Thermodynamic equilibrium with acceleration and the Unruh effect

We address the problem of thermodynamic equilibrium with constant acceleration along the velocity field lines in a quantum relativistic statistical mechanics framework. We show that for a free scalar quantum field, after vacuum subtraction, all mean values vanish when the local temperature $T$ is as low as the Unruh temperature $T_U=A/2\pi$ where $A$ is the magnitude of the acceleration four-vector. We argue that the Unruh temperature is an absolute lower bound for the temperature of any accelerated fluid at global thermodynamic equilibrium. We discuss the conditions of this bound to be applicable in a local thermodynamic equilibrium situation.

Figure 1

2D Minkowski space-time diagram with the field lines (dashed) of the four-temperature $\beta$ Killing field (13) in the right Rindler wedge (RRW) and the left Rindler wedge (LRW). In these wedges the vector field (drawn with arrows) is timelike and it is future-directed only in the RRW. Also shown a hyperplane $\mathrm{\Sigma }$ perpendicular to $\beta$ which can be used as spacelike hypersurface to define the Klein-Gordon inner product and calculate the operators ${\widehat{\mathrm{\Pi }}}_R$,${\widehat{\mathrm{\Pi }}}_L$. The quantum field in the RRW is causally disconnected from the LRW.

Figure 2

In the $T-|A|$ phase diagram of an accelerated fluid at global equilibrium there is a forbidden region delimited by the line $T=(h/4{\pi }^{2}cK)|A|$.