Hawking Versus Unruh Temperature as a Measure of the Health of the Equivalence Principle by Douglas Alexander Singleton

Dear Azzam,

Thanks for reading my essay and for your comments. I will try to fully read you essay and send comments.

Best,

Doug

Dear Jayakar,

Yes string theory may have something interesting to say about the EP. Thanks for your post and for reading my essay.

Best,

Doug

Dear Vladimir,

Thanks for reading my essay and your comments. I will try to read your essay and send comments as well. The illustrations look very nice.

Best,

Doug

I am thinking of bending some metal on this. I would feel better if my concern over whether this reflects a breakdown in an approximation is answered.

The accelerated frame near the event horizon observes the outside world "speed up," for units of proper time on this frame correspond to very large units of time is asymptopia. As the accelerated frame approaches the horizon, which requires a larger acceleration, we may think of adiabically moving this situation to the stretched horizon. The distinction between this frame and the frozen appearance of a freely falling frame on the stretched horizon as observed from the outside are indistinguishable. This flies in the face of some standard physics, where physics is written according to inertial frames. There is an asymmetry between inertial and accelerated frames, which I think is in greater generality removed. I reiterate what I wrote the other day on the accelerated frame below.

For the accelerated observer the life time of the exterior world races by in a flash. For a stellar mass black hole it requires billions of g-forces to remain a few meters from the horizon, and to get within centimeters requires about a billion billion g's of acceleration. If by some means an observer could do this the outside world would be racing by, say for a small proper time with t = g^{-1} cosh(gs). So the proper time element is s ~ g^{-1}ln(gt) for a time unit t outside. As a result for t the lifetime of the black hole ~ 10^{67}year, g in units of distance ~ 1cm ~ 10^{-10} sec ~ 10^{-17}year the proper time the observer on the accelerated frame observes the BH to evaporate is

s ~ 5x10^{18} years.

This is much shorter than the BH life time measured by the exterior world. Assume you get that acceleration up to 10^{33}cm^{-1} or 10^{43}sec^{-1} or 10^{50}year^{-1}, then you are hovering practically on the stretched horizon. The BH evaporates in about 10^{-42} seconds, or close to the Planck unit of time! Bang!; which means all that ingoing and outgoing radiation which interacts with the black hole hits you at once is a colossal thunderclap. The event horizon appears for larger g close in to be more of a singularity, or a surface region of huge energy density that is radiating and absorbing energy at a ferocious rate.

Cheers LC

Hello Mr Singleton,

And if we arrive to put a light near this BH and if we superimpose the holographic concept ? we could see it :)

You know I beleive strongly in the equivalence principle. It is an important parameter at all scales you know.

The works of hawking are indeed relevant about the hawking radiations correlated with heat and temperature.If I could, I will say him(Mr Hawking) that it is possible to see what is really our BH. first , what is really our central BH of our galaxy. we cannot really see what are their productions. I beleive that they are productors also of complementary matters.But we cannot see them just due to their mass. My equations are relevant. I ask me how we can see it really. I think that we can lighting up it. Of course it is not really easy.But it is possible.

The violations of the equivalence principle are not really rational you know.

That said, I liked to read your essay.

Regards

Douglas,

The equivalence principle is quite useful in that it converts the gravitational redshift into Doppler shift and vice versa, as shown in this quotation:

http://galileo.phys.virginia.edu/classes/252/general_relativity.html

Michael Fowler, University of Virginia: "What happens if we shine the pulse of light vertically down inside a freely falling elevator, from a laser in the center of the ceiling to a point in the center of the floor? Let us suppose the flash of light leaves the ceiling at the instant the elevator is released into free fall. If the elevator has height h, it takes time h/c to reach the floor. This means the floor is moving downwards at speed gh/c when the light hits. Question: Will an observer on the floor of the elevator see the light as Doppler shifted? The answer has to be no, because inside the elevator, by the Equivalence Principle, conditions are identical to those in an inertial frame with no fields present. There is nothing to change the frequency of the light. This implies, however, that to an outside observer, stationary in the earth's gravitational field, the frequency of the light will change. This is because he will agree with the elevator observer on what was the initial frequency f of the light as it left the laser in the ceiling (the elevator was at rest relative to the earth at that moment) so if the elevator operator maintains the light had the same frequency f as it hit the elevator floor, which is moving at gh/c relative to the earth at that instant, the earth observer will say the light has frequency f(1 + v/c) = f(1+gh/c^2), using the Doppler formula for very low speeds."

You don't think this analysis based on the equivalence principle should be abandoned do you?

Pentcho Valev pvalev@yahoo.com

Dear Mr. Dufourny,

Many thanks for your comments. You are correct that from the thought experimetn I present in my essay one could as well take the stand that the EP is correct and then the implication would be that there is something "wrong" with Hawking/Unruh radiation (e.g. Hawking radiation does not exist, or does not have the form given by Hawking). In fact there are researchers who questin the existence of Hawking radiation. For exmaple

"Do black holes radiate?"

by Adam D. Helfer Rept.Prog.Phys. 66 (2003) 943-1008

e-Print: gr-qc/0304042 [gr-qc]

also there are researchers who question the existence of the Unruh effect

"An Example of a uniformly accelerated particle detector with nonUnruh response",

by A.M. Fedotov, N.B. Narozhny, V.D. Mur, V.A. Belinski

Phys.Lett. A305 (2002) 211-217

e-Print: hep-th/0208061 [hep-th]

Thus if one wanted to take the EP as exact under all conditions then my thought experiment would be an argument (along the lines above) of the non-existence of Hawking/Unruh radiation.

However, my choice was to assume the EP was violated and look at the consequences.

Best regards,

Doug

Dear Pentcho,

Prof. Fowler was at UVA during my time there (and I think he still is). Great professor and researcher.

I have no problem with the example of the EP that you give. It is completely correct. But also it treats the photon classically. The suggested violation of the EP that I give in my essay occurs when one treats fields (inlcuding the E&M) field quantum mechanically. If one treats the E&M field classically then one would not have Hawking or Unruh radiation and my thought experiment breaks down. It is only when one treats the E&M field as a quantum field that the possible viiolation of the EP arises.

By the way there is a suggestion that the EP *is* violated even classically. In the article

"Nonequivalence of a uniformly accelerating reference frame and a frame at rest in a uniform gravitational field", Edward A. Desloge

Am. J. Phys. 57, 1121 (1989)

where Desloge writes down what he claims is a uniform gravitational field in 1+1 dimensions and then shows that the geodesics in this metric are not the same as in the 1+1 Rindler metric -- the metric seen by an observer accelerating through Minkowski (flat) space-time.

Best regards,

Doug

Hi Douglas. Quantum gravity (and the union of gravity and electromagnetism as well) definitely require observer and observed, instantaneity, and a fundamental balancing and equivalency of inertia and gravity. Balance and completeness and the combining, balancing, and inclusion of opposites is essential.

Ultimately, in a truly unified and FUNDAMENTAL understanding of physics, space must be shown as invisible, not visible, and visible in a balanced fashion in keeping with the above paragraph. Temperature is basically averaged [FUNDAMENTALLY] given such a unification.

What are your thoughts on this please? Thanks.

Dear Douglas Alexander Singleton,

I hope to discuss with my idea which is related to my MSRT http://fqxi.org/community/forum/topic/1272 and is related to your paper.

According to my theory, since time dilation and Lorentz factor is depending on the difference of the vacuum energy. Thus I found, in the case of the train moving with constant speed v, then for the stationary earth observer the vacuum energy of this train must be higher than the vacuum energy of the earth surface. Thus the temperature inside the boundaries of the moving train will be increased. And if the velocity increased also, the temperature will increase also. This increase in the temperature is because of the increase of the vacuum energy which is related to the velocity of the train. But according to my MSRT, there is no Unruh radiation, where I proof that. What is your opinion for my idea? I hope to hear from you.

Azzam

Hi Frank,

Thanks for your comments on my essay. I'm not sure I completely follow the question but let me give it a try. You seem to be saying that spatial dimensions should become "invisible" in a unified theory. Generally in things like string theory the trend goes the other way -- as one probes higher energy scales the extra dimensions which had been hidden, compactified or invisible "open" up and become visible. Thus in theories with extra dimensions like string theory more spatial dimensions open up/become visible as one goes toward the unification scale.

However, there is recent work which postulates that dimensions compactify or curl up at larger energy scales. This idea can be found in

"Detecting Vanishing Dimensions Via Primordial Gravitational Wave Astronomy",

Jonas R. Mureika, Dejan Stojkovic, Phys.Rev.Lett. 106 (2011) 101101

e-Print: arXiv:1102.3434 [gr-qc]

and the references therein. In this scheme (which is based on the causal dynamical triangulations of GR) the spatial dimensions reduce at larger energy scales.

Something along these lines might be what you had in mind?

Best regards,

Doug

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

Dear Douglas Alexander Singleton

Thank you very much for your previous comment.

In my theory (the equivalence principle) difference temperature is one term that affected on the difference of the vacuum energy, and thus affecting on the Lorentz factor. The other terms are the effective density and pressure. Most of the quantum tunneling experiments and entanglement are performed in a very low temperature. That means according to my theory, in the very low temperature, events and the motion of clocks will be moved on in a faster rate than at higher temperature. For example, in the case of tube of length L in the lab. and the temperature of the tube is very low compared to the temperature of the lab. Thus according to my theory the motion of the clock inside the tube will be faster than the motion of the clock of the lab. That is because according to my equivalence principle, the observer of the lab is equivalent to move with speed v relative to the frame of the low temperature of the tube. Thus from the difference of the temperature of the tube and the lab we can determine the difference of the vacuum energy and thus determining the Lorentz factor, and then determining how the clock inside the tube moving faster. This case is same as when the rider of the moving train with constant v determined by his clock the time t', and when he stopped his train he will find the clock of the earth surface computed more time t where t=gama*t' where gama is the Lorentz factor. Thus if I'm right in my theory and equivalence principle, that led me to ask myself, if the increasing of the temperature of the medium led to the vacuum energy to increase, and thus led to the Lorentz factor to increase, thus for the moving train with constant velocity v, relative a stationary earth observer must observe there is an equivalent temperature increase inside the medium of the moving train, where this increase in temperature can be determined from the Lorentz factor. Thus we can develop this idea in the case of gravity, and thus we can illustrate the Hawking radiation is right. Where from this principle, for any mass, from the Schwarzschild geometry we can determine the Lorentz factor which is depending on the distance from the center of mass. Thus from this Lorentz factor we can determine the equivalent temperature at any distance r from the center of mass. Relative to Hawking radiation, since the Lorentz factor is very high near the event horizon, there must exist a high temperature, and each time we approach from the event horizon the Lorentz factor will increase, thus the temperature will increase also.

Sincerely,

Azzam

Dr. Singleton,

I noticed your statement in the abstract :"To date there has been no definitive, experimental evidence for a problem with general relativity as the proper theory of gravity."

Although the following is not definitive, what is your opinion of the following statement?

"An alternative explanation of the accelerating expansion of the Universe is that general relativity or the standard cosmological model is incorrect. We are driven to consider this prospect by potentially deep problems with the other options." REPORT OF THE

DARK ENERGY TASK FORCE

Regards,

Jeff Baugher

Dear Mr. Baugher,

I agree with this statement. As I read the statement it says that we need to consider that GR and/or the standard cosmological model may be wrong. The current cosmological picture is that GR and the current cosmological picture are correct and that one only needs to add some field/fluid/stuff, which is generically called dark energy, to account for accelerated expansion rate of the Universe. But as well it could be possible that the accelerated expansion is due to some modification of GR. In some sense my essay leans in this direction although I focus on modification at short distances rather than cosmological distances.

And even if there is no *definitive* experimental evidence, currently, that GR is wrong there are plenty of hints that GR needs to be modified: (i) the inability to quantize gravity; (ii) dark matter (this might be due to some modification of GR at galactic distance a la modified Newtonian dynamics); (iii) dark energy (again this might be evidence for modifcation of GR rather than the existence of some substance with the odd properties of dark energy); (iv) the singularities at the center of a BH. Thus there is strong circumstantial evidence the GR will need to be modified at short and/or long distance scale.

Best,

Doug

Doug,

The fact that Hawking radiation and Unruh radiation are different does not show any failure of the equivalence principle (EP). The EP is local, while Hawking radiation is generated in an extended region. If you are in a small elevator, and see radiation coming at you, you have no way of knowing whether it is Hawking radiation or radiation due to some other source. The EP only applies to what goes on within the elevator (local), not to what external boundary conditions the elevator is likely to have.

It is no different from the standard example of an observer standing on a planet (near 0 Kelvin). This observer observes no Unruh or Hawking radiation, but he feels a force on his feet.

Regards,

Jack

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

Doug,

The most important implication of the equivalence principle is that, in a gravitational field, the speed of light varies like the speed of any projectile. If, in Fowler's example, "the earth observer will say the light has frequency f(1+v/c)", then the same observer will say the light has speed c'=c+v. Do you agree?

Pentcho Valev pvalev@yahoo.com

Variable speed of light in a gravitational field - explicit derivation:

http://www.youtube.com/watch?v=ixhczNygcWo

"Relativity 3 - gravity and light"

Pentcho Valev pvalev@yahoo.com