Renormalization of the energy-momentum tensor and the validity of the equivalence principle at finite temperature

Using the techniques of finite-temperature field theory we renormalize the electromagnetic and gravitational couplings of an electron which is immersed in a heat bath with $T\ll m_e$. By taking the nonrelativistic limit, we demonstrate that the inertial and gravitational masses are unequal. The implications for the equivalence principle are discussed.