# A question of string v.s LQG derivation of BH entropy -

One question I have for LQG's derivation of BH entropy is this:

would any arbitrarily enclosed sphere in space, whether it's that of the surface of planet earth, or the moon, or the sun, not to mention BH's also have the same BH entropy since as I understand the calculation, it is merely counting spin network microstates on the surface of a black hole, but couldn't this also apply to any enclosed sphere? I understand that it is not volume extensive entropy but only surface entropy.

As for string theory, is the computation of bh entropy include the volume of the BH? Why hasn't it been able to reproduce Hawking results for ordinary astrophysical BH's if string theory is indeed a theory of QG.

## Answers and Replies

The derivation of black hole entropy I know in LQG is the one by Ashtekar, Baez, Corichi, Krasnov (http://arxiv.org/abs/gr-qc/9710007) [Broken]. Their derivation applies to "isolated horizons", which are essentially black holes in equilibrium. This constraint is applied at the classical level and causes some simplification that allows them to count the horizon states. I believe that most subsequent work is based on this approach.

In fact, I don't think any existing derivation of black hole entropy in LQG would apply to arbitrary horizons, let alone any enclosed sphere. But I would very much like to be proven wrong on this point.

Last edited by a moderator:
josh1
Why hasn't [string theory] been able to reproduce Hawking results for ordinary astrophysical BH's if string theory is indeed a theory of QG.

Youve got it backwards. Its string theory and not lqg that has been able to produce the black hole area-entropy relation. In fact, Its only string theory that has been successful in this regard. All lqg has been able to do is produce the proportionality between entropy and area, but not the correct constant of proportionality of 1/4, and it is the latter that presents the real challenge for any quantum theory of gravity. This is just one of many reasons why hardly anyone in the physics community takes lqg seriously. If you really want to understand whats going on in high energy theory, learn string theory and learn it from string theorists and not lqg people.

Youve got it backwards. Its string theory and not lqg that has been able to produce the black hole area-entropy relation. In fact, Its only string theory that has been successful in this regard. All lqg has been able to do is produce the proportionality between entropy and area, but not the correct constant of proportionality of 1/4, and it is the latter that presents the real challenge for any quantum theory of gravity. This is just one of many reasons why hardly anyone in the physics community takes lqg seriously. If you really want to understand whats going on in high energy theory, learn string theory and learn it from string theorists and not lqg people.

The original papers in string theory was only in the special case of extremel BH, which do not exist in reality. The results are protected by SUSY. The results have been extended to near-extremel BH, but I am not aware that they have been generalized to all BH's.

josh1
The original papers in string theory was only in the special case of extremel BH, which do not exist in reality. The results are protected by SUSY. The results have been extended to near-extremel BH, but I am not aware that they have been generalized to all BH's.

You've failed to appreciate the significance of this, which is that this result must be produced for any black hole solution in string theory or any other theory of quantum gravity, whether or not the solution is that of an ordinary astrophysical black hole. If string theory hadnt been able to produce the correct result, even if it is for unrealistic black holes, it would be inconsistent with the most important clue to quantum gravity we have. Like all other tests of consistency string theory has ever been subjected to, it passed this test with flying colors. The same cannot be said of any other competing theory of quantum gravity including lqg.

Demystifier
Gold Member
LQG predicts that entropy of ANY surface (not necessarily a surface of an isolated horizon) is proportional to the surface. The constant of proportionality is always the same, and can be adjusted so that it agrees with that in semiclassical gravity. However, LQG does not provide any new explanation of why entropy of a black hole should coincide with that of its surface.

String theory really explains why entropy of the black hole is proportional to its surface. Quantitatively, the constant of proportionality can be calculated explicitly only in some special (unrealistic) cases, but there are qualitative arguments for proportionality with the surface even in the general case.

Demystifier, could you give a url for the calculation of black hole entropy that does not depend on the isolated horizon constraints? The papers I have read on the arxiv all rely on the isolated horizon constraint, including those from this year.

marcus
Gold Member
Dearly Missed
LQG predicts that entropy of ANY surface (not necessarily a surface of an isolated horizon) is proportional to the surface. The constant of proportionality is always the same, and can be adjusted so that it agrees with that in semiclassical gravity. However, LQG does not provide any new explanation of why entropy of a black hole should coincide with that of its surface.

String theory really explains why entropy of the black hole is proportional to its surface. Quantitatively, the constant of proportionality can be calculated explicitly only in some special (unrealistic) cases, but there are qualitative arguments for proportionality with the surface even in the general case.

I had the same question as W.D. So I will just repeat for emphasis

Demystifier, could you give a url for the calculation of black hole entropy that does not depend on the isolated horizon constraints? The papers I have read on the arxiv all rely on the isolated horizon constraint, including those from this year.

Corichi has a recent series of papers on LQG BH entropy including one this year. There have been developments. Corichi also has a monograph in preparation on that topic. Nothing I've seen over the years corresponds to Demy.'s assertion as far as I can remember. So I think Demy. owes us a url. (would be interesting to have :-))

Last edited:
josh1
LQG predicts that entropy of ANY surface (not necessarily a surface of an isolated horizon) is proportional to the surface.

But this demonstration is a near-trivial result of the way that spin networks include degrees of freedom called links which by definition measure area.

A rather nontrivial example is the way that the entropy of highly excited strings is area-extensive. This is the string-black hole correspondence principle due to susskind. This is another one of many important clues string theory is giving us about black hole holography.

The constant of proportionality is always the same, and can be adjusted so that it agrees with that in semiclassical gravity.

The whole game is to produce the correct factor of 1/4. If a theory cant do so, its wrong. What lqg needs is new physical input that fixes the immirzi parameter to produce the correct value. Not a single attempt has met with anything but utter failure.

The thing is that - and this is very important - lqg is a very simple theory and seems to offer no additional theoretical elbow room where such new physics could lurk. Precisely the opposite is the case with string theory.

However, LQG does not provide any new explanation of why entropy of a black hole should coincide with that of its surface.

It doesnt provide any explanation, new or old.

String theory really explains why entropy of the black hole is proportional to its surface.

Well, it offers clues anyway.

Quantitatively, the constant of proportionality can be calculated explicitly only in some special (unrealistic) cases, but there are qualitative arguments for proportionality with the surface even in the general case.

The idea is to embed black hole physics in string theory by invoking the geometry of d-brane solutions with event horizons and then to use our knowledge of the spectrum of physical states of d-branes to explcitly count the number of states and compare the result to the area of the event horizon. However, this is not really an explanation of black hole holography, but it is no doubt an important clue.

Demystifier
Gold Member
William Donnelly and Marcus, see the derivation of black-hole entropy in the book "Quantum Gravity" of Rovelli. This book was also available online on the Rovelli's home page, but I do not know if this is still the case. I have not been reading the corresponding original papers of Rovelli, but I pressume that the same derivation can be seen there. The idea is very simple. The entropy of a surface is essentially the number of the degrees of freedom associated with the surface, so it is hardly surprising that the result is proportional with the surface (at least when the surface is much larger than the Planck surface).

Last edited:
I have been reading the derivation of black hole entropy in string theory and there are a few aspects that I don´t see clear.

For example, the problem of uniquines. You can, as has explained josh, make somethin which resembles a black hole using a few kinds of d-branes and F-strings, ok. And you can count the number of states using CFT, o.k.

But, how can you be sure that there are not diferent ways to make the same black hole with another kind of branes/strings configuratiosn. If so the number of microstates would increase (Maybe there is some uniquines theorem anyway and simply I don´t know it).

marcus
Gold Member
Dearly Missed
William Donnelly and Marcus, see the derivation of black-hole entropy in the book "Quantum Gravity" of Rovelli. This book was also available online on the Rovelli's home page...

Demy, I have the book in hardcopy and back in 2003 used to read it online. Great book! but I would not consult it for BH entropy, there has been quite a bit written since Jan 2004 when Rovelli's text went to the publisher.

I could be wrong but I think if you check out 2006 and 2007 papers by Corichi (and references therein) you will be doing yourself a favor (if you are at all interested in LQG and BH entropy.)

It is certainly NOT the case that in LQG the entropy is simply proportional to the eventhorizon area! and the relation to area seems to depend somewhat on the size of the BH.

Corichi is preparing a review article on this. A lot has been written. The review article is not yet available but you should check out his recent papers in arxiv. Hope this helps

josh1
It is certainly NOT the case that in LQG the entropy is simply proportional to the eventhorizon area!

As a principled person who always argues honestly based on the facts as I understand them, I would only have included the exclamation mark if I believed I understood precisely what lqg does say about the area-entropy relation and was prepared to make my case in detail. You should note that one of the posting guidelines in PF requires members to back up their claims when asked and ignoring such requests when they are reasonable - especially when they do so on a regular basis - is a definite no no here. So please explain this strangely vague statement and enlighten us.

josh1
...how can you be sure that there are not different ways to make the same black hole with another kind of branes/strings configuration? If so the number of microstates would increase...

Yes, the entropy of a particular black hole may always be viewed as counting all of the ways it couldve been made. But by a black hole we mean a particular solution of the theory. If a black hole involves different ingredients, the solution and hence the black hole - including the way its degrees of freedom are counted - will be different.

Yes, the entropy of a particular black hole may always be viewed as counting all of the ways it couldve been made.

Do you have a reference for this argument? Specifically, why should the logarithm of the number of ways a black hole could have formed be the same as the entropy of the thermal state as seen by an observer outside the black hole?

I understand the statistical mechanical definition of entropy, I just don't see how "ways a black hole could be made" are the same thing as "black hole microstates".

josh1
Do you have a reference for this argument? Specifically, why should the logarithm of the number of ways a black hole could have formed be the same as the entropy of the thermal state as seen by an observer outside the black hole? I understand the statistical mechanical definition of entropy, I just don't see how "ways a black hole could be made" are the same thing as "black hole microstates".

Ive come across this way of saying things a number of times, but only in the context of black holes, but I dont have a reference. But its not meant in a literal way. As you know, a microstate is a collection of excitations that represent one possible configuration of a system. We then think in terms of "building" the system out of the excitations of a particular microstate with the complete collection of all possible micrsostate then representing in this sense the "different ways of building" the system.

Last edited by a moderator:
Demystifier
Gold Member
Demy, I have the book in hardcopy and back in 2003 used to read it online. Great book! but I would not consult it for BH entropy, there has been quite a bit written since Jan 2004 when Rovelli's text went to the publisher.
Marcus, could you specify what exactly is wrong with the simple analysis presented by Rovelli in his book? The fact that later other people made a more complicated analysis does not convince me that the earlier simpler analysis is wrong. Can you explain, in simple terms, in what sense the later results represent an improvement?

In my opinion, scientific papers are often too polite, in the sense that they often hesitate to say explicitly what are the drawbacks of earlier papers on that subject. For that reason you can understand the point of a new paper only if you already studied all earlier papers on that subject.

marcus
Gold Member
Dearly Missed
In my opinion, scientific papers are often too polite, in the sense that they often hesitate to say explicitly what are the drawbacks of earlier papers on that subject. For that reason you can understand the point of a new paper only if you already studied all earlier papers on that subject.

Yes! But maybe this politeness has a good reason.
Remember that this is an area of expertise of Corichi---maybe he has the best judgment. He was one of the authors of the original 1997 paper Asht. Baez Corichi Krasnov, and of those four he is the only one who is currently working on it.

Moreover he is writing a review of the whole LQG/BH entropy subject.

Corichi has chosen not to directly attack earlier work and point out its limitations in an obvious way. I would understand if you think it "too polite". But maybe he is behaving with correct amount of politeness.

This business has been going on for 10 years. There have been gradual refinements. There is still room for people to disagree.

Where I personally stand (as an observer, not researcher) is not so important but I will tell you frankly that at first I was dubious of Corichi's work and gradually I am finding it more and more convincing. Partly this is because of the work by Ghosh and Mitra around the same time (2005) which seems to corroborate Corichi. Partly it is the fact that he is using computer models and brute force. I have a lot of respect for brute force number crunching as an alternative to theoretical/analytical----both are fallible so one should try to do both as a check on each other.

The Indians, Ghosh and Mitra, are very analytical, so Corichi numerical approach complements theirs.

Corichi is a prominent organizer of the Loops '07 international LQG conference this summer in Morelia. I am sure that this summer's conference will have a lot about BH!

Probably the earlier work of ten years back should be honored and quietly improved on

josh1
Hi marcus,

You posted the following:

Demy, I have the book in hardcopy and back in 2003 used to read it online. Great book! but I would not consult it for BH entropy, there has been quite a bit written since Jan 2004 when Rovelli's text went to the publisher.

I could be wrong but I think if you check out 2006 and 2007 papers by Corichi (and references therein) you will be doing yourself a favor (if you are at all interested in LQG and BH entropy.)

It is certainly NOT the case that in LQG the entropy is simply proportional to the eventhorizon area! and the relation to area seems to depend somewhat on the size of the BH.

I think that the site guidelines (not to mention considerations of simple respect of other members) obligate you to respond in some meaningful way to the following question posted by Demystifier:

Marcus, could you specify what exactly is wrong with the simple analysis presented by Rovelli in his book?

Last edited by a moderator:
I can't speak for Marcus, but I find the Krasnov/Rovelli approach to black hole entropy a little circular. They start by assuming the area is the only relevant macroscopic quantity for the black hole and so count the number of states for a fixed area. This quantity must be a function of the area by definition.

josh1
I find the Krasnov/Rovelli approach to black hole entropy a little circular. They start by assuming the area is the only relevant macroscopic quantity for the black hole and so count the number of states for a fixed area. This quantity must be a function of the area by definition.

It could turn out that the correct degrees of freedom of black holes really are explicitly area-extensive. But then being the correct theory, it would produce the correct factor of 1/4. The failure of lqg to achieve this emphasizes that producing the correct factor of 1/4 can only follow from a theory that identifies the correct degrees of freedom of a black hole and that these degrees of freedom are certainly not the links of spin networks.

Failure seems a bit strong. Admitted, loop quantum gravity has its problems, but I don't think the presence of a single free parameter is one of them. I don't know of any reason to believe the Immirzi parameter should not take on the right value to reproduce the Bekenstein-Hawking entropy.

josh1
Failure seems a bit strong.

Maybe "failure-ish"?:tongue2:

I don't know of any reason to believe the Immirzi parameter should not take on the right value to reproduce the Bekenstein-Hawking entropy.

Some piece of new physics is needed to fix the immirzi parameter, and I cant deny that its possible that this physics lives in some undiscovered part of lqg. Its just that lqg seems too simple for there to be anywhere such new principles could hide.

You know, the euclidean quantum gravity program lost its popularity because it predicted a gravitational phenomenon called wormholes which occurred below the planck scale. If correct, euclidean quantum gravity would have to produce a constant that tells us at what scale below the planck scale wormholes do occur, but it couldnt. Now euclidean quantum gravity is used only because it seems to capture some properties that could very well be relevant to quantum gravity or quantum cosmology.

Another example is the consistent histories approach to quantum cosmology. It aimed to show how classical physics emerges from quantum mechanics. To do so it introduced the notion of quasi-classicity as an interpolater between quantum mechanical behaviour and purely classical behaviour. The problem was there was nothing in this theory that could be used to identify under what conditions quasi-classical behaviour appears.

The same is true of lqg. It is seductive, but without an argument to establish how it connects with ordinary gravitational physics, it cannot succeed, and a rather large majority of physicists in this field are quite certain it never will.

Last edited by a moderator:
Some piece of new physics is needed to fix the immirzi parameter
I'm listening tell me more.
jal

marcus
Gold Member
Dearly Missed
one trouble with outsiders discussing LQG is that the field doesn't hold still---various "LQG-like" off-shoots develop.

Some currently-researched LQG-like approaches don't have an Immirzi parameter. Others make it have physical meaning so that it could in principle be determined experimentally in lab (not from Black Holes! from other stuff I gather). One doesnt know which of the approaches are correct

it would be a big job to say in an accurate and balanced way what the situation is

I cant take the time to look up the arxiv links but I will give author names and you or anyone interested can use the search engine at arxiv to find the papers if you want.

1. Rovelli and Freidel have several papers together or separately where the Immirzi has a physical meaning, tells something about nature that is in principal measurable in laboratory---this doesnt have to do with black holes, it has to do with particle physics, keyword torsion comes up, have to see the articles

2. a defining event for the LQG community is the QGQG school now in progress. The person who is teaching the LQG series of lectures is Thomas Thiemann. He has a new version of LQG called AQG and I am not sure that it has an Immirzi parameter. Maybe we can ask some PF people who are attending the school: Francesca is one and F-H is another.

3. Sergei Alexandrov and Etera Livine have a variant of LQG with no Immirzi, they call it "covariant LQG". Recently Alexandrov Buffenoir Roche came out with a new version, just in the past 6 months---no Immirzi.

4. a lot of the LQG community's research is spinfoam based or GFT (group field theory)----so there would be no Immirzi AFAIK. I could be wrong.
At the QGQG school, the people teaching the spinfoam approach are Laurent Freidel and Etera Livine.

Last edited:
marcus
Gold Member
Dearly Missed
to a lot of people it is the LQG community which they want to talk about because it is the main rival to string research

but then they get talking about vintage 1995 canonical LQG or whatever, and it misses the point, because the community is already 10 years down the pike and moving fast.

So a lot of confusion and misleading talk arises, which i will try to straighten out some.

An important issue is CONTACT WITH ORDINARY GRAVITY. Everybody should have noticed when Rovelli and team derived gravitons in a spinfoam context in 2005. After one of their papers in August 2006 he said what I put in sig

essentially we got Newton gravity starting from scratch

This was published in major journal articles and also in popular media, so I would guess anybody who follows QG knows about it.

But there have been a lot of other cases of this by other LQG people in the past 2 years---in several different special cases and contexts: different from Rovelli's spinfoam context. More an more often when i scan the literature i am seeing the phrase correct classical limit turning up. Always so far in a special case, not yet a totally general solution to the problem.

It is old news that you get the correct classical limit in LQC. But it came up last year in a particular case Bojowald was working on where it wasnt even LQC it was a new version he'd developed of LQG! I also saw it in a recent paper of Magueijo. And the correct classical limit came up in Thiemann's talk at KITP if i remember correctly. It seemed to interest folks so they gave him a second hour.

So it seems that you get a bunch of critics of the LQG community who stonewall it and say "LQG has nothing to do with gravity!"
But they are talking about something that doesnt have much to do with the actual stuff the LQG community is working on! The approaches the community actually works on DO seem to be reaching out and contacting classical gravity here and there these days.

A lot has happened in the past 2 years. A lot of new faces besides the original pioneers (Rovelli, Smolin, Ashtekar). A lot of variations, new approaches. so it is hard to get it right if you have a static picture.

It may be that vintage 1995 LQG has nothing to do with gravity---I dont know I watch progress on a lot of different fronts. But the LQG community's job is to adapt and evolve the stuff they work with. Which they have over the past 10 years almost beyond recognition. And the past 2 years progress seems to me to have been especially rapid.

this year we SHOULD be seeing a new book from Cambridge University Press called Approaches to Quantum Gravity, towards a new understanding of space time and matter which includes work by 20 or so people. It is edited by Dan Oriti (in Renate Loll's group at Utrecht) and it should give us some better perspective on what the LQG community is doing
(the mention of matter in the title is important---LQG people are very much into discovering matter and QFT Feynman diagrams in their spinfoams these days----matter turning out to be a facet of geometry----look up Laurent Freidel papers he has many about this starting in 2005)

so anyway, Oriti's book will help clarify and define the field----and gradual increasing contact with classic gravity as one aspect.
And we will hear more from the Loops '07 conference in Morelia, this summer. And hopefully we'll hear from some of the people now attending that QGQG school.

Last edited:
Demystifier
Gold Member
I can't speak for Marcus, but I find the Krasnov/Rovelli approach to black hole entropy a little circular. They start by assuming the area is the only relevant macroscopic quantity for the black hole and so count the number of states for a fixed area. This quantity must be a function of the area by definition.
I agree. But can you explain in simple terms how other LQG approaches (presumably with isolated horizons) avoid this circularity?

josh1
one trouble with outsiders discussing LQG is that the field doesn't hold still

So now your holding yourself out as an lqg insider and the rest of us as outsiders? Boy youve got alot of nerve given the lack of specifics in your posts.

...various "LQG-like" off-shoots develop. Some ...approaches don't have an Immirzi parameter.

The problem isnt the immirzi parameter. The problem is the failure to recover general relativity. We can express this difficulty in terms of the immirzi parameter in those variants of lqg that have one, or in different terms in those that dont.

On the other hand, string theory reduces in the low energy limit to various classical supergravities from which one can write down a classical metric involving event horizons of various types and then analyze these solutions, including in some cases calculating the entropy which yields the correct value in all cases considered so far.

More an more often when i scan the literature i am seeing the phrase correct classical limit turning up.

And from this were suppose to learn what exactly?

Others make it have physical meaning so that it could in principle be determined experimentally in lab (not from Black Holes! from other stuff I gather).

This has already been discussed since it is just another way of trying to identify new physics in lqg that would fix the value of immirzi to the correct one. None of these approaches have succeeded either.

The fact that so many different approaches to lqg are available and none of them have succeeded is screaming at us that these ideas are fundamentally wrong.

By contrast there are no variants of string theory, just as one would expect from a correct theory.

Last edited by a moderator:
josh1
Hi jal,

You asked me to expand on my remark that some piece of new physics is needed to fix the immirzi parameter. Unfortunately, in the absence of any real progress on this issue, there isnt a hell of a lot more to say. Maybe you had something more specific in mind?

Marcus}
Some currently-researched LQG-like approaches don't have an Immirzi parameter. Others make it have physical meaning so that it could in principle be determined experimentally in lab (not from Black Holes! from other stuff I gather). One doesnt know which of the approaches are correct
Thanks. I am looking in the paths that you have mentioned.
josh1
Maybe you had something more specific in mind?
Well!!!.... I am gathering in my blog what I think would end being citations for a new approach.
Do we need to open up a new thread to discuss another paper?

http://arxiv.org/PS_cache/gr-qc/pdf/0703/0703135.pdf [Broken]
Loop quantization of spherically symmetric midi-superspaces
Miguel Campiglia1, Rodolfo Gambini1, Jorge Pullin2
27 March 2007
We quantize the exterior of spherically symmetric vacuum space-times using a midi-superspace reduction within the Ashtekar new variables. Through a partial gauge fixing we eliminate the diffeomorphism constraint and are left with a Hamiltonian constraint that is first class. We complete the quantization in the loop representation. We also use the model to discuss the issues that will arise in more general contexts in the “uniform discretization” approach to the dynamics.

p.12 If one adopts the point of view commonly used in loop quantum cosmology, that the quantum of distance should have a minimum value, then one would not expect to take the limit ρ going to zero, but to keep the parameters at a minimum value. In such a case one could expect to eliminate the singularity. This is plausible since then the triads would likely not go to zero.
I would think that this can be achieved by applying/obeying the Quantum Minimum Length Structure (QMLS)
Since a 2d surface is being analysed... would it be too hard to use these results to get a better understanding of a membrane?
jal

Last edited by a moderator:
I agree. But can you explain in simple terms how other LQG approaches (presumably with isolated horizons) avoid this circularity?

Sure. The isolated horizon approach starts by quantizing in the same manner as ordinary lqg, but the manifold has a boundary. Then they apply the isolated horizon constraint - this is a classical constraint on the boundary that forces it to be a horizon. Then they count the number of states of the boundary Hilbert space. In essence this space should contain all the observables of the horizon, not just the area.

The other advantage of the isolated horizon calculation is that it seems to match the value derived from the quasinormal modes of a Schwarzschild black hole, under certain assumptions about the edges intersecting the horizon.

Of course the down side is that they have applied an extra classical constraint that shouldn't be there. It really restricts the applicability of their derivation.

Regarding the ways to fix the Immirzi parameter, those I know of are:
- Black hole entropy
- Quasinormal modes of Schwarzschild black holes (Dreyer)
- Fermion scattering (Perez and Rovelli)
The entropy has been studied a lot, the quasinormal modes less so, and the fermion scattering very little as far as I know. I don't think this constitutes "no progress".

josh1
Regarding the ways to fix the Immirzi parameter, those I know of are:
- Black hole entropy
- Quasinormal modes of Schwarzschild black holes (Dreyer)
- Fermion scattering (Perez and Rovelli)
The entropy has been studied a lot, the quasinormal modes less so, and the fermion scattering very little as far as I know. I don't think this constitutes "no progress".

Why? Whats your definition of progress? Failing in many ways rather than only one?

In your words "some piece of new physics is needed to fix the immirzi parameter". This is exactly what the papers I referenced provide. I didn't want those in this thread to misinterpret your post as saying that nobody is thinking about or trying to solve this problem.

josh1
In your words "some piece of new physics is needed to fix the immirzi parameter". This is exactly what the papers I referenced provide. I didn't want those in this thread to misinterpret your post as saying that nobody is thinking about or trying to solve this problem.

Fair enough, but are they really any closer now to a solution then theyve been in the recent past?

Last edited by a moderator: