  The relativistic Langevin equation poses a number of technical and conceptual
problems related to its derivation and underlying physical assumptions.
Recently, a method has been proposed in [A. Petrosyan and A. Zaccone, J. Phys.
A: Math. Theor. 55 015001 (2022)] to derive the relativistic Langevin equation
from a first-principles particle-bath Lagrangian. As a result of the
particle-bath coupling, a new ``restoring force'' term appeared, which breaks
translation symmetry. Here we revisit this problem aiming at deriving a fully
translation-invariant relativistic Langevin equation. We successfully do this
by adopting the renormalization potential protocol originally suggested by
Caldeira and Leggett. The relativistic renormalization potential is derived
here and shown to reduce to Caldeira and Leggett's form in the non-relativistic
limit. The introduction of this renormalization potential successfully removes
the restoring force and a fully translation-invariant relativistic Langevin
equation is derived for the first time. The physically necessary character of
the renormalization potential is discussed in analogy with non-relativistic
systems, where it emerges due to the renormalization of the tagged particle
dynamics due to its interaction with the bath oscillators (a phenomenon akin to
level-repulsion or avoided-crossing in condensed matter). We discuss the
properties that the corresponding non-Markovian friction kernel has to satisfy,
with implications ranging from transport models of the quark-gluon plasma, to
relativistic viscous hydrodynamic simulations, and to electrons in graphene.
