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**Christian Corda**: *on* 10/31/13 at 17:05pm UTC, wrote Dears Colleagues, Congrats for the Prize. You, Jennifer Nielsen and...

**Douglas Singleton**: *on* 8/10/13 at 6:52am UTC, wrote Hi Willard, I missed your original reply. I haven't figure out how to get...

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FQXi FORUM

December 6, 2021

CATEGORY:
It From Bit or Bit From It? Essay Contest (2013)
[back]

TOPIC: Self-similarity, conservation of entropy/bits and the black hole information puzzle by Douglas Singleton, Elias Vagenas, & Tao Zhu [refresh]

TOPIC: Self-similarity, conservation of entropy/bits and the black hole information puzzle by Douglas Singleton, Elias Vagenas, & Tao Zhu [refresh]

John Wheeler coined the phrase "it from bit" or "bit from it" in the 1950s. However, much of the interest in the connection between information, i.e. "bits", and physical objects, i.e. "its", stems from the discovery that black holes have characteristics of thermodynamic systems having entropies and temperatures. This insight led to the information loss problem {what happens to the "bits" when the black hole has evaporated away due to the energy loss from Hawking radiation? In this essay we speculate on a conservative answer to this question using the assumption of self-similarity of quantum correction to the gravitational action and the requirement that the quantum corrected entropy be well behaved in the limit when the black hole mass goes to zero.

Douglas Singleton is a professor at California State University Fresno and currently on a leave at ITB in Bandung, Indonesia. Elias Vagenas is a professor at Research Center for Astronomy and Applied Mathematics, Academy of Athens. In Sept. 2013, he moved to Kuwait University as Associate Professor and joined the Theoretical Physics Group in the Department of Physics. Tao Zhu is a post-doc at Baylor University, and holds a position at the Institute for Advanced Physics & Mathematics,Zhejiang University of Technology, Hangzhou. All three work in the area of gravitational physics, high energy/particle physics and the interface between the two.

Dear Prof Douglas,

Thank you for presenting your nice essay. I saw the abstract and will post my comments soon. So you can produce material from your thinking. . . .

I am requesting you to go through my essay also. And I take this opportunity to say, to come to reality and base your arguments on experimental results.

I failed mainly because I worked against the main stream....

view entire post

Thank you for presenting your nice essay. I saw the abstract and will post my comments soon. So you can produce material from your thinking. . . .

I am requesting you to go through my essay also. And I take this opportunity to say, to come to reality and base your arguments on experimental results.

I failed mainly because I worked against the main stream....

view entire post

report post as inappropriate

Dear Dr.Douglas Singleton,

I have down loaded your essay and soon post my comments on it. Meanwhile, please, go through my essay and post your comments.

Regards and good luck in the contest.

Sreenath BN.

http://fqxi.org/community/forum/topic/1827

report post as inappropriate

I have down loaded your essay and soon post my comments on it. Meanwhile, please, go through my essay and post your comments.

Regards and good luck in the contest.

Sreenath BN.

http://fqxi.org/community/forum/topic/1827

report post as inappropriate

A good essay, that I must reread with attention.

I am thinking that if the Hawking radiation transfer information from a naked, and not rotating, black hole, then each emitted photon change locally the curvature of the horizon (the change of the emitted entropy must change the surface information).

If this is true, then the surface is not smooth, and the infrared emission must contain information of the surface (spectrum and polarization information).

If there are two near naked black hole, then the change of curvature of one emission, change the other black hole absorption curvature.

Then the temperature of the black hole can be not uniform.

The problem that I have is that if you have an uniform emission (like a black body radiation) then there is only a spectrum information (so you can determine only the mass of the black hole): is the information contained in the different time of the emissions?

report post as inappropriate

I am thinking that if the Hawking radiation transfer information from a naked, and not rotating, black hole, then each emitted photon change locally the curvature of the horizon (the change of the emitted entropy must change the surface information).

If this is true, then the surface is not smooth, and the infrared emission must contain information of the surface (spectrum and polarization information).

If there are two near naked black hole, then the change of curvature of one emission, change the other black hole absorption curvature.

Then the temperature of the black hole can be not uniform.

The problem that I have is that if you have an uniform emission (like a black body radiation) then there is only a spectrum information (so you can determine only the mass of the black hole): is the information contained in the different time of the emissions?

report post as inappropriate

Hi,

If I understand correctly you are worried about the spectrum of emitted Hawking radiation being purely thermal? As you correctly point out this is a problem and in fact is a key point in the information loss puzzle of BHs. The approach we use to Hawking radiation is the tunneling approach pioneering by Wilczek&Parikh, Srinvinsan&Padmanhban, Volovik, Berezin&Boyarsky&Neronov (this last group was actually the first to propose a detailed calculation of Hawking radiation as tunneling but since the paper was published in a good but not so read Russian journal it does not really get the credit it deserves).

In the tunneling approach it is possible to take into account the back reaction of space-time geometry of the BH due to the energy loss due to Hawking radiation. The effect of this back reaction can be seen in the omega^2 term in our equation (13) in the essay. For a purely thermal spectrum one only has the omega term. What we show is that this back reaction along with the assumption of self-similarity of the quantum corrections leads to a resolution to the information loss puzzle. In our essay information/entropy is not lost but comes out in the non-thermal radiation.

Best, Doug

If I understand correctly you are worried about the spectrum of emitted Hawking radiation being purely thermal? As you correctly point out this is a problem and in fact is a key point in the information loss puzzle of BHs. The approach we use to Hawking radiation is the tunneling approach pioneering by Wilczek&Parikh, Srinvinsan&Padmanhban, Volovik, Berezin&Boyarsky&Neronov (this last group was actually the first to propose a detailed calculation of Hawking radiation as tunneling but since the paper was published in a good but not so read Russian journal it does not really get the credit it deserves).

In the tunneling approach it is possible to take into account the back reaction of space-time geometry of the BH due to the energy loss due to Hawking radiation. The effect of this back reaction can be seen in the omega^2 term in our equation (13) in the essay. For a purely thermal spectrum one only has the omega term. What we show is that this back reaction along with the assumption of self-similarity of the quantum corrections leads to a resolution to the information loss puzzle. In our essay information/entropy is not lost but comes out in the non-thermal radiation.

Best, Doug

Hello,

Congratulations on an excellent essay! I have a question (actually, a set of related questions), which is prompted by P. Nicolini and B. Niedner's "Hausdorff dimension of a particle path in a quantum manifold" (arXiv:1009.3267; Phys Rev D83:024017, 2011). They argue that the existence of a minimal length -- which may reasonably be regarded as a feature of quantum gravity itself -- breaks self-similarity at the minimal-length scale. In view of their argument, it's not clear that self-similarity can or should be invoked to conclude that information/entropy is well-behaved as M approaches zero; hence, there may be a problem with your proposed resolution of the black hole information paradox..

So, my set of questions is: do you reject the idea that a minimal length exists (and if so, on what do you base your rejection), or do you reject Nicolini and Niedner's argument as flawed in some way (and if so, in what way is it flawed)?

Thank you, and good luck,

Willard Mittelman

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Congratulations on an excellent essay! I have a question (actually, a set of related questions), which is prompted by P. Nicolini and B. Niedner's "Hausdorff dimension of a particle path in a quantum manifold" (arXiv:1009.3267; Phys Rev D83:024017, 2011). They argue that the existence of a minimal length -- which may reasonably be regarded as a feature of quantum gravity itself -- breaks self-similarity at the minimal-length scale. In view of their argument, it's not clear that self-similarity can or should be invoked to conclude that information/entropy is well-behaved as M approaches zero; hence, there may be a problem with your proposed resolution of the black hole information paradox..

So, my set of questions is: do you reject the idea that a minimal length exists (and if so, on what do you base your rejection), or do you reject Nicolini and Niedner's argument as flawed in some way (and if so, in what way is it flawed)?

Thank you, and good luck,

Willard Mittelman

report post as inappropriate

Dear Dr. Mittleman,

Thanks for your excellent question. Also I hope that the proposal in our essay does not contradict Piero's (i.e. Dr. Nicolini's) and his co-worker's paper since it was Piero who got me thinking about self-similarity and who explained to me some issues with self-similarity and QM/QFT when I visited him last fall. We should have cited the paper you mention and let me...

view entire post

Thanks for your excellent question. Also I hope that the proposal in our essay does not contradict Piero's (i.e. Dr. Nicolini's) and his co-worker's paper since it was Piero who got me thinking about self-similarity and who explained to me some issues with self-similarity and QM/QFT when I visited him last fall. We should have cited the paper you mention and let me...

view entire post

Hello Doug,

Thanks for your reply; you don't need to apologize for its length! Sorry for not replying earlier; I've been distracted by various extraneous issues.

At any rate, there's one thing that still puzzles me. I grant your point about the horizon radius increasing with the energy scale. What I'm not sure about is whether this point is applicable to the case of an evaporating black hole, in which the black hole's mass M goes to zero. For, as the horizon radius grows, it would seem that M increases, so that we're not dealing with an evaporating black hole at all. Or, if evaporation does occur here, it (arguably) happens "all at once" in a sudden burst of radiation that reflects instability of the black hole at (or below?) the Planck scale; and this doesn't seem to fit your model of an evaporation process characterized by self-similarity. Such an instability leading to evaporation is mentioned on p. 4 of Spallucci and Ansoldi's "Regular black holes in UV self-complete quantum gravity" (arXiv:1101.2760), in which the authors note the positive correlation between horizon size and energy scale. They address the above instability by arguing that the Planck scale represents a minimal size for black holes, which are stable with respect to this scale. In other words, they conclude that the horizon/energy correlation is associated with stable black holes of minimal size, rather than with evaporating black holes -- which brings us back to my question above concerning the (im)possibility of using this correlation in the case of evaporating black holes.

I apologize if I'm missing something here, or if I've misunderstood your ideas.

Thanks again for your reply.

-Willard

report post as inappropriate

Thanks for your reply; you don't need to apologize for its length! Sorry for not replying earlier; I've been distracted by various extraneous issues.

At any rate, there's one thing that still puzzles me. I grant your point about the horizon radius increasing with the energy scale. What I'm not sure about is whether this point is applicable to the case of an evaporating black hole, in which the black hole's mass M goes to zero. For, as the horizon radius grows, it would seem that M increases, so that we're not dealing with an evaporating black hole at all. Or, if evaporation does occur here, it (arguably) happens "all at once" in a sudden burst of radiation that reflects instability of the black hole at (or below?) the Planck scale; and this doesn't seem to fit your model of an evaporation process characterized by self-similarity. Such an instability leading to evaporation is mentioned on p. 4 of Spallucci and Ansoldi's "Regular black holes in UV self-complete quantum gravity" (arXiv:1101.2760), in which the authors note the positive correlation between horizon size and energy scale. They address the above instability by arguing that the Planck scale represents a minimal size for black holes, which are stable with respect to this scale. In other words, they conclude that the horizon/energy correlation is associated with stable black holes of minimal size, rather than with evaporating black holes -- which brings us back to my question above concerning the (im)possibility of using this correlation in the case of evaporating black holes.

I apologize if I'm missing something here, or if I've misunderstood your ideas.

Thanks again for your reply.

-Willard

report post as inappropriate

Hi Willard,

Another good question. Actually Euro and Ansoldi’s paper has some of the same features as our paper as well as an important difference which I will try to touch on. First if one looks at figure 1 of Euro’s paper this is exactly the same point I was making about the UV/IR correspondence in my earlier reply. The hyperbola in the figure is the usual inverse relationship between...

view entire post

Another good question. Actually Euro and Ansoldi’s paper has some of the same features as our paper as well as an important difference which I will try to touch on. First if one looks at figure 1 of Euro’s paper this is exactly the same point I was making about the UV/IR correspondence in my earlier reply. The hyperbola in the figure is the usual inverse relationship between...

view entire post

A very nice essay,

The presentation of the relation between the its(physical objects) and bits(information/entropy) as you mentioned it was indeed very interesting and captivating.

good luck ,

Salvish Goomanee

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The presentation of the relation between the its(physical objects) and bits(information/entropy) as you mentioned it was indeed very interesting and captivating.

good luck ,

Salvish Goomanee

report post as inappropriate

Dear three Author

An essay represents collaboration at a high level, unfortunately I did not have enough expertise to understand, nonetheless wish you success.

report post as inappropriate

An essay represents collaboration at a high level, unfortunately I did not have enough expertise to understand, nonetheless wish you success.

report post as inappropriate

Dear Singleton, Vagenas and Zhu,

Superb paper! I just gave you guys a big shot in the arm with a top vote score. The expansion of the action S = S_0 + sum_n γ_nħ^nS_0 leads to results similar to yours. This is also interesting for in string theory the action is an expansion of the form

S = sqrt(-g)(R + α’R_{abcd}R^{abcd} + O(α’^2)

where α’ is the string parameter that is O(ħ). In the vacuum solution for the Schwarzschild black hole with R_{ab} ~ g_{ab} this will lead to an expansion similar to the one in equation 7 of this paper.

I have long been interested in the idea there are connections between string theory and LQG. This expansion, both by you and with string theory appear to assume a classical background. The connection to LQG though might remove some of the issues with background dependency of string theory. Also the AdS/CFT correspondence indicates that four dimensional spacetimes have embedded within them the same data as in 10 dimensional supergravity or superstring theory. This means that 4-dim quantum gravity with loops or knots should contain the same information as in superstring theory.

Cheers LC

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Superb paper! I just gave you guys a big shot in the arm with a top vote score. The expansion of the action S = S_0 + sum_n γ_nħ^nS_0 leads to results similar to yours. This is also interesting for in string theory the action is an expansion of the form

S = sqrt(-g)(R + α’R_{abcd}R^{abcd} + O(α’^2)

where α’ is the string parameter that is O(ħ). In the vacuum solution for the Schwarzschild black hole with R_{ab} ~ g_{ab} this will lead to an expansion similar to the one in equation 7 of this paper.

I have long been interested in the idea there are connections between string theory and LQG. This expansion, both by you and with string theory appear to assume a classical background. The connection to LQG though might remove some of the issues with background dependency of string theory. Also the AdS/CFT correspondence indicates that four dimensional spacetimes have embedded within them the same data as in 10 dimensional supergravity or superstring theory. This means that 4-dim quantum gravity with loops or knots should contain the same information as in superstring theory.

Cheers LC

report post as inappropriate

Hi LC,

Many thanks for reading our essay and your nice comments. I will certainly try to read your essay soon. As I mentioned I will be moving in a weak (leaving my temporary position at ITB) but should have time after this.

In regard to your comments you are correct -- our proposal still has the feature of background dependence, which as you point out is also a drawback of string theory. LQG might have some advantage in this regard but I have not followed the developments here closely enough to say anything useful -- but in the LQG talks I have seen the speakers always mention background independence as a good feature of LQG.

Best, Doug

Many thanks for reading our essay and your nice comments. I will certainly try to read your essay soon. As I mentioned I will be moving in a weak (leaving my temporary position at ITB) but should have time after this.

In regard to your comments you are correct -- our proposal still has the feature of background dependence, which as you point out is also a drawback of string theory. LQG might have some advantage in this regard but I have not followed the developments here closely enough to say anything useful -- but in the LQG talks I have seen the speakers always mention background independence as a good feature of LQG.

Best, Doug

Doug et al,

Every time I hear that string theory is background dependent, I want to reach for my gun. :-) I think that even by Lee Smolin's criteria -- and *especially* applying the principle of self-similarity that you have so elegantly incorporated into your thesis, string theory forms a complete quantum field of spacetime that is self organizing and self annihilating.

I agree with Lawrence that your essay's position in the ratings does not at all reflect the quality of the work. I hope my high score to come will help.

And I hope you find interesting my own essay that agrees with your conjecture of infinitely self similar curves. Of course, I'm sure you've seen Professor Corda's take on unitary evolution of pure quantum states, with which I also agree.

(Lawrence, I'll hopefully get to your essay soon, as well.)

All best,

Tom

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Every time I hear that string theory is background dependent, I want to reach for my gun. :-) I think that even by Lee Smolin's criteria -- and *especially* applying the principle of self-similarity that you have so elegantly incorporated into your thesis, string theory forms a complete quantum field of spacetime that is self organizing and self annihilating.

I agree with Lawrence that your essay's position in the ratings does not at all reflect the quality of the work. I hope my high score to come will help.

And I hope you find interesting my own essay that agrees with your conjecture of infinitely self similar curves. Of course, I'm sure you've seen Professor Corda's take on unitary evolution of pure quantum states, with which I also agree.

(Lawrence, I'll hopefully get to your essay soon, as well.)

All best,

Tom

report post as inappropriate

Hi Tom,

Thanks for reading our essay and as well reminding me that I promised read some essays starting with Lawrence's. I did download and read his essays and liked it very much so I'm heading over there to rate it now. Prof. Corda's essay is very close to our own and as well I had promised to read this as well -- it is downloaded but still needs reading :-(. Also if I understand correctly your essay has some connection with self-similarity so I should read this as well. Again thanks for your comments are reading our essay.

Best,

Doug

Thanks for reading our essay and as well reminding me that I promised read some essays starting with Lawrence's. I did download and read his essays and liked it very much so I'm heading over there to rate it now. Prof. Corda's essay is very close to our own and as well I had promised to read this as well -- it is downloaded but still needs reading :-(. Also if I understand correctly your essay has some connection with self-similarity so I should read this as well. Again thanks for your comments are reading our essay.

Best,

Doug

Dears Colleagues,

Congrats for this beautiful Essay. It is very well written and complementary to my one. You solve the black hole information paradox by showing that the number of microstates between the initial and final states is the same and this implies unitary evolution. My approach finds directly the unitary evolution through a time dependent Schrödinger equation. This permits me to write down explicitly the final state like a pure state rather than a mixed one.

If you are interested on my approach you can see my Essay here: Christian Corda

Best wishes and good luck in the Contest,

Ch.

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Congrats for this beautiful Essay. It is very well written and complementary to my one. You solve the black hole information paradox by showing that the number of microstates between the initial and final states is the same and this implies unitary evolution. My approach finds directly the unitary evolution through a time dependent Schrödinger equation. This permits me to write down explicitly the final state like a pure state rather than a mixed one.

If you are interested on my approach you can see my Essay here: Christian Corda

Best wishes and good luck in the Contest,

Ch.

report post as inappropriate

Hi Dr. Corda,

Thanks for taking a look at our essay and your comments. I agree that from reading the title and abstract it appears we are dealing with similar issues in a complementary way. There has been actually a lot of recent interest in the BH information loss puzzle. We will certainly have a look at your essay and send any useful comments/questions. As I mention above I am moving this week from ITB so it may be after one week.

In any case thanks for the comments and reading our essay and best of luck likewise. Doug

Thanks for taking a look at our essay and your comments. I agree that from reading the title and abstract it appears we are dealing with similar issues in a complementary way. There has been actually a lot of recent interest in the BH information loss puzzle. We will certainly have a look at your essay and send any useful comments/questions. As I mention above I am moving this week from ITB so it may be after one week.

In any case thanks for the comments and reading our essay and best of luck likewise. Doug

Please excuse me Professor Singleton, Professor Vagenas and Dr. Zhu,

I am a decrepit old realist. I do not mean to be critical of your technically perfect essay; I would just like to comment on it.

“In this essay we look at the connection between physical objects, i.e. “its” and information/entropy i.e. “bits,” in the context of black hole physics."

As I have pointed out in my essay BITTERS, one real unique universe can only be eternally occurring, once. Real unique, once cannot be connected. Every real “it” is unique, once. What you appear to be doing in your essay is simply finding an abstract connection between abstract objects and abstract information/entropy in the context of abstract black hole physics.

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I am a decrepit old realist. I do not mean to be critical of your technically perfect essay; I would just like to comment on it.

“In this essay we look at the connection between physical objects, i.e. “its” and information/entropy i.e. “bits,” in the context of black hole physics."

As I have pointed out in my essay BITTERS, one real unique universe can only be eternally occurring, once. Real unique, once cannot be connected. Every real “it” is unique, once. What you appear to be doing in your essay is simply finding an abstract connection between abstract objects and abstract information/entropy in the context of abstract black hole physics.

report post as inappropriate

Can we Wheeler a black hole?

Is the nothing inside of a black hole real? No

Is the something outside of a black hole real? Yes.

Energy is made of particles and waves. What are black holes made of?

Good luck, Joe

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Is the nothing inside of a black hole real? No

Is the something outside of a black hole real? Yes.

Energy is made of particles and waves. What are black holes made of?

Good luck, Joe

report post as inappropriate

Send to all of you

THE ADDITIONAL ARTICLES AND A SMALL TEST FOR MUTUAL BENEFIT

To change the atmosphere "abstract" of the competition and to demonstrate for the real preeminent possibility of the Absolute theory as well as to clarify the issues I mentioned in the essay and to avoid duplicate questions after receiving the opinion of you , I will add a reply to you :

1 . THE...

view entire post

THE ADDITIONAL ARTICLES AND A SMALL TEST FOR MUTUAL BENEFIT

To change the atmosphere "abstract" of the competition and to demonstrate for the real preeminent possibility of the Absolute theory as well as to clarify the issues I mentioned in the essay and to avoid duplicate questions after receiving the opinion of you , I will add a reply to you :

1 . THE...

view entire post

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If given the time and the wits to evaluate over 120 more entries, I have a month to try. My seemingly whimsical title, “It’s good to be the king,” is serious about our subject.

Jim

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Jim

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Doug, et al,

This is a copy of my respond to Doug Singleton on my essay blog.

I have read several papers by Vladimir Dzhunshaliev on octonion field theory, and Merab Gogberashvili is a familiar name as well. Trying to understand how nonassociative mathematics of operators fits into physics is really the hard part. I think that quantum mechanics is purely complex, or C. Of...

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This is a copy of my respond to Doug Singleton on my essay blog.

I have read several papers by Vladimir Dzhunshaliev on octonion field theory, and Merab Gogberashvili is a familiar name as well. Trying to understand how nonassociative mathematics of operators fits into physics is really the hard part. I think that quantum mechanics is purely complex, or C. Of...

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Dear Douglas Elias & Tao,

I enjoyed your essay and agree that "energy conservation and the tunneling picture of black hole radiation allow us to show how the original "bits" of black hole information encoded in the horizon were transformed into the "its" of the outgoing correlated Hawking photons, thus providing a potential all orders in ~ solution to the black hole information loss puzzle".

It makes perfect sense and was were I was going with my essay, but omitted some further detail which agrees with this. It would have taken another 10 pages in my case to add this aspect.

However, part of the process is apparent when we note that both sides of 0 of the Fibonacci sequence, 2 remains positive. My representation of how Hawking Radiation releases energy is the -1, such that information sits evenly at the horizon and is maintained there. But also in 1-dimensional string like structures that may become Hawking radiation.

Anyway please take a look, and I'm delighted to see such a fantastic essay doesn't contradict mine!

Best wishes and kind regards,

Antony

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I enjoyed your essay and agree that "energy conservation and the tunneling picture of black hole radiation allow us to show how the original "bits" of black hole information encoded in the horizon were transformed into the "its" of the outgoing correlated Hawking photons, thus providing a potential all orders in ~ solution to the black hole information loss puzzle".

It makes perfect sense and was were I was going with my essay, but omitted some further detail which agrees with this. It would have taken another 10 pages in my case to add this aspect.

However, part of the process is apparent when we note that both sides of 0 of the Fibonacci sequence, 2 remains positive. My representation of how Hawking Radiation releases energy is the -1, such that information sits evenly at the horizon and is maintained there. But also in 1-dimensional string like structures that may become Hawking radiation.

Anyway please take a look, and I'm delighted to see such a fantastic essay doesn't contradict mine!

Best wishes and kind regards,

Antony

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Hi Anthony,

Thanks for the kind words and we will try to have a look at your essay since your description sounds like there may be some connection. Since you mention "...1-dimensional string like structures that may become Hawking radiation." you may be interested in the work of Samir Mathur. He has a host of papers which treat the microscopic degrees of freedom near the horizon as string derived states which he calls "fuzzballs". I do not fully understand Mathur's approach but it is interesting a has the good feature of giving explicit microstates for the black hole.

Best regards,

Doug

Thanks for the kind words and we will try to have a look at your essay since your description sounds like there may be some connection. Since you mention "...1-dimensional string like structures that may become Hawking radiation." you may be interested in the work of Samir Mathur. He has a host of papers which treat the microscopic degrees of freedom near the horizon as string derived states which he calls "fuzzballs". I do not fully understand Mathur's approach but it is interesting a has the good feature of giving explicit microstates for the black hole.

Best regards,

Doug

Doug,

Many thanks for the suggested reading - I will take a look. No panic about my essay - there are so many to get through - it is an enjoyable, but massive task.

Best wishes,

Antony

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Many thanks for the suggested reading - I will take a look. No panic about my essay - there are so many to get through - it is an enjoyable, but massive task.

Best wishes,

Antony

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Dear Doug,

Very nice essay. Not that I dont have reservations about the black hole idea but your thoughts were well presented. So a good score from me.

On the Planck length, what likelihood that it is physically real? My essay is here you may take a look.

Best regards,

Akinbo

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Very nice essay. Not that I dont have reservations about the black hole idea but your thoughts were well presented. So a good score from me.

On the Planck length, what likelihood that it is physically real? My essay is here you may take a look.

Best regards,

Akinbo

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Hi Akinbo,

First sorry it took some time to reply -- I'm at my wife's family's home in a remote part of Thailand and my USB modem is slow. Anyway this is an interesting question about the reality of the Planck length. If one takes the physics we know at low energy as characterized by G, hbar and c (which can be combined to give a PLanck length, mass and time) one inevitably comes up with the Planck scale of 10^19 GeV, 10^{-44} sec or 10^{-35} m. But *experimentally* the only high energy scale we have directly tested is the weak scale which is at ~250-300 GeV. This was the jumping off point for the large extra dimension work of ADD which postulated that the true Planck scale coming from large extra dimensions was/could be much lower than 10^19 GeV. The last run of the LHC did not find evidence for these large extra dimensions.

I'll mention more in this regard on your discussion page.

Best,

Doug

First sorry it took some time to reply -- I'm at my wife's family's home in a remote part of Thailand and my USB modem is slow. Anyway this is an interesting question about the reality of the Planck length. If one takes the physics we know at low energy as characterized by G, hbar and c (which can be combined to give a PLanck length, mass and time) one inevitably comes up with the Planck scale of 10^19 GeV, 10^{-44} sec or 10^{-35} m. But *experimentally* the only high energy scale we have directly tested is the weak scale which is at ~250-300 GeV. This was the jumping off point for the large extra dimension work of ADD which postulated that the true Planck scale coming from large extra dimensions was/could be much lower than 10^19 GeV. The last run of the LHC did not find evidence for these large extra dimensions.

I'll mention more in this regard on your discussion page.

Best,

Doug

Hi Doug

Richard Feynman in his Nobel Acceptance Speech (http://www.nobelprize.org/nobel_prizes/physics/laureates/19

65/feynman-lecture.html)

said: “It always seems odd to me that the fundamental laws of physics, when discovered, can appear in so many different forms that are not apparently identical at first, but with a little mathematical fiddling you can show the relationship. And example of this is the Schrodinger equation and the Heisenberg formulation of quantum mechanics. I don’t know why that is – it remains a mystery, but it was something I learned from experience. There is always another way to say the same thing that doesn’t look at all like the way you said it before. I don’t know what the reason for this is. I think it is somehow a representation of the simplicity of nature.”

I too believe in the simplicity of nature, and I am glad that Richard Feynman, a Nobel-winning famous physicist, also believe in the same thing I do, but I had come to my belief long before I knew about that particular statement.

The belief that “Nature is simple” is however being expressed differently in my essay “Analogical Engine” linked to http://fqxi.org/community/forum/topic/1865 .

Specifically though, I said “Planck constant is the Mother of All Dualities” and I put it schematically as: wave-particle ~ quantum-classical ~ gene-protein ~ analogy- reasoning ~ linear-nonlinear ~ connected-notconnected ~ computable-notcomputable ~ mind-body ~ Bit-It ~ variation-selection ~ freedom-determinism … and so on.

Taken two at a time, it can be read as “what quantum is to classical” is similar to (~) “what wave is to particle.” You can choose any two from among the multitudes that can be found in our discourses.

I could have put Schrodinger wave ontology-Heisenberg particle ontology duality in the list had it comes to my mind!

Since “Nature is Analogical”, we are free to probe nature in so many different ways. And you have touched some corners of it.

Good Luck,

Than Tin

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Richard Feynman in his Nobel Acceptance Speech (http://www.nobelprize.org/nobel_prizes/physics/laureates/19

65/feynman-lecture.html)

said: “It always seems odd to me that the fundamental laws of physics, when discovered, can appear in so many different forms that are not apparently identical at first, but with a little mathematical fiddling you can show the relationship. And example of this is the Schrodinger equation and the Heisenberg formulation of quantum mechanics. I don’t know why that is – it remains a mystery, but it was something I learned from experience. There is always another way to say the same thing that doesn’t look at all like the way you said it before. I don’t know what the reason for this is. I think it is somehow a representation of the simplicity of nature.”

I too believe in the simplicity of nature, and I am glad that Richard Feynman, a Nobel-winning famous physicist, also believe in the same thing I do, but I had come to my belief long before I knew about that particular statement.

The belief that “Nature is simple” is however being expressed differently in my essay “Analogical Engine” linked to http://fqxi.org/community/forum/topic/1865 .

Specifically though, I said “Planck constant is the Mother of All Dualities” and I put it schematically as: wave-particle ~ quantum-classical ~ gene-protein ~ analogy- reasoning ~ linear-nonlinear ~ connected-notconnected ~ computable-notcomputable ~ mind-body ~ Bit-It ~ variation-selection ~ freedom-determinism … and so on.

Taken two at a time, it can be read as “what quantum is to classical” is similar to (~) “what wave is to particle.” You can choose any two from among the multitudes that can be found in our discourses.

I could have put Schrodinger wave ontology-Heisenberg particle ontology duality in the list had it comes to my mind!

Since “Nature is Analogical”, we are free to probe nature in so many different ways. And you have touched some corners of it.

Good Luck,

Than Tin

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Hi Than Tin,

A very nice and correct quote by Feynman. And after all one of his biggest achievements was to re-formulate quantum mechanics/quantum field theory in the path integral formalism i.e. he found another way to "look at" QM.

I very much like and appreciate analogies. One of my first works after grad school was to write down a Schwarzschild-like solution to the Yang-Mills equations using the analogy "GR and Yang-Mills theory are in some sense both non-Abelian gauge theories thus they should have similar solutions." The idea was that BHs provide confinement and thus these Schwarzschild-like solution of Yang-Mills theory might also provide the long sought after proper of confinement in QCD. The paper is "Exact Schwarzschild - like solution for Yang-Mills theories", D. Singleton (Virginia U.), Phys.Rev. D51 (1995) 5911-5914

e-Print: hep-th/9501052. It turns out a Soviet physicist had discovered similar solutions about 20 years earlier ("Exact Classical Solutions of Yang-Mills Sourceless Equations" by A.P. Protogenov, Phys.Lett. B67 (1977) 62-64)). Actually (half jokingly) this is one thing one finds out about theoretical physics -- everything was done earlier and better by some Soviet/Russian physicist. Again this is half jokingly.

About the use of analogies/dualities this is a very rich area of work. Almost every other paper these days has some reference to AdS/CFT which is a duality/analogy between Anti-de-Sitter space-time and conformal field theories. Thus many people appreciate dualities. I do not understand in which sense Planck's constant represents a duality so I'll go over and have a look at your essay. It may take a 1-2 days.

Best,

Doug

A very nice and correct quote by Feynman. And after all one of his biggest achievements was to re-formulate quantum mechanics/quantum field theory in the path integral formalism i.e. he found another way to "look at" QM.

I very much like and appreciate analogies. One of my first works after grad school was to write down a Schwarzschild-like solution to the Yang-Mills equations using the analogy "GR and Yang-Mills theory are in some sense both non-Abelian gauge theories thus they should have similar solutions." The idea was that BHs provide confinement and thus these Schwarzschild-like solution of Yang-Mills theory might also provide the long sought after proper of confinement in QCD. The paper is "Exact Schwarzschild - like solution for Yang-Mills theories", D. Singleton (Virginia U.), Phys.Rev. D51 (1995) 5911-5914

e-Print: hep-th/9501052. It turns out a Soviet physicist had discovered similar solutions about 20 years earlier ("Exact Classical Solutions of Yang-Mills Sourceless Equations" by A.P. Protogenov, Phys.Lett. B67 (1977) 62-64)). Actually (half jokingly) this is one thing one finds out about theoretical physics -- everything was done earlier and better by some Soviet/Russian physicist. Again this is half jokingly.

About the use of analogies/dualities this is a very rich area of work. Almost every other paper these days has some reference to AdS/CFT which is a duality/analogy between Anti-de-Sitter space-time and conformal field theories. Thus many people appreciate dualities. I do not understand in which sense Planck's constant represents a duality so I'll go over and have a look at your essay. It may take a 1-2 days.

Best,

Doug

Dear Doug, Elias and Tao,

I just finished reading your essay. Your starting point is the previous result of Banerjee and Majhi (arXiv:0808.3688, arXiv:0805.2220) for the temperature of a black hole (eqn 7 in your essay) where backreaction is taken into account. The crucial point is your ansatz (eqn 9) for the form of the coefficients in the expansion of the quantum action in a power series in hbar, from which you obtain the regularized form of the entropy (eqn 10).

If correct (and I haven't checked your calculations) then your result is of great importance. It would put on a firm foundation the notion that quantum mechanics and general relativity **can be** reconciled in a self-consistent framework. However, the work remains incomplete in that underlying your calculations is the implicit assumption that the continuous, commutative nature of spacetime remains the same at all scales. As Mittelman noted in previous comments, at some point non-commutative geometry should take over, leading to a black hole remnant as the end result of black hole evaporation. There should be some simple way to reconcile your calculations with those from the NC approach.

Your work also likely has some connections with Ansari's work on black hole radiation in the framework of LQG (hep-th/0607081, arXiv:0711.1879). Overall your essay is very readable (except for the "public" who have a hard time following any mathematical arguments) and arguably deserves a place in the top three - if not at the very top.

Cheers,

Deepak

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I just finished reading your essay. Your starting point is the previous result of Banerjee and Majhi (arXiv:0808.3688, arXiv:0805.2220) for the temperature of a black hole (eqn 7 in your essay) where backreaction is taken into account. The crucial point is your ansatz (eqn 9) for the form of the coefficients in the expansion of the quantum action in a power series in hbar, from which you obtain the regularized form of the entropy (eqn 10).

If correct (and I haven't checked your calculations) then your result is of great importance. It would put on a firm foundation the notion that quantum mechanics and general relativity **can be** reconciled in a self-consistent framework. However, the work remains incomplete in that underlying your calculations is the implicit assumption that the continuous, commutative nature of spacetime remains the same at all scales. As Mittelman noted in previous comments, at some point non-commutative geometry should take over, leading to a black hole remnant as the end result of black hole evaporation. There should be some simple way to reconcile your calculations with those from the NC approach.

Your work also likely has some connections with Ansari's work on black hole radiation in the framework of LQG (hep-th/0607081, arXiv:0711.1879). Overall your essay is very readable (except for the "public" who have a hard time following any mathematical arguments) and arguably deserves a place in the top three - if not at the very top.

Cheers,

Deepak

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Dear Douglas,

One single principle leads the Universe.

Every thing, every object, every phenomenon

is under the influence of this principle.

Nothing can exist if it is not born in the form of opposites.

I simply invite you to discover this in a few words,

but the main part is coming soon.

Thank you, and good luck!

I rated your essay accordingly to my appreciation.

Please visit My essay.

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One single principle leads the Universe.

Every thing, every object, every phenomenon

is under the influence of this principle.

Nothing can exist if it is not born in the form of opposites.

I simply invite you to discover this in a few words,

but the main part is coming soon.

Thank you, and good luck!

I rated your essay accordingly to my appreciation.

Please visit My essay.

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Hi Doug,

Thanks for the very helpful and interesting comments over on my page and for the suggested reading here. I really liked your essay and think you deserve to do well, so hope my rating helps.

Best wishes & kind regards,

Antony

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Thanks for the very helpful and interesting comments over on my page and for the suggested reading here. I really liked your essay and think you deserve to do well, so hope my rating helps.

Best wishes & kind regards,

Antony

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Hi Anthony,

No problem and if you read through the dimensional reduction stuff and want to discuss feel free although this is not my area but rather than my my colleague Jonas Mureika who got me interested in this and did explain to me some of the ins and outs of the idea. Actually Jonas also mentions that certain high energy cosmic ray events seem to be anomalously planar which they claim can be taken as evidence for dimensional reduction (if at high energy one has dimensional reduction then ones claims the decay products would only be distributed in a plane or line at higher energy. I'm not sure why kinematic effects would not do this, but Jonas and the authors of the paper he cites in the paper seem to think these events are odd and Jonas's explanation is dimensional reduction.

Thanks and Best of luck,

Doug

No problem and if you read through the dimensional reduction stuff and want to discuss feel free although this is not my area but rather than my my colleague Jonas Mureika who got me interested in this and did explain to me some of the ins and outs of the idea. Actually Jonas also mentions that certain high energy cosmic ray events seem to be anomalously planar which they claim can be taken as evidence for dimensional reduction (if at high energy one has dimensional reduction then ones claims the decay products would only be distributed in a plane or line at higher energy. I'm not sure why kinematic effects would not do this, but Jonas and the authors of the paper he cites in the paper seem to think these events are odd and Jonas's explanation is dimensional reduction.

Thanks and Best of luck,

Doug

Hello again Doug - sorry about the messages above too - going stir crazy trying to touch back with every thread I've commented on.

I certainly will look at this.

Thanks again and best wishes,

Antony

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I certainly will look at this.

Thanks again and best wishes,

Antony

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Dear Douglas Singleton, Elias Vagenas, & Tao Zhu :

I am an old physician and I don’t know nothing of mathematics and almost nothing of physics,

But maybe you would be interested in my essay over a subject which after the common...

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I am an old physician and I don’t know nothing of mathematics and almost nothing of physics,

But maybe you would be interested in my essay over a subject which after the common...

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Hi Hector,

Yes there are many ways to approach "time" -- Newtonian time, relativistic time, psychological time (how a person perceives time), etc. One of the most profound (in my opinion) people to deal with how people perceive and "store"/remember time was the French novelist Marcel Proust in his massive work "Remembrance of Things Past" which if one does a more direct rather than poetic translation is "Recovery of Lost Time".

A colleague of mine at CSU Fresno, Bob Levine, has done studies of how people perceive time. He has a thesis which in simplified form (hopefully not so simplified that I misrepresent) is that all/most of what people do is done in order to "buy" more time or make a given amount of time more pleasant, exciting, relaxing etc.

Anyway I will try to have a look at your essay.

Best regards,

Doug

Yes there are many ways to approach "time" -- Newtonian time, relativistic time, psychological time (how a person perceives time), etc. One of the most profound (in my opinion) people to deal with how people perceive and "store"/remember time was the French novelist Marcel Proust in his massive work "Remembrance of Things Past" which if one does a more direct rather than poetic translation is "Recovery of Lost Time".

A colleague of mine at CSU Fresno, Bob Levine, has done studies of how people perceive time. He has a thesis which in simplified form (hopefully not so simplified that I misrepresent) is that all/most of what people do is done in order to "buy" more time or make a given amount of time more pleasant, exciting, relaxing etc.

Anyway I will try to have a look at your essay.

Best regards,

Doug

Dear All

A standard-issue big city all-glass high-rise stands across the street from my usual bus stop. When I look up the high-rise facade, I can see the reflections of the near-by buildings and the white clouds from the sky above. Even when everything else looks pretty much the same, the reflections of the clouds are different, hour to hour and day to day.

After I boarded the bus,...

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A standard-issue big city all-glass high-rise stands across the street from my usual bus stop. When I look up the high-rise facade, I can see the reflections of the near-by buildings and the white clouds from the sky above. Even when everything else looks pretty much the same, the reflections of the clouds are different, hour to hour and day to day.

After I boarded the bus,...

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Having read so many insightful essays, I am probably not the only one to find that my views have crystallized, and that I can now move forward with growing confidence. I cannot exactly say who in the course of the competition was most inspiring - probably it was the continuous back and forth between so many of us. In this case, we should all be grateful to each other.

If I may, I'd like to...

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If I may, I'd like to...

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Dear Douglas Singleton, Elias Vagenas, & Tao Zhu :

I am an old physician and I don’t know nothing of mathematics and almost nothing of physics,

But maybe you would be interested in my essay over a subject which after the common...

view entire post

I am an old physician and I don’t know nothing of mathematics and almost nothing of physics,

But maybe you would be interested in my essay over a subject which after the common...

view entire post

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Dear All

Let me go one more round with Richard Feynman.

In the Character of Physical Law, he talked about the two-slit experiment like this “I will summarize, then, by saying that electrons arrive in lumps, like particles, but the probability of arrival of these lumps is determined as the intensity of waves would be. It is this sense that the electron behaves sometimes like a particle and sometimes like a wave. It behaves in two different ways at the same time.

Further on, he advises the readers “Do not keep saying to yourself, if you can possibly avoid it. ‘But how can it be like that?’ because you will get ‘down the drain’, into a blind alley from which nobody has yet escaped. Nobody knows how it can be like that.”

Did he says anything about Wheeler’s “It from Bit” other than what he said above?

Than Tin

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Let me go one more round with Richard Feynman.

In the Character of Physical Law, he talked about the two-slit experiment like this “I will summarize, then, by saying that electrons arrive in lumps, like particles, but the probability of arrival of these lumps is determined as the intensity of waves would be. It is this sense that the electron behaves sometimes like a particle and sometimes like a wave. It behaves in two different ways at the same time.

Further on, he advises the readers “Do not keep saying to yourself, if you can possibly avoid it. ‘But how can it be like that?’ because you will get ‘down the drain’, into a blind alley from which nobody has yet escaped. Nobody knows how it can be like that.”

Did he says anything about Wheeler’s “It from Bit” other than what he said above?

Than Tin

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Dears Colleagues,

I have just ended to re-read your intriguing Essay. As I previously told you, it is complementary to my one and, in my opinion, it further improves the elegant approach by Zhang, Cai, Zhan and You. Thanks also to have stressed the importance of the black hole information paradox within the theme of this Essay Competition. In fact, there is some guy who strangely claims that it does not address the core questions of this Essay Contest.

In any case, I enjoyed a lot in reading your Essay and I think that your results are of fundamental importance, not only for the core of this Essay Contest, but also for the whole theoretical physics. Thus, I am going to give you the top rate.

Cheers,

Ch.

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I have just ended to re-read your intriguing Essay. As I previously told you, it is complementary to my one and, in my opinion, it further improves the elegant approach by Zhang, Cai, Zhan and You. Thanks also to have stressed the importance of the black hole information paradox within the theme of this Essay Competition. In fact, there is some guy who strangely claims that it does not address the core questions of this Essay Contest.

In any case, I enjoyed a lot in reading your Essay and I think that your results are of fundamental importance, not only for the core of this Essay Contest, but also for the whole theoretical physics. Thus, I am going to give you the top rate.

Cheers,

Ch.

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Douglas, Elias, Tao - congratulations on an outstanding essay. I’m glad I managed to read it and get my rating in to help before the end of the contest.

I see a background of time assumption in your equations. I wonder what correlated photons of Hawking radiation would look like under the subtime interpretation?:

http://fqxi.org/data/forum-attachments/Borri

ll-TimeOne-V1.1a.pdf

(sorry if the fqxi web site splits this url up, I haven’t figured out a way to not make it do that).

Kind regards, Paul

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I see a background of time assumption in your equations. I wonder what correlated photons of Hawking radiation would look like under the subtime interpretation?:

http://fqxi.org/data/forum-attachments/Borri

ll-TimeOne-V1.1a.pdf

(sorry if the fqxi web site splits this url up, I haven’t figured out a way to not make it do that).

Kind regards, Paul

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Hi Paul,

A technical measure of the correlation, C, between photons of energy E_1 and E_2 emitted is given by the somewhat opaque expression

C(E_1+E_2 ; E_1, E_2) = ln(Gamma(E_1+E_2))-ln(Gamma(E_1)Gamma(E_2))

where Gamma(X) is the probability for a photon of energy X to tunneling through the horizon and appear as Hawking radiation. Essentially one is comparing emitting one photon with energy E_1+E_2 versus two separate photons E_2 and E_1. This is discussed in more detail in an early articl eon the subject by my co-author Elias. The exact reference is

"Hawking radiation as tunneling through the quantum horizon",

Michele Arzano, A.J.M. Medved, Elias C. Vagenas, JHEP 0509 (2005) 037

e-Print: hep-th/0505266

For a truly thermal spectrum the correlation function above is zero -- C=0 -- and one says the photons are uncorrelated. If C=/=0 there is some correlation.

Now the next question how woul done check this experimentally -- until recently I would have said "I don't know" and in fact I still have to say this, but now I know where to look. There is a recent PRD article

"Towards experimentally testing the paradox of black hole information loss"

Baocheng Zhang, Qing-yu Cai, Ming-sheng Zhan, Li You (Tsinghua U., Beijing). Phys.Rev. D87 (2013) 4, 044006

e-Print: arXiv:1302.1341 [gr-qc]

which from the title and also abstract (the paper is downloaded and sitting on my laptop but still needs reading) promises to give a way to test the correlations of emitted photons. I'm not sure if the proposal is in terms of analogy systems or what but it is interesting and might be address your question about what correlated photons would look like in the lab.

Best,

Doug

A technical measure of the correlation, C, between photons of energy E_1 and E_2 emitted is given by the somewhat opaque expression

C(E_1+E_2 ; E_1, E_2) = ln(Gamma(E_1+E_2))-ln(Gamma(E_1)Gamma(E_2))

where Gamma(X) is the probability for a photon of energy X to tunneling through the horizon and appear as Hawking radiation. Essentially one is comparing emitting one photon with energy E_1+E_2 versus two separate photons E_2 and E_1. This is discussed in more detail in an early articl eon the subject by my co-author Elias. The exact reference is

"Hawking radiation as tunneling through the quantum horizon",

Michele Arzano, A.J.M. Medved, Elias C. Vagenas, JHEP 0509 (2005) 037

e-Print: hep-th/0505266

For a truly thermal spectrum the correlation function above is zero -- C=0 -- and one says the photons are uncorrelated. If C=/=0 there is some correlation.

Now the next question how woul done check this experimentally -- until recently I would have said "I don't know" and in fact I still have to say this, but now I know where to look. There is a recent PRD article

"Towards experimentally testing the paradox of black hole information loss"

Baocheng Zhang, Qing-yu Cai, Ming-sheng Zhan, Li You (Tsinghua U., Beijing). Phys.Rev. D87 (2013) 4, 044006

e-Print: arXiv:1302.1341 [gr-qc]

which from the title and also abstract (the paper is downloaded and sitting on my laptop but still needs reading) promises to give a way to test the correlations of emitted photons. I'm not sure if the proposal is in terms of analogy systems or what but it is interesting and might be address your question about what correlated photons would look like in the lab.

Best,

Doug

Best of Luck for the Magnificent Eight !

I am throught the 180 essays, all rated. For me 2/3 of them were poor and other 1/6 curious. The rest (1/6) have I rated over 4/10.

You are among the authors of the top essays from my sight - alphabetically :

Corda, D'Ariano, Maguire, Rogozhin, Singleton, Sreenath, Vaid, Vishwakarma,

and I hope one of you will be the winner. (Please, don't rate my essay.)

David

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I am throught the 180 essays, all rated. For me 2/3 of them were poor and other 1/6 curious. The rest (1/6) have I rated over 4/10.

You are among the authors of the top essays from my sight - alphabetically :

Corda, D'Ariano, Maguire, Rogozhin, Singleton, Sreenath, Vaid, Vishwakarma,

and I hope one of you will be the winner. (Please, don't rate my essay.)

David

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Hi Doug,

I've just posted the remarks below at my own essay; but because they're so late, you may have stopped looking for them there, so I'm posting them here too.

I apologize for being so late in responding to your stimulating comments and thoughts; but hopefully, it's still better late than never! I had written a somewhat lengthy response earlier, but for some reason it wouldn't post; so, I'm going to try a condensed version to see if I can get anything posted.

Taking your "further unrelated question" first, my "N" is related to that of causal set theory, in which N is the total number of causet elements (CEs) in the four-volume constituted by the past light cone of a suitable observer. This number is way too large for each CE to be a degree of freedom (dof), and hence it is not of the same magnitude as the cosmological entropy S. However, on my account, the number of uncanceled volume-fluctuations of CEs, which is on the order of N's square root at any given time t, is roughly comparable to S. (I don't view these uncanceled CEs as actual dof's, but they are, in some ways, "something like" dof's, at least).

I agree with your point about the desirability of unifying inflation and late-time dark energy, and I find your idea of linking inflation with Hawking radiation very interesting. My own account of dark energy appeals to the nature of discrete spacetime and its fluctuations, which may also be relevant to the explanation of inflation, as your own work suggests. Of course, the particular aspects of discrete spacetime that are crucial to inflation may be (partly) different than those connected with dark energy; but even so, there's still a significant amount of "unification" here.

Good luck; hope you win a prize!

-Willard

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I've just posted the remarks below at my own essay; but because they're so late, you may have stopped looking for them there, so I'm posting them here too.

I apologize for being so late in responding to your stimulating comments and thoughts; but hopefully, it's still better late than never! I had written a somewhat lengthy response earlier, but for some reason it wouldn't post; so, I'm going to try a condensed version to see if I can get anything posted.

Taking your "further unrelated question" first, my "N" is related to that of causal set theory, in which N is the total number of causet elements (CEs) in the four-volume constituted by the past light cone of a suitable observer. This number is way too large for each CE to be a degree of freedom (dof), and hence it is not of the same magnitude as the cosmological entropy S. However, on my account, the number of uncanceled volume-fluctuations of CEs, which is on the order of N's square root at any given time t, is roughly comparable to S. (I don't view these uncanceled CEs as actual dof's, but they are, in some ways, "something like" dof's, at least).

I agree with your point about the desirability of unifying inflation and late-time dark energy, and I find your idea of linking inflation with Hawking radiation very interesting. My own account of dark energy appeals to the nature of discrete spacetime and its fluctuations, which may also be relevant to the explanation of inflation, as your own work suggests. Of course, the particular aspects of discrete spacetime that are crucial to inflation may be (partly) different than those connected with dark energy; but even so, there's still a significant amount of "unification" here.

Good luck; hope you win a prize!

-Willard

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Hi Willard,

I missed your original reply. I haven't figure out how to get announcements from other threads I'm involved with. Thanks for the well wishes.

With the new Planck results there is a lot of papers on inflation trying to match u with this data. Steinhardt and colleagues even published a recent PLB questioning the inflationary picture in light of the Planck data.

Best,

Doug

I missed your original reply. I haven't figure out how to get announcements from other threads I'm involved with. Thanks for the well wishes.

With the new Planck results there is a lot of papers on inflation trying to match u with this data. Steinhardt and colleagues even published a recent PLB questioning the inflationary picture in light of the Planck data.

Best,

Doug

Just a couple of addenda to the comments I recently posted: (i) Barrow & Shaw themselves seem to take an agnostic stance about the cause(s) of inflation, which makes some sense in view of the fact that it's simply unclear whether vacuum energy alone can explain, e.g., the origin of intergalactic magnetic fields, as well as being unclear whether such fields should be connected with inflation at all. It's possible, at least, that any inflationary theory that's able to explain a wide range of phenomena will be somewhat lacking with respect to simplicity and "unity."

(ii) I admit that the idea of nonlocal cancellation of spacetime fluctuations on a cosmological scale, and the related idea of the spacetime causal set of our universe as a kind of "particle," is rather extreme and crazy-sounding (and possibly just downright crazy!). Fwiw, I actually arrived at this idea by trying to address some issues connected with the account of dark energy in Ahmed, Dodelson, Greene and Sorkin's "Everpresent Lambda" paper. In particular, I was concerned about the criticism made by John Barrow ("A strong constraint on ever-present Lambda") regarding this account's conflict with cosmological observations. It may be that the ideas I'm advancing create too many new issues and problems; but I thought it was at least worth making an attempt to articulate and develop such ideas.

Anyway, thanks for your insightful comments, and for reading my essay. And, again, best of luck.

Willard

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(ii) I admit that the idea of nonlocal cancellation of spacetime fluctuations on a cosmological scale, and the related idea of the spacetime causal set of our universe as a kind of "particle," is rather extreme and crazy-sounding (and possibly just downright crazy!). Fwiw, I actually arrived at this idea by trying to address some issues connected with the account of dark energy in Ahmed, Dodelson, Greene and Sorkin's "Everpresent Lambda" paper. In particular, I was concerned about the criticism made by John Barrow ("A strong constraint on ever-present Lambda") regarding this account's conflict with cosmological observations. It may be that the ideas I'm advancing create too many new issues and problems; but I thought it was at least worth making an attempt to articulate and develop such ideas.

Anyway, thanks for your insightful comments, and for reading my essay. And, again, best of luck.

Willard

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Dears Colleagues,

Congrats for the Prize.

You, Jennifer Nielsen and Cistnel Stoica are the only positive news on the ridiculous and shameful "results" of this Essay Contest.

Cheers,

Ch.

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Congrats for the Prize.

You, Jennifer Nielsen and Cistnel Stoica are the only positive news on the ridiculous and shameful "results" of this Essay Contest.

Cheers,

Ch.

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