First Order Corrections to the Unruh Effect

First order corrections to the Unruh effect are calculated from a model of an accelerated particle detector of finite mass. We show that quantum smearing of the trajectory and large recoil essentially do not modify the Unruh effect. Nevertheless, we find corrections to the thermal distribution and to the Unruh temperature. In a certain limit, when the distribution at equilibrium remains exactly thermal, the corrected temperature is found to be T = T U (1 − T U /M), where T U is the Unruh temperature. We estimate the consequent corrections to the Hawking temperature and the black hole en-tropy, and comment on the relationship to the problem of trans-planckian frequencies.