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

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

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

Doug,

The measurement is local, but your interpretation of that measurement depends on your assumed knowledge of the situation over extended distances. You see radiation, so you assume that it's Hawking radiation; but you would see the same radiation if in place of a black hole there were a low mass object of the appropriate size and temperature there. The two situations are locally equivalent at the detector, and such a possibility is all that the EP requires.

If you put a radiation shield around the detector, that could block much of the Hawking radiation, so clearly what you are measuring is not independent of the situation external to the detector volume.

Physics within the detector volume still 'looks like' physics in flat space but with appropriate boundary conditions (with the usual caveats about tidal forces and so on if your detector has finite size). By contrast, suppose that an object within your detector had a gravitational mass that was that was not equal to its inertial mass. There is no set of boundary conditions which can be imposed external to your detector volume that could 'spoof' or cause you to falsely reproduce the set of measurements that could confirm this. It would therefore be a true violation of the local EP.

Sincerely,

Jack

Doug,

You wrote: "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."

Yet he derives the fundamental equation of Newton's emission theory of light, c'=c(1+gh/c^2), in the form dc/dh=g/c:

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

"Relativity 3 - gravity and light"

That is, in a gravitational field the speed of photons varies exactly like the speed of cannonballs. This prediction of the emission theory is confirmed by the Pound-Rebka experiment:

http://www.einstein-online.info/spotlights/redshift_white_dwarfs

Albert Einstein Institute: "One of the three classical tests for general relativity is the gravitational redshift of light or other forms of electromagnetic radiation. However, in contrast to the other two tests - the gravitational deflection of light and the relativistic perihelion shift -, you do not need general relativity to derive the correct prediction for the gravitational redshift. A combination of Newtonian gravity, a particle theory of light, and the weak equivalence principle (gravitating mass equals inertial mass) suffices. (...) The gravitational redshift was first measured on earth in 1960-65 by Pound, Rebka, and Snider at Harvard University..."

Pentcho Valev pvalev@yahoo.com

Doug,

I am interested in your thoughts on the equivalence principle. My essay is based on a simple premise. If the function F1 is the Newtonian gravitational field strength, then F1' is the gravitational force. How do we know we have been anti-differentiating the Newtonian field correctly since we could have just been mistaking F1' for (C-F2)' following the rules concerning arbitrary constants of integration. This should lead back to a substitution in the Einstein field equation of [math]G_{\mu\nu}=\Omega g_{\mu\nu}-L_{\mu\nu}[/math]. If the constant term is equated to the potential energy of the vacuum, then the Luv term is just equated to the dynamic residual energy tensor. This would seem to solve the cosmological constant magnitude problem, make it look like gravity is attractive but also allow for a repulsion after a certain radius.

Thoughts welcome.

Regards,

Jeff

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

Dear Douglas,

I really enjoyed your essay! A couple of questions come to mind.

1. I am trying to understand how to frame your ideas in the context of the general principle of covariance. In SR, covariance may be viewed as symmetry under the Poincare group, which is also viewed as determining many of the properties of particles in QFT. In GR, covariance may be viewed as a local group symmetry. Would you say, then, that quantum effects break covariance? Or would you describe it in some other way?

By the way, I prefer NOT to view covariance as a local group symmetry, but as an order-theoretic concept; I explain this briefly in my essay:

On the Foundational Assumptions of Modern Physics

2. You mention that dominance of gravitation mass over inertial mass might be an alternative explanation to the dark matter hypothesis. What about dark energy? Here the relevant scale is even larger, so one might naively expect an even greater enhancement of the "gravitational force" (which is obviously not what we observe); however, in this case we are no longer dealing with a "central force" as in the case of a black hole or galaxy, so I'm not sure what to expect. What do you think about this?

Take care,

Ben Dribus

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

Dear Douglas:

I enjoyed reading your well-written essay and a creative approach to resolve the black hole singularity in GR. Have you given any consideration to use the black hole evaporation concept to solve the singularity issue?

You may be interested in my paper - - -" From Absurd to Elegant Universe" wherein I use the mass evaporation not only to resolve the black hole singularity but also the dark energy (Cosmological Constant) problem.

I would greatly appreciate your comments on my paper.

Best of Luck and Regards

Avtar Singh

Avtar,

Thanks for your comments and for looking at my essay. I will try to have a look at your essay. It may take a while since next week I'm traveling to a conference and must still prepare my talk.

I have not looked specifically at using Hawking radiation as a means for resolving the issue of the singularity of a Schwazschild BH, however in two recent papers with Sujoy Modak

"Hawking Radiation as a Mechanism for Inflation",

Sujoy Kumar Modak, Douglas Singleton,

e-Print: arXiv:1205.3404 [gr-qc]

and

"Inflation with a graceful exit and entrance driven by Hawking radiation",

Sujoy Kumar Modak, Douglas Singleton,

e-Print: arXiv:1207.0230 [gr-qc]

we propose using the Hawking-like radiation of FRW space-time as a possible mechanism for inflation. This model is along the line of the work by Prigogine et al

I. Prigogine, J. Geheniau, E. Gunzig, P. Nardone, Gen. Rel. Grav. 21, 767 (1989).

who consider generic particle creation models (rather than the specific Hawking radiation particle creation) as a means of driving inflation. These models in general do resolve the cosmological singularity problem (i.e. there is not cosmological singularity in Prigogine's model). Thus it may well be that the same kind of thinig would work for BH singularities. But I would need to think about this some more.

Best,

Doug

Hi Jeff,

Sorry for the delay in reply, but I missed this post. Anyway by F1 do you mean gravitational field strength or gravitational potential? It seems that in order to get the gravitational force (F1') you are differentiating F1, which would mean F1 is the potential rather than the field (the field is essentially just the force but one of the charges/masses divided out). Also is ' a time derivative or spatial derivative/gradient? It should be gradient right since force field = grad(potential).

Anyway at the Newtonian level adding a constant to the (gravitational) potential will not change anything since grad(potential)=grad(potential +K) since grad(K)=0.

Best,

Doug

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

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

Hi Hải.Caohoàng,

Yes the recent discovery of the Higgs-like particle could be relevant to the discussion of the local equivalence of inertia and gravity since the Higgs mechanism gives particles their inertial mass. On the other hand one might argue that to understand gravitational mass one would need to understand gravity at its natural scale i.e. the Planck scale. If one buys this hand waving argument (i.e. inertial mass is set at the Higgs scale ~ 125 GeV and the gravitational mass is connected with the Planck scale) then one has a puzzle of how two such different scales can lead to masses (inertial vs. gravitational) which are so similar. This is then just another form of the heirarchy problem.

Thanks for your comments and reading my essay.

Best,

Doug