Hi Jack,
Yes it is interesting and important to the point I'm making that not only the temperature but the Greens function and as well the related transition rate per unit time depend on the situation one is considering (i.e. an Unruh-DeWitt detector in a Hawking background vs. Rindler background vs. temperature bath etc.) One does not have to look far for the physical reason for this. The Unruh-DeWitt detector is the same in each case so the reason for the difference has to do with the basis of field modes one uses i.e. it depends on the vacuum. Note also that the field modes do "feel" more than just the local point at which the UD detector sits and this is the conceptual reason that one has a problem between the local EP and non-local QM/QFT.
One thing that may help you understand this point better is to answer the simpler classical "violation" of the EP: "An accelerating charge radiates yet if you place the same charge in a gravitational field which locally has the same 'acceleration' as that of the actual accelerating charge it won't radiate. Isn't this a violation of the EP? And if not why not?" This is an old paradox whose answer is known. Answering this will help you start to understand the more complex QM/QFT case.
OK but rather than give the spoiler answer immediately let's move on to the other questions. Next you said.
"My claim is that you'd see the same thing if an ordinary low-mass blackbody at the right temperature and of the right diameter were in place of the black hole. For both these cases, the time correlation function should be similar, but would *not* match that for the Unruh effect, nor does it need to."
First one technical point - the spectrum from a low mass blackbody at the right temperature still would not be the same as that of a Schwarzschild black hole. This is because the spectrum of the Hawking radiation is not exactly thermal but due to back-reaction (i.e. the radiation spectrum changes due to the emission of Hawking radiation) deviates from thermal. This was first shown in
"Hawking radiation as tunneling"
Maulik K. Parikh and Frank Wilczek Published in Phys.Rev.Lett. 85 (2000) 5042-5045
Very well you may say "Then I'll just contrive to make the spectrum of the low mass almost blackbody match that of the blackhole." This may not be possible since the black hole has a negative specific heat i.e. it gets hotter as it radiates. I'm not sure you could even make something (other than a black hole) that had this property. Even if you could there is still the criticism that this would be completely contrived. But anyway the first step would be to show that you could even make something (other than a black hole) with negative specific heat.
My next point/question is that I'm not even sure why you want to replace a black hole with a "low mass blackbody". For the EP one doesn't want to compare a black hole with a gravitating body that happens to emitted an almost thermal spectrum of a black hole (assuming this is even possible), but one should compare the observer/UD detector in a gravitational field with the Rindler observer/UD detector. At the end you say that in fact there will be a difference in temperature, transition rate so this would seems to support violation of the EP. Maybe you meant that the low mass blackbody would have the temperature of the "Unruh/Rindler" observer?
Also as a final technical point the UD detector measures the transition rate per unit time not the "time correlation function" which is not applicable here.
Next you say
"In order to establish a violation of equivalence, what you'd need to do is *not* just to find a discrepancy with one such proposal for an example like that. What you'd need to do is to prove that *no* set of possible boundary conditions on a small box around your detector could produce the same effects that you'd measure from the Hawking radiation. That you have certainly not done."
Actually for my answer here to make more sense it will help if you answer the earlier "classical violation" of the EP of the accelerating charge vs. the charge in a gravitational field. However you are correct that one should consider different boundary conditions, different vacua, etc. This was actually done by Ginzburg and Frolov in
V.L. Ginzburg and V.P. Frolov, Sov. Phys. Usp. 30, 1073 (1987)
Here they consider various boundary conditions, vacua, bloacking radiation with shielding etc. Their point is that there are some simple "violations" of the EP if one considers incorrect boundary conditions and vacua when comparing the gravitating observer/UD detector and the accelerating observer/UD detector. They emphasize that when comparing different observers (gravitating vs. accelerating) one needs to make sure one is using the correct boundary conditions and vacua. My comparison in terms of boundary conditions and vacua is for one that Ginzburg and Frolov claim are equivalent (in particular I'm using the situation given in figure 8c of their paper - which qualitatively satisfies the EP since both UD detect radiation, but quantitatively there is a violations since they detect radiation at a different rate/temperature). Note also that as the UD detector/observer approaches the horizon the two results *do* become equivalent thus restoring the EP. Anyway in some sense you're right one needs to take into account different BCs and also different vacua, but I have done this implicitly since from the Ginzburg and Frolov paper I am going immediately the non-trivial case (there are other cases but for these the EP is trivially violated). Anyway if you answer the "classical violation" of EP I gave earlier it will help understand the more complex QM/QFT case.
Lastly you say
"In fact, Hawking derived his radiation using a semi-classical approximation which has the equivalence principle built in by assumption, as does any metric model of gravity. Thus, by definition, the equivalence principle is obeyed in his model."
If you're talking about Hawking's original derivation of Hawking radiation
Hawking, S.W. Commun.Math.Phys. 43 (1975) 199-220, Erratum-ibid. 46 (1976) 206-206
this is not correct. I'm not sure what you mean by "semi-classical approximation" but what Hawking did in this paper was to study ingoing modes and outgoing modes in the presence of the collapsing surface of the star. In this derivation the collapsing surface is crucial. Then by comparing the ingoing and outgoing modes at past and future infinity he obtains particle production. But nothing here depends on the fact that this is coming from a metric theory. This is more clearly seen in that many analog systems have been suggested (acoustic "black holes", optical "black holes") which absolutely do not require a metric and nevertheless exhibit "Hawking radiation". Maybe you meant another of Hawkings papers - his paper with Hartle or perhaps Gibbons? But in any case in regard to the original paper the assertion you make is absolutely incorrect. Have a look at derivation which runs from equation 2.2 to 2.29. The result depends crucially on a Bogliubov transformation between the ingoing and outgoing basis.
Best,
Doug
