Alternative formulation of the macroscopic field equations in a linear magneto-dielectric medium: Lagrangian field theory and spacetime setting

A transparent linear magneto-dielectric material in free space that is illuminated by a finite quasimonochromatic field is a thermodynamically closed system, definitively, regardless of what field and material subsystems that one defines. The energy--momentum tensor that is formally derived from the Maxwell--Minkowski field equations is inconsistent with both angular and linear momentum conservation in this closed system; this very solid fact is the foundational and continuing issue of the Abraham--Minkowski controversy. The extant resolution of the Abraham--Minkowski dilemma is to treat Maxwellian continuum electrodynamics as being a subsystem and to write the total energy--momentum tensor as the sum of a Maxwellian electromagnetic subsystem energy--momentum tensor and a phenomenological material subsystem energy--momentum tensor. We prove that fundamental principles of physics are violated by Maxwellian continuum electrodynamics and that fundamental principles of physics are violated by Maxwellian continuum electrodynamics supplemented by the material subsystem conjecture. We use field theory to derive legitimate equations for macroscopic electromagnetic fields in a transparent linear magneto-dielectric medium. The new field equations are a part of a self-consistent formulation of macroscopic electrodynamics, conservation laws, special relativity, and invariance in a continuous linear medium. In the new formulation, the temporal and spatial coordinates are renormalized by the continuous linear medium instead of the permittivity and permeability being carried as independent material parameters. Then an isotropic, homogeneous, flat, four-dimensional, continuous, linear, non-Minkowski spacetime is the proper setting for the continuum electrodynamics of a simple linear medium in which the effective speed of light is $c/n$ and each medium will be associated with a different spacetime.