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

Douglas Singleton

Essay Abstract

The equivalence principle is the conceptual basis for our best theory of gravity, general relativity. To date there has been no definitive, experimental evidence for a problem with general relativity as the proper theory of gravity. In this essay we present a thought experiment, involving the quantum effects of Hawking and Unruh radiation, which indicates a violation of the equivalence principle. The way in which the equivalence principle is violated by this thought experiment points toward a possible way to resolve some of the short comings of general relativity - the lack of a consistent quantum theory of gravity and the existence of singularities in black hole space-times.

Author Bio

The author completed his SB degree in physics at MIT and obtained his PhD from the University of Virginia. He taught physics for one year at a high school while also working as a parking garage attendant, before moving to Virginia Commonwealth University for three years as a lecturer. In 1998 he moved to California State University, Fresno. He has held visiting faculty positions at Peoples' Friendship University Russia in 2004 and 2009 and at the Universidad de Costa Rica in 2005. His recent work has focused on quantum fields in gravitational backgrounds.

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[deleted]

Dear Dr. Douglas Alexander Singleton

I really want to thank you more than words for your great essay. You discussed an important issues regarded my essay http://fqxi.org/community/forum/topic/1272, the equivalence principle in SRT, the relative velocity in SRT, and the reciprocity principle. You illustrated my interpretation to the length contraction and the time dilation in my MSRT which depending on the concepts and principles of quantum theory is right, and thus my interpretation to faster than light and quantum tunneling, Casimir effect and entanglement is right. In my theory, in the case of measuring faster than light there is no violation for Lorentz transformation or causality. Wormholes in GRT can be interpreted according to my theory same as quantum tunneling and entanglement. I hope to read my essay and then discuss more details.

AZZAM

[deleted]

Dear Azzam,

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

Best,

Doug

Jayakar Joseph

Dear Douglas Singleton,

When gravity emerges from an eigen-rotational string, that gravity is proportional to the total mass representational by that string which includes inertial and gravitational masses of the objects in that string; and thus consistency of quantum gravity is expressional in this paradigm.

With best wishes,

Jayakar

[deleted]

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

Lawrence Crowell

Dear D. Singleton,

Lawrence Crowell

Dear D. Singleton,

I forgot that this system does not like backwards facing carrot symbols, and it cuts things off. So I repost this with I hope things fixed.

I think there might be some problems with this. The substitution of the gravitational acceleration g = (1 - 2m/r)(m/r^2) into the Unruh temperature equation may only apply in the near horizon condition where r = 2m d, 2m >> d. The question is whether this is really a violation of the equivalence principle or a breakdown in the appropriate approximation for this substitution.

It is further difficult to know what is meant by Unruh effects in the region 2m > r. There is no comparable physics for the Unruh effect at a distance d = c^2/g, for d outside the Rinder wedge, or 0 > d.

I will try to give this some thought, but I think there is a question of whether this represents a real departure from the equivalence principle or whether this reflects T_H =~ T_U is correct only with the near horizon approximation.

Cheers LC

[deleted]

Dear LC,

Thanks for reading my essay and your questions. You are correct that inside the horizon there are difficulties. However I do note this by saying that on crossing the horizon the analysis will proceed by heuristic arguments and conjecture. One point is that at and inside the horizon one can not "stand still". This is already indicated by the fact that the local acceleration for an observer who is at rest near the horizon approaches \infty as one goes to the horizon.

But outside the horizon one is in a regime where one can trust the calculation of the local acceleration. For an observer at a fixed distance from a gravitating object one can straightforwardly calculate the local acceleration. Then taking this finite acceleration one can think of a Rindler observer with this acceleration. In turn this Rindler observer will detect an Unruh temperature proportional to a=(GM)/R^2. This temperature will be lower than the Hawking temperature of an observer in the field of a gravitating body. Thus the tow observer will measure the same local acceleration but will measure different temperatures.

Best,

Doug

Lawrence Crowell

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

Vladimir Tamari

Dear Douglas,

Although the technicalities of comparing Hawking and Unruh's results are beyond me, I would like to comment on two things in your interesting essay: If one sets aside Einstein's rather arbitrary decision in SR to have c constant and the resulting spacetime concept, the Equivelance Principle in GR reduces to the curvature of the energy streamlines (or of the equipotential surfaces). This reduces it to simple refraction with variable speed of light in a region of gradient index of refraction (Eddington 1920). In other words the mechanics of classical deceleration resulting in curvature!

My other point is that in my theory the Holographic Principle can be explained as the resultant of physical node-node inductive forces within the ordered ether lattice of space, including a black hole. See attached Fig. 27 of the following paper to illustrate GR as refraction, and Fig. 11 to show the mechanism by which the Holographic Principle can be understood. Beautiful Universe Theory I hope these two concepts will somehow one day mesh in with your interesting research. If you have time I hope you will also have a look at my fqxi essay Fix Physics!

With best wishes,

VladimirAttachment #1: 1_BUFIG11.jpg Attachment #2: BUFIG27.jpg

[deleted]

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

Steve Dufourny

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

[deleted]

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

Steve Dufourny

Hello ,

Is it necessary to invest in already knwown results ? the monney is a tool and it must be utilized witht he biggest wisdom.

The violations are not really foundamental you know.

The Bh radiates probably but their motions are above the fermions at my humble opinion.In fact the real interest is to fid the fractal above the weak and the strong int. more the electromagnetism. I beleive that the volumes of spheres are the secret......see the stabilities of informations.

In fact the aim is to class the BH and the volumes of stabilities. I beleive that the works of Hawking are relevant considering a pure thermodynamical correlation. The BH can be classed. The BH have a lot of properties correlated with the rotating spheres. The kinetic energy and the potential energy are always relevant.

In fact, these BH are like the stars, they produce the matter with all its fractalization and its complexity. The quarks, the neutrinos, the gravitons,muons,.... but not bosons at my humble opinion. In fact they are above the SR. It is intringuing considering the sortings and the synchronizations of informations in a pure general point of vue.

I beleive strongly that the volumes of BH increases more we go towards our main central BH of our Universal sphere. It is relevant considering the other productions of matters from these central spheres. The steps appear when the volumes and the rotations are inserted. The pure heat and thermodynamics can showing the road towards these central spheres. The system is a finite serie.

Regards

Pentcho Valev

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

[deleted]

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

Pentcho Valev

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

Pentcho Valev

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

[deleted]

Hi Pentcho,

Very nice YouTube video. From my brief viewing of this it seems correct. Note though that the author mentions that the derivation/motivation he gives is not rigorous. He is able to derive the Schwarzschild radius using these heuristic arguments (but also he says that there is a bit of luck involved similar to the Rutherford scattering formula being the same both classically and quantum mechanically). In terms of the bending of light around a massive object these heuristic arguments get the angle of derivation wrong by the factor of two.

You also asked if I agree with c'=c+v (which seems to imply c'>c) which comes up in the intermediate steps in the video. First, the special relativistic rule that nothing can travel faster than c assumes that one has a global inertial reference frame. Once one allows non-inertial references frames and/or gravity things can get interesting. Now *locally*, even in a gravitational field, the local speed of light is limited by c since one can (by the equivalence principle) always locally go to Minkowski space-time. Even in the presence of a gravitational field one usually gets the speed of light (as measured by an observer that is far from the center of the black hole) to be c or less. In the example given in the video (light speed measured in the background of a Schwarzschild black hole by an external observer) the speed of light goes to zero at the horizon and then increases to c as one moves away from the black hole toward the region of the observer. But for an observer in a box in free fall toward the Schwarzschild black hole the local speed of light will just be c. By the way there are more or less standard examples of "speeds greater than c". I think (need to check this) that for a Kerr black hole one can arrange things so that some observers will get an effective speed greater than c. Or even more simply if one considers a rotating reference frame in Minkowski space-time (i.e. no gravitational field) then there is the possibility to measure an effective speed greater than c (but again a rotating reference frame is not an inertial reference frame). This is discussed in the article

"Relativistic description of a rotating disk",

Ø. Grøn, Am. J. Phys. 43, 869 (1975)

which explicitly mentions the "greater than c behavior" for the rotating reference frame.

Best,

Doug

[deleted]

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.

[deleted]

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

[deleted]

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

[deleted]

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