Unruh temperature appears to depend on reference frame, how is this resolved? I am wondering about observer-(in)dependence of Unruh temperature. Specifically, consider two observers, A and B, initially at rest in the same position. Then, at some point, A starts uniformly accelerating with some acceleration $a$, thus observing a thermal bath with a temperature of (in natural units)
$$ T = \frac{a}{2\pi}~.$$
Observer B is still at rest and should therefore observe a temperature equal to zero. However, in the accelerating reference frame of A, it seems as if B is getting ever closer to the Unruh horizon, so that it should be receiving ever hotter Unruh radiation. If B carries a thermometer, then it seems that the thermometer should give a different reading in the reference frames of A and B, respectively. It seems that the latter statement is a logical inconsistency. How is this apparent paradox resolved?
 A: Let me restate with some additional considerations.
Consider two observers, A and B, where A is uniformly accelerated and B is in rest. Both of them carry a thermometer. Without losing generality I will assume that the observers are in contact with a photon field in its ground state (vacuum) and the thermometers are just a single atom coupled to the photon field. In order to get a temperature measurement all they have to do is check what is the spectrum of the atom.
According to the Unruh effect observer A, accelerating, will see a thermal distribution of electrons in the atom he carries. But observer B sees just a vacuum. How does he explain A's thermometer? According to B's reference frame the accelerated thermometer is in contact with a photonic vacuum but accelerates due to an external force. This external force will modify the atom-photon coupling in the precise way as to generate a planckian distribution of electrons. In another way, the external force induces transitions between electronic levels in just the way as to reproduce thermal behavior.
Now for the converse: Observer A accelerating uses his thermometer and reaches the conclusion that he is immersed in a thermal bath of electrons. Yet observer B's thermometer registers zero temperature. A then reasons that if he is uniformly accelerating then he is in free fall in a constant gravitational field. According to him if B does not move then he must be subject to an external force that counterbalances the gravitational field. Analyzing B's thermometer he reaches the conclusion that the atom is indeed exchanging photons with the field around him, yet the external force keeping him in place is inducing another set of electronic transitions, whose effect is exactly equal and opposite to the thermal bath, thus maintaining the atom in its ground level and resulting in the thermometer giving a zero temperature reading.
So both A and B agree that A's thermometer gives a non-zero temperature and B's a zero temperature. However since both of them disagree on the state of the photonic field and who is subject to an external force, they describe the physics of the thermometers in different fashion.
