Relativistic covariance of Ohm's law

TL;DR

This paper derives a Lorentz-covariant form of Ohm's law using a response tensor framework, establishing its validity in all inertial frames and providing a relativistic transformation law for conductivity.

Contribution

It introduces a first-principles, Lorentz-covariant approach to Ohm's law, connecting microscopic conductivity with relativistic response tensors.

Findings

Proves Ohm's law is valid in all inertial frames.

Derives a relativistic transformation law for conductivity.

Reproduces standard textbook generalization of Ohm's law for scalar conductivity.

Abstract

The derivation of Lorentz-covariant generalizations of Ohm's law has been a long-term issue in theoretical physics with deep implications for the study of relativistic effects in optical and atomic physics. In this article, we propose an alternative route to this problem, which is motivated by the tremendous progress in first-principles materials physics in general and ab initio electronic structure theory in particular. We start from the most general, Lorentz-covariant first-order response law, which is written in terms of the fundamental response tensor $\chi^\mu_\nu$ relating induced four-currents to external four-potentials. By showing the equivalence of this description to Ohm's law, we prove the validity of Ohm's law in every inertial frame. We further use the universal relation between $\chi^\mu_\nu$ and the microscopic conductivity tensor $\sigma_{k\ell}$ to derive a fully relativistic transformation law for the latter, which includes all effects of anisotropy and relativistic retardation. In the special case of a constant, scalar conductivity, this transformation law can be used to rederive a standard textbook generalization of Ohm's law.