Hi Azzam,

In the case you are considering (a train moving with constant velocity) you are correct -- there is not Unruh radiation. To observe Unruh radiation one must go to an accelerated frame. Also although there is no definite, undisputed evidence for the Hawking or Unruh radiation there have been recent claims that analog Hawking radiation has been detected in a system where ultra...

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Hi Azzam,

In the case you are considering (a train moving with constant velocity) you are correct -- there is not Unruh radiation. To observe Unruh radiation one must go to an accelerated frame. Also although there is no definite, undisputed evidence for the Hawking or Unruh radiation there have been recent claims that analog Hawking radiation has been detected in a system where ultra short, high intensity lasers pulses are aimed at a certain type of fused silicate glass. This creates *two* optical, analog event horizons from which something like Hawking radiation was detected. The link for this is

http://phys.org/news204866995.html

and the technical paper was published in PRL in 2010. There are also claims that the Unruh effect can be (and in fact has been) detected in through the shifting of expected populations of electrons in storage rings of particle accelerators. This work is

"Electrons As Accelerated Thermometers",

J.S. Bell, J.M. Leinaas (CERN). Nucl.Phys. B212 (1983) 131

CERN-TH-3363

Both of these claimed experimental detections of Hawking radiation and Unruh radiation have some loop holes so that the claims are not completely accepted, but at least there is some work in the direction of experimental verification of these effects.

Now your second point seems to revolve around the behavior of temperature under Lorentz transformations. This is an open and still debated subject. A relatively recent paper on this subject is

"Inverse Temperature 4-vector in Special Relativity", Zhong Chao Wu

Europhys.Lett.88:20005 (2009) ; arXiv:0804.3827v4 [gr-qc]

I'm not sure I agree with the conclusions of this paper but it does list all three possible transformations of temperature namely:

T=T_0 *gamma

T=T_0 /gamma

T=T_0

(where gamma is the usual SR gamma-factor and these possible transformations and the references where they were proposed are given in eqns. (1)(2) (3) at the beginning of the paper. Thus your proposal above -- that the temperature increases with relative velocity -- seems to be in line with the suggestion T=T_0*gamma.

There is also the suggestion (which can be found in the beautiful but nonstandard physics "textbook" http://www.motionmountain.net/) that it only makes sense to define a temperature in the frame of reference where the center of mass of the object whose temperature is to be measured is at rest.

I'll try to read your essay more closely to see if I have additional comments.

Best regards,

Doug

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Hi Jack,

Your questions touch on some subtle issues that the essay format did not really lend itself to. However, the measurement of the Hawking and Unruh temperature that I have in mind is *local*. The "thermometer" to be used is an Unruh-DeWitt detector - this is a two-state quantum system which is placed into some curved space-time or moved along some space-time trajectory in flat...

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Hi Jack,

Your questions touch on some subtle issues that the essay format did not really lend itself to. However, the measurement of the Hawking and Unruh temperature that I have in mind is *local*. The "thermometer" to be used is an Unruh-DeWitt detector - this is a two-state quantum system which is placed into some curved space-time or moved along some space-time trajectory in flat space-time. One simple version of an Unruh-DeWitt detector is an electron in a magnetic field. This has two energy levels. By looking at the transition per unit time to occupy the upper energy level one can determine the temperature. Details of Unruh-DeWitt detectors can be found in

N.D. Birrell and P.C.W. Davies, "Quantum fields in curved space", (Cambridge University Press, Cambridge 1982)

or mor eeasily accesible review of the relevant parts of the the Unruh-DeWitt detector can be found in

Emil T. Akhmedov and Douglas Singleton

Int.J.Mod.Phys. A22 (2007) 4797-4823

e-Print: hep-ph/0610391 [hep-ph]

In any case the Unruh-DeWitt detector is a point detector and thus gives a local measurement of the temperature.

Now in regard to the last statement/question you pose (which is also related to boundary condition and the choice of vaccuum) it is not necessarily true that an observer on a planet without a horizon will not detect Hawking radiation. This also depends on the choice of vacuum (i.e. boundary conditions). This is most clearly described in

V.L. Ginzburg and V.P. Frolov, Sov. Phys. Usp. 30, 1073 (1987)

In particular see figure 8a,b,c,d which shows an Unruh-DeWitt detector either accelerating (the left side of the figures) or at rest in the gravitational field without a horizon (the right side of the figures). Note that whether the detector gets exctied or not depends on the choice of vacuum (the vacua considered by Ginzburg and Frolov are the Minkowski vacuum, Rindler vacuum, Boulware vaccuum, and Hartle-Hawking vacuum). Note the cases I consider correspond to figures 8c and 8d where the detectors are fixed (this is the "string" shown in the figure). The case of a detector at rest in the gravitational field of a planet corresponds to the right side of figure 8d which is the Boulware vacuum. For a gravitaional field without a horizon one can choose the Boulware vaccum, but for a BH space-time one encounters divergences at the horizon in the Boulware vacuum which makes it un-physical. This is the reason I compare an Unruh-DeWitt "thermometer" accelerating with an Unruh-DeWitt "thermometer" fixed in a BH background. For this case no matter what vacuum one chooses there will be violation of the EP. These are subtle issues so I would be happy to discuss this further.

Best regards,

Doug

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Hi Jack,

If I understand correctly you are saying "If an observer feels an acceleration to the bottom of the Einstein elevator and measures a temperature there is no way to tell if the elevator is near a black hole and the observer is detecting Hawking radiation *or* if the observer is in the Einstein elevator near a gravitating body that is not a black hole (e.g. the Earth) and is...

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Hi Jack,

If I understand correctly you are saying "If an observer feels an acceleration to the bottom of the Einstein elevator and measures a temperature there is no way to tell if the elevator is near a black hole and the observer is detecting Hawking radiation *or* if the observer is in the Einstein elevator near a gravitating body that is not a black hole (e.g. the Earth) and is simultaneously embedded in a thermal bath." If this is the contention this it turns out to be wrong for a subtle reason. The reason is that the Greens function for the Hawking and Unruh effect are *different* from the thermal Greens function of a thermal bath at rest in Minkowski space-time. This in turn means that the transition probability per unit time of the Unruh-Dewitt detector is different. Thus by measuring the transition rate (in addition to the temperature) one *can* distinguish between and Hawking or Unruh thermal bath and a "true" thermal bath.

The details of this can be found in

"Hawking radiation, Unruh radiation and the equivalence principle",

Douglas Singleton, Steve Wilburn

Phys.Rev.Lett. 107 (2011) 081102; e-Print: arXiv:1102.5564 [gr-qc]

also important is the comment and reply

"Comment on 'Hawking Radiation, Unruh Radiation, and the Equivalence Principle'",Luis C.B. Crispino et al Phys.Rev.Lett. 108 (2012) 049001

and

"Reply to 'Comment on 'Hawking Radiation, Unruh Radiation, and the Equivalence Principle''", Douglas Singleton, Steve Wilburn, Phys.Rev.Lett. 108 (2012) 049002

The point discussed in the two replies is exactly the fact that by looking not only at the temperature but also at the response function one does in fact get a violation of the equivalence principle, since not only are the temperatures different but so are the response Grrens functions. In fact, even for Hawking radiation but with different vacua (e.g. Unurh vacuum vs. Hartle Hawking vacuum) the transition rate will be different by a factor of two.

As these comments discuss you are correct that if one only specifies the temperature, this does not distinguish between a "true" thermal bath and a thermal bath due to Hawking/Unruh radiation. However the Greens functions and response functions for the two case are certainly different which allows one to distinguish the two.

The details of the Greens functions for the two cases (Hawking/Unruh vs. "true" thermal) can be found in the article

J.S.Bell and J.M.Leinaas, Nucl.Phys. B 212, 131 (1983)

If you look at equation (21) of this article you find that the Greens function for the Hawking/Unruh case is ~ sinh^{-4}(..). On the other hand the thermal Greens function in Minkowski space is given by equation (26) of this article and is ~ sinh^{-4}(...) + sinh^{-2}(...). And as well the transition probability per unit time will be different between the two cases.

Anyway the upshot is that if one measures the local temperature and local transition probability one *can* tell the difference between an Unruh-Dewitt detector at rest in a thermal bath and a spherical gravitating body without a horizon and an Unruh-DeWitt detector at rest in the gravitational field of a black holes.

Best,

Doug

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Doug,

Thanks for the detailed reply. I do find it interesting that the time correlation function is different for a thermal bath as opposed to the Unruh effect. What do you think is the physical reason for the difference?

However, that does not affect the point about equivalence. Of course a thermal bath is in no way equivalent to Hawking radiation, if only because Hawking radiation...

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Doug,

Thanks for the detailed reply. I do find it interesting that the time correlation function is different for a thermal bath as opposed to the Unruh effect. What do you think is the physical reason for the difference?

However, that does not affect the point about equivalence. Of course a thermal bath is in no way equivalent to Hawking radiation, if only because Hawking radiation is highly directional: At a large distance from the event horizon (which is the region that you claim is most strongly affected by your argument) you could see (if you had the right equipment) the radiation coming from the direction of the event horizon. 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.

Of course that is just an example of a situation where you'd probably see the same thing as with the black hole. 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.

The fact that you could block the Hawking radiation by putting shielding around your detector shows that the effect of the Hawking radiation is just to change local boundary conditions around your detector.

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.

I don't know if full quantum gravity will obey the equivalence principle or not, but I don't see the kind of measurements of Hawking radiation that you propose as being able to address the question.

Sincerely,

Jzck

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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...

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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

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Hi Ben,

Thanks for reading my essay. Also very good/interesting questions. In regard to quantum effects breaking general covariance this is indeed the case. In a nice paper by Wilczek and Robinson

"Relationship between Hawking Radiation and Gravitational Anomalies"

Phys. Rev. Lett. 95, 011303 (2005).

they show that Hawking radiation can be thought of as arising out of...

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Hi Ben,

Thanks for reading my essay. Also very good/interesting questions. In regard to quantum effects breaking general covariance this is indeed the case. In a nice paper by Wilczek and Robinson

"Relationship between Hawking Radiation and Gravitational Anomalies"

Phys. Rev. Lett. 95, 011303 (2005).

they show that Hawking radiation can be thought of as arising out of the need to cancel the 1+1 gravitational anomaly in a BH space-time. Interestingly this anomaly derivation does not work for the Unruh effect as shown in

"Comments on anomaly versus WKB/tunneling methods for calculating Unruh radiation" Valeria Akhmedova et al Phys.Lett. B673 (2009) 227-231

e-Print: arXiv:0808.3413

Thus this may provide a further motivation/reason for the break down of EP when comparing Hawking and Unruh radiation. I need to think about this some more. But in any case the breaking of general covariance by quantum effects has been studied (by Bertlemann and Kolprath, as well as Witten -- Robibson and Wilczek use teh anomaly found by these reseachers to give a new interpretation to Hawking radiation).

In regard to your second question on the face of it my proposed running of inertial vs. gravitational mass goes in the wrong direction (at the surface) to explain dark energy. In my picture gravitational effects get stronger as one goes further from the center of the galaxy and for dark energy one wants gravitational repulsion not stronger attraction. Also dark matter is attractive whereas dark energy is repulsive. But truely I have not thought about this on the cosmological scale. My example in the essay when one is far from the event horizon in a simple way indicates that this would not be a way to explain dark energy. But really I should think about this some more. Also there is good evidence that there is dark matter (the Bullet cluster, gravitational lensing) so my feeling is that if my proposed effect has any role to play it one would still need to take into account dark matter (of course the researchers who study MOND think they can do away with dark matter all together, but I have not studied this issue deeply enough to say for sure).

Best,

Doug

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