It is worth stressing that the Unruh effect only says that, according to uniformly-accelerated observers, the vacuum state is a thermal state specified by its temperature, and one should resist reading too much into the thermal state or bath. Different thermal states or baths at the same temperature need not be equal, for they depend on the Hamiltonian describing the system. In particular, the thermal bath seen by accelerated observers in the vacuum state of a quantum field is not the same as a thermal state of the same field at the same temperature according to inertial observers. Furthermore, uniformly accelerated observers, static with respect to each other, can have different proper accelerations (depending on their separation), which is a direct consequence of relativistic red-shift effects. This makes the Unruh temperature spatially inhomogeneous across the uniformly accelerated frame.