Lorentz transformation in Maxwell equations for slowly moving media

TL;DR

This paper investigates the Lorentz transformation in Maxwell equations under small velocity approximation, showing the importance of relativistic effects even at low speeds and deriving consistent integral forms for slowly moving media.

Contribution

It demonstrates the necessity of Lorentz transformation at O(v/c) in Maxwell equations for slowly moving media and derives covariant integral forms of Faraday and Ampere laws.

Findings

Lorentz transformation affects time and charge density in SVA

Derived deformed Maxwell equations in lab frame

Presented covariant integral form of Faraday law

Abstract

We use the method of field decomposition, a technique widely used in relativistic magnetohydrodynamics, to study the small velocity approximation (SVA) of the Lorentz transformation in Maxwell equations for slowly moving media. The "deformed" Maxwell equations derived under the SVA in the lab frame can be put into the conventional form of Maxwell equations in the medium's comoving frame. Our results show that the Lorentz transformation in the SVA up to $O(v/c)$ ($v$ is the speed of the medium and $c$ is the speed of light in vacuum) is essential to derive these equations: the time and charge density must also change when transforming to a different frame even in the SVA, not just the position and current density as in the Galilean transformation. This marks the essential difference of the Lorentz transformation from the Galilean one. We show that the integral forms of Faraday and Ampere equations for slowly moving surfaces are consistent with Maxwell equations. We also present Faraday equation the covariant integral form in which the electromotive force can be defined as a Lorentz scalar independent of the observer's frame. No evidences exist to support an extension or modification of Maxwell equations.