On the thermodynamics of moving bodies

We model the situation in which a photon detector moves with constant velocity in a Minkowski heat-bath by considering an Unruh-DeWitt detector, with variable energy gap $\hbar \omega$, moving on a brane, at a non-zero (Unruh) temperature $T$ due to acceleration in an orthogonal direction. We compute the angular response for a 2-brane. At low velocity we find agreement with the standard Doppler shift formula in the limit $\hbar\omega \gg T$ in which the photon gas is classical; otherwise there is a discrepancy, which we attribute to angular-dependence of the induced emission rate. At relativistic velocities our result disagrees with the standard formula even when $\hbar\omega \gg T$, and above a critical velocity there is no response from the detector within a `backward' cone. We discuss potential implications for observations of the cosmic microwave background radiation.