  We show how a relativistic Langevin equation can be derived from a
Lorentz-covariant version of the Caldeira-Leggett particle-bath Lagrangian. In
one of its limits, we identify the obtained equation with the Langevin equation
used in contemporary extensions of statistical mechanics to the
near-light-speed motion of a Brownian particle in non-relativistic dissipative
fluids. The proposed framework provides a more rigorous and first-principles
form of the Langevin equation often quoted or postulated as ansatz in previous
works. We then refine the aforementioned results by considering more terms in
the particle-bath coupling, which improves the precision of the approximation
for fully relativistic settings where not only the tagged particle but also the
thermal bath motion is relativistic. We discuss the implications of the
apparent breaking of space-time translation and parity invariance, showing that
these effects are not necessarily in contradiction with the assumptions of
statistical mechanics. The intrinsically non-Markovian character of the fully
relativistic generalized Langevin equation derived here, and of the associated
fluctuation-dissipation theorem, is also discussed.
