Question

Mach's principle says that it is impossible to tell if something is accelerating unless there is something else in the universe to compare that motion to, which seems reasonable. However, if you had one detector in the universe, you seem to be able to tell if it is accelerating because an accelerating detector would record radiation where a non-accelerating detector would not, due to the Unruh effect. So, my question is, does the Unruh effect provide a way to tell if something is accelerating, even if it is the only thing in otherwise "empty" space, thereby violating Mach's principle? (At least Mach's principle in its form stated above.)

Answer 1

Yes, the Unruh effect violates Mach's principle.

In effective quantum field theory, the Unruh effect arises because the ground state of the Hamiltonian naturally associated with an accelerating observer is different than the ground state of the Hamiltonian generating time translations in a "static" frame. The static and accelerating frames have different time coordinates t, and because the Hamiltonian generates infinitesimal changes of t, it is different. And different operators have different eigenstates, including the lowest-eigenvalue one (the ground state).

One vacuum may be viewed as a squeezed/coherent state built upon the other, which may be interpreted as particle production, and creation operators have to be accordingly mixed with the annihilation operators according to the other frame - by the so-called Bogoliubov transformation.

Although Mach's principle has never been properly defined or turned into a realistic theory, it seems pretty clear that according to all of its interpretations, it postulated that it was impossible to have two different notions of a "vacuum" depending on the acceleration of the observer. However, that's exactly the case because of the Unruh effect.

An acceleration-dependent definition of the ground state (the vacuum), as realized in the derivation of the Unruh effect, exactly means that the acceleration may be distinguished from no acceleration even in the vacuum - which is exactly what Mach's principle would like to prohibit.

The Unruh effect therefore violates Mach's principle. More precisely, it refutes it because Mach's principle is wrong and the Unruh effect is real. However, one doesn't really have to go to the quantum theory to see that Mach's principle has been showed incorrect by subsequent developments. The existence of a dynamical metric tensor - even in the classical, non-quantum theory - is enough. The gravitational waves are the simplest classical entities that show that the gravitational field is real even in the vacuum, and it allows the particles moving through it to distinguish free fall from accelerating motion.