Electrodynamics in a Filled Minkowski Spacetime with Application to Classical Continuum Electrodynamics

TL;DR

This paper develops a relativistic electrodynamics framework in a dielectric-filled Minkowski spacetime, deriving invariant equations of motion for macroscopic fields that incorporate the medium's refractive index.

Contribution

It introduces a modified relativistic principle of relativity for media with refractive index n, and derives consistent equations of motion for electromagnetic fields in such media.

Findings

Effective signal velocity scales with 1/n

Derived invariant equations of motion for macroscopic fields

Ensured consistency with quantum and boundary conditions

Abstract

Minkowski spacetime is a convenient setting for the study of the relativistic dynamics of particles and fields in the vacuum. In order to study events that occur in a dielectric or other linear medium, we adopt the familiar continuum assumption of a linear, isotropic, homogeneous, transparent medium of refractive index n filling all space and seek the principle of relativity that applies in the filled spacetime. Applying the Einstein postulates with c/n as the speed of light, we show how the effective signal velocity results in a scaling of the proper time by the refractive index and examine the consequences for D'Alembert's principle, the Lagrange equations, and the canonical momentum field. The principles of dynamics in the filled spacetime are then applied to the electromagnetic Lagrangian and we derive equations of motion that are invariant with respect to a material Lorentz transformation. The new representation of the dynamics of macroscopic fields is shown to be consistent with the equal-time commutation relation for quantized macroscopic fields, quantum--classical correspondence, the principle of superposition, and electromagnetic boundary conditions.