Derek, not sure I quite understand, please bear with me.

The Hawking radiation near a black hole comes on top of Unruh Davies - U-D does not cease to be valid near a BH or any other extraneous energy source.

So by monitoring temperature you can tell whether you are in free fall near a BH or away from a BH, or for that matter near any extraneous energy source of any kind affecting you. This does not violate the principle of equivalence. In your energy change DT, DT is an aggregate of DT1 + DT2, DT i.e. 2 separate components, one due to U-D, and one due to your being the recipient of extraneous energy.

Did I misunderstand the question ?

I believe you understood the question. However, your answer did not appear to explain why equivalence is not violated. That was the essence of the question.

I don´t quite understand the problem here. First, The Unruh effect assumes a uniform acceleration, which is far from being the case for a BH. Secondly, Unruh and Hawking temperatures do have the same functional form when the latter is written in terms of the surface gravity of the BH --see e.g. http://en.wikipedia.org/wiki/Unruh_effect.

The question asks if there is a way to recover the equivalence principle and I think you just answered the question!!

To my knowledge, an observer will not register any Hawking radiation when falling freely towards a black hole. His inertial system is no different from other inertial systems in that respect.

Of course, acceleration by free fall and by rocket thrust are not equivalent situations in general relativity. In free fall, you stay in an inertial system, whereas when accelerating by rockets, you are noninertial -- and you immediately notice via the pseudoforces due to acceleration. In free fall, there are no forces acting on you (gravity is not a force), whereas in an accelerating rocket, there is the thrust by your engine.

So the basic assumption of the question is already wrong.

Hawking Radiation comes from the spontaneous decay of space into matter and anti-matter. I don't think it's relevant to your discussion

@James: "Hawking Radiation comes from the spontaneous decay of space into matter and anti-matter." I would say, that is a fancy description but not necessarily a correct one. It is not space that decays.

One way to look at Hawking radiation is to say there is a horizon, in the vicinity of which particles and antiparticles are momentarily created and destroyed as allowed by Heisenberg's uncertainty relation. If one of the partners of a creation process disappears behind the horizon and the other stays on this side, then the horizon is seen to emit a particle. But these particles are not created out of space, they appear in a short-time violation of energy conservation, so to speak. (Literally out of nothing.) Since a long-time violation of energy conservation is not allowed by Heisenberg's uncertainty relation, there must be energy balance and hence the energy content of the black hole decreases.

One way to look at Unruh radiation is to say that there is a horizon, too. The analysis of constant acceleration within the framework of special relativity shows that this motion has two asymptotes in a Minkowski diagram, which are light cones. Nothing beyond these light cones can reach our accelerated observer (if he continues to accelerate forever, that is). At this horizon, we can again have particle-antiparticle creation and one of the partners can fall below the horizon while the other reaches our observer. This is the origin of Unruh radiation. So the idea of the original poster of comparing Hawking and Unruh radiation in an attempt to verify the equivalence principle was quite legitimate. And the answer to this question is probably that you cannot disregard the global nature of the two horizons. They are geometrically quite different. When applying Einstein's relativity principle to inhomogeneous gravitational fields, it usually applies only locally. But you cannot restrict yourself to local considerations when discussing what happens far from the respective horizons.

For an inertial observer, the horizon of the accelerating observer is absent, so an inertial observer will not see any Unruh radiation. A freely falling observer near a black hole is also inertial and hence one might expect him not to see any Hawking radiation. And so it is. An observer hovering at rest above the event horizon and hence feeling the gravity of the black hole does see Hawking radiation, a freely falling observer does not.

So the question of the original poster should apply not to a comparison of a freely falling observer near a black hole and and accelerating observer. What can be compared in terms of the equivalence principle are a) a freely falling observer near the black hole and an inertial observer far away and b) a stationary observer near a black hole and an accelerating observer far away. In the case of a), the equivalence principle is satisfied as neither of the two sees any radiation. In the case of b), the equivalence principle cannot be applied to the comparison of the radiation spectra, etc., because these depend on global properties (shape of the horizon), and the principle holds only locally.

Literally out of nothing and space decaying appear to me to follow some other equivalency principle!

However, I do not see any way that what you call a free falling observer cannot see Hawking Radiation. In fact, you are pretty much a free falling observer as are just about all entities that are not being crushed by acceleration. On earth we are very nearly in gravitational equilibrium. I only feel my 100 kg due to the earth. If I were at the center of the earth, I would be in perfect gravitational equilibrium as felt by my portable scale that survives the trip down some miraculous Tube Alloy conduit I've developed. Therefore, I still don't see the relevance.

It is a statement of the theory that a freely falling observer will not see Hawking radiation. I can't help it. That is what the theory says and what experts agree on.

I'd just like to point out that there is a big controversy today about free falling observers at the edge of a black hole, and whether they will "burn up" at what is called a firewall, or not know anything unusual is going on.

Dear all,

I think I agree that with Derek about the non-triviality of the issue.

As pointed out by Kassner, you can compare for example a stationary observer near a black hole and an accelerating observer in Minkowski spacetime: the equivalence principle states that a local observer cannot distinguish between the two situations (local means that the observer is unaware of the surrounding geometry, but experiences a flat geometry in his/her "local patch", right?).

The problem is that 1) an observer with constant acceleration in Minkowski spacetime perceives the thermal Unruh radiation while 2) an observer which stays at fixed distance from a black hole (let's suppose he is far enough to prevent excessive tidal stresses) will perceive both Hawking radiation from the stationary horizon and Unruh radiation (due to its proper acceleration, needed to stay at constant distance from the black hole).

This result can be obtained, for example, with the "Unruh-DeWitt detector" techinque and for instance it has been studied by Singleton in http://arxiv.org/abs/1102.5564 (also on Phys. Rev. Lett.).

Note that the U-DW detector is a local measuring device, that is a local observer.

Greetings!

Giovanni

If this paper is invoked, it should also be stated that there is a comment to it claiming to point out some errors (and an answer by the authors refuting parts of the comment...) So to be fair, one should note that there are calculations (in four dimensions) purporting that the equivalence principle holds near a black hole and other calculations (in two dimensions) purporting that it fails. Interestingly, as the horizon is approached by an observer, the principle is supposed to become correct again.

Now, there is also a philosophical side to this. Both the Unruh and Hawking radiation are not local effects. They are created at a distance from the observer. So it is debatable whether any difference in the two detected in a gravitational field can be interpreted as a violation of the (local) equivalence principle at all. After all, it is not the locality of the detectors that matters but the locality of the whole experiment. Of course, I can detect my acceleration with respect to a star, whether I am in free fall or not, just by observing that star. This can be done with a telescope and a spectrometer, both local devices. The measurement of the radiation coming from the star however does not qualify as a local measurement, by which I would be unable to determine velocities or accelerations. (Both are possible by observation of the Doppler effect of the radiation from the star.)

Dear Kassner,

thanks for the references to the comments (which I would like to include here for completeness: http://prl.aps.org/abstract/PRL/v108/i4/e049001 and http://prl.aps.org/abstract/PRL/v108/i4/e049002)

I'm not sure I understand your comment about Unruh and Hawking radiation being not local effects. What do you mean by "They are created at a distance from the observer"?

The thought-experiment involves two different setups (accelerated observer in vacuum on one hand, and static observer in an accelerated "spaceship" near a black hole on the other hand) which on a classical level are locally not distinguishable. Is it right?

By *local* experiments, they should be indistinguishable. But Hawking radiation is created at the event horizon of the black hole. So it always has to travel a finite distance to an observer not directly at the horizon (and the papers we were discussing agree that directly at the horizon, the "equivalence principle is reestablished"). Similarly, Unruh radiation is created at the horizon behind an accelerated observer, not at his actual position.

I do not think that one can state a violation of the equivalence principle on the basis of nonlocal observations. Clearly, any experimental device that allows to measure the curvature of space-time will be able to distinguish between an observer near a black hole and an observer accelerating in Minkowski space-time. But such experiments need to take into account extended laboratories, they do not satisfy the locality condition.