], [42], [44]. Here we are interested mostly in the skewon effects on the light cone structure.…”

Generally covariant Maxwell theory for media with a local response: Progress since 2000

“…Symbol and hyperbolic polynomials of the wave equation and their relation to premetric electrodynamics were discussed by Beig [5]. Itin [35] derived premetric covariant jump conditions for the initial value hypersurface, the light cone, and a boundary between two media.…”