  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.
