  It is well known that radiative corrections evaluated in nontrivial
backgrounds lead to effective dispersion relations which are not Lorentz
invariant. Since gravitational interactions increase with energy,
gravity-induced radiative corrections could be relevant for the trans-Planckian
problem. As a first step to explore this possibility, we compute the one-loop
radiative corrections to the self-energy of a scalar particle propagating in a
thermal bath of gravitons in Minkowski spacetime. We obtain terms which
originate from the thermal bath and which indeed break the Lorentz invariance
that possessed the propagator in the vacuum. Rather unexpectedly, however, the
terms which break Lorentz invariance vanish in the high three-momentum limit.
We also found that the imaginary part, which gives the rate of approach to
thermal equilibrium, vanishes at one loop.
