# Thread: Entropy and Discreetness of Space-Time

1. The event horizon of a black hole has a finite surface area, i.e. it is a more or less well defined area of space-time. Black hole thermodynamics now tell us that we can associate entropy with that surface area, and by definition entropy is in turn related to the number of micro-states of a system. So, the event horizon is not a material object but rather a region of space-time, and we can associate a well-defined entropy with such a region, which, in turn, allows us to compute a number associated to micro-states, and that number will be finite. Would this not automatically imply that the space-time region of the event horizon must be fundamentally discreet since it has a finite number of micro-states ? In the absence of anything but space-time ( the event horizon is not a tangible surface as such ) the entropy cannot be associated with anything but micro-states of space-time itself, so it ought to be impossible for such a space-time to be a continuum because the entropy would then be infinite.

Furthermore, since the entropy depends only on the area of the horizon and not on any other property of the black hole, we should in principle be able to choose any closed surface / connected region of space-time, regardless of whether that is an event horizon or not, and associate entropy to it, and that entropy should correspond to a finite number of micro-states. That would mean that space-time in general ought to be discreet on a microscopic level, and not a continuum.

What do you think ?

2. This reminded me of the holographic principle and the Bekenstein bound. But I don't really have a good understanding of either...

I am not aware that anyone else has explicitly used this as an argument for discreteness of space-time. Although it may be implicit in some of the analysis.

3. Well, the argument is almost too simple to be true, so I was hoping someone can point out where and why it is wrong...

4. Err, well, I knew that as soon as I stepped into this room I should out of my depth!

But I do recall from Phys Chem courses that the distribution of micro states for a classical mechanical system is continuous - or do I?

But since entropy has something to do with energy, which in quantum mechanics is quantized, essentially by definition, then your claim that "the distribution of micro states is discrete at the event horizon (i.e. boundary) of a black hole is discrete" , and since further that black holes have something to do with gravity, this seems to imply you have a quantum theory of gravity

I am sure I am wrong - I certainly hope so

Suppose an arbitrary n- manifold which has no boundary

Question: Does there exist a sub-manifold that has a boundary i.e. does there exist a sub-mainifold such that where the manifold is an n-1 manifold i.e. an "embedded" surface in

Question: Since by definition, if and is of class , we must have a diffeomorphism i.e. an invertible mapping

Question: In the case of space-time, where we may assume that all diffeomorphisms respect the space-time metric, does the theory of black holes, whether bounded or not, likewise respect the metric? What little I know about black holes suggests the answer NO.

Once again, I am talking out of my ear (or worse)

5. But I do recall from Phys Chem courses that the distribution of micro states for a classical mechanical system is continuous - or do I?
Yes, the distribution is continuous, but the total number of micro states is finite and well defined. Recall that the entropy in general is connected to the number of corresponding microscopic states via

and on the other hand a black hole's entropy is

This would give us a well defined number of micro states for a given region of space-time A, meaning that space-time would have to be discreet on some level since otherwise no micro-states could exist. After all, those micro states must exist somewhere, so even otherwise empty space-time would need to have a microscopic structure of some sort, or else the notion of entropy on such a space-time region would have no meaning.

As to what this says about gravity - I don't know. I haven't really thought this through yet, it was just something that occurred to me when I was reading up about black hole thermodynamics. Of course if one considers A to be the area of an event horizon I suppose it would connect back to gravity eventually. All I am saying right now is that the very fact that one can associate a region of space-time with entropy could conceivably imply that such a region must have a discreet microstructure. It is just an idea, so I shall be happy to be shown wrong.

topic is at 30,000 ft...
(IOW this is just a bump to subscribe to the thread..)

7. Originally Posted by Markus Hanke
Yes, the distribution is continuous, but the total number of micro states is finite and well defined. Recall that the entropy in general is connected to the number of corresponding microscopic states via

and on the other hand a black hole's entropy is

This would give us a well defined number of micro states for a given region of space-time A, meaning that space-time would have to be discreet on some level since otherwise no micro-states could exist. After all, those micro states must exist somewhere, so even otherwise empty space-time would need to have a microscopic structure of some sort, or else the notion of entropy on such a space-time region would have no meaning.

As to what this says about gravity - I don't know. I haven't really thought this through yet, it was just something that occurred to me when I was reading up about black hole thermodynamics. Of course if one considers A to be the area of an event horizon I suppose it would connect back to gravity eventually. All I am saying right now is that the very fact that one can associate a region of space-time with entropy could conceivably imply that such a region must have a discreet microstructure. It is just an idea, so I shall be happy to be shown wrong.

I think you might be mixing things up. A state is well, as such a state. Not necessarily a location in space time. It is a quantum state. Just as there are finite states in any given surface. The definition of S use present is one that stems from statistical physics. Now a black hole, and particulary it's even't horizon has a huge degree of symmetry. After all, light is captured and this makes the microstates rather confined. For a discrete amount of states to be present one doesn't need a discrete amount of space (as our particles do not live as point particles) But I'll ask someone anyway.

8. Originally Posted by Guitarist
Suppose an arbitrary n- manifold which has no boundary
I was going to ask, "Does possess all the requisite properties to meet the definition of being a manifold?"
But I think the answer is "Yes"?
Originally Posted by Markus Hanke
What do you think ?
I like the idea of a discreet spacetime. However, discreet intervals do exist in without implying the discreetness of .
But I do like the idea of discreet spacetime and would readily bet \$5 on it.

By the way, does anybody know if a black hole has a de Broglie wavelength?

9. Originally Posted by Kerling
I think you might be mixing things up. A state is well, as such a state. Not necessarily a location in space time. It is a quantum state. Just as there are finite states in any given surface. The definition of S use present is one that stems from statistical physics. Now a black hole, and particulary it's even't horizon has a huge degree of symmetry. After all, light is captured and this makes the microstates rather confined. For a discrete amount of states to be present one doesn't need a discrete amount of space (as our particles do not live as point particles) But I'll ask someone anyway.
True, but the event horizon is just a region of space-time; so, what are the quantum states of space-time ? Must there not be a micro-structure of some kind to enable us to to define the notion of "states" ?

10. Originally Posted by GiantEvil
By the way, does anybody know if a black hole has a de Broglie wavelength?
A black hole is a body with a well defined mass, so I suppose one could associate a matter wave with it, which has the frequency :

11. Originally Posted by Markus Hanke
True, but the event horizon is just a region of space-time; so, what are the quantum states of space-time ? Must there not be a micro-structure of some kind to enable us to to define the notion of "states" ?
I think we have hit the whole thermodynamics / information theory entropy problem - you have to be really smart to pick the right set of states to use in the entropy calculation to match with the entropy you would calculate from statistical mechanics and I do not believe we have any idea which states the entropy of a black hole and its event horizon correspond to the thermodynamic entropy of the black hole.

Scholarpedia has a long list of possible ideas : Bekenstein-Hawking entropy - Scholarpedia

12. Originally Posted by Markus Hanke
True, but the event horizon is just a region of space-time; so, what are the quantum states of space-time ? Must there not be a micro-structure of some kind to enable us to to define the notion of "states" ?
Well I'd say no. Microstates are unique states. And unique states have a lot to do, with the impossibilities of particles to have the same state. (hence they must be unique).
Thing is, your even horizon, is mostly empty. There isn't very much inside it. And it is aboundary. It is like the ultimate open-boundary conditions, whereas for the persepective of the black hole it is the ultimate closed boundary condition. So, looking at it like that, the actual even horizon is nothing but some sort of shell. A mathematical entity almost which appears black due to the simple lack of its observability. (directly) So first of, the event horizon would always be 2Dimensional, by definition at least. (hence the holographic principle being derived from it, I asked a friend for some help here)
And since everything that ends up in the event horizon falls through, and everything at even this strong spatial dillitation has finite size. The even horizon is essentially perfectly empty. So the only state which could be there, is mearly a description of its position. (and perhaps electromagnetic field lines, not sure about the flux)
This puts us in the deterministic limit of quantum mechanics (2D, here Bell inequalities are in fact not violated) And then further quantization is no longer required in order to uphold the laws of physics. This leaves the question of whether or not space-time is quantized, but for the special case of an even horizon. This wouldn't be the required case. And from a Copenhagen perspective the entire black hole is a single giant super-state. As it cannot be observed. And from that simple assertion, we know that an entangled super-state can hold more information then it's seperate parts. It just can't realise it. I'm not saying that there couldn't be a micro-space-time structure. I just don't think we need it.

13. Originally Posted by river_rat
I think we have hit the whole thermodynamics / information theory entropy problem - you have to be really smart to pick the right set of states to use in the entropy calculation to match with the entropy you would calculate from statistical mechanics and I do not believe we have any idea which states the entropy of a black hole and its event horizon correspond to the thermodynamic entropy of the black hole.

Scholarpedia has a long list of possible ideas : Bekenstein-Hawking entropy - Scholarpedia
Very interesting, thank you.

14. Surprisingly, I have found a paper on arXiv which appears to make the exact same point as myself :

http://arxiv.org/pdf/1302.2849.pdf

The paper is even quite new, dated Feb 2013. Allow me to quote ( page 4 of the paper ) :

"But it is the notion of black hole
entropy (or more generally of horizon entropy [16]) itself that challenges in a very serious way the usual
continuum picture of spacetime. A black hole horizon is nothing but a particular region of spacetime, so it
is spacetime itself that has entropy. And something that has entropy has a microstructure, whose number
of degrees of freedom that entropy counts. If spacetime was a continuum, that entropy would be infinite.
So the finite value for the horizon entropy is a clear indication that there exist a discrete microstructure for
spacetime [17].
"

How cool is that

15. Originally Posted by river_rat
I think we have hit the whole thermodynamics / information theory entropy problem - you have to be really smart to pick the right set of states to use in the entropy calculation to match with the entropy you would calculate from statistical mechanics and I do not believe we have any idea which states the entropy of a black hole and its event horizon correspond to the thermodynamic entropy of the black hole.

Scholarpedia has a long list of possible ideas : Bekenstein-Hawking entropy - Scholarpedia
This is very esoteric and speculative stuff. I you are interested, Wald wrote a short book called Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics. QFT on curved spacetimes is sufficiently weird that not only is it not well-defined in the mathematical sense, but in fact one loses the very idea of particles. It is the first tentative step toward a quantum theory of gravity and there are a lot of problems with it.

Note that the prediction of Hawking radiation is based on QFT on curved spacetime so you can very quickly see that the explanation in popular books, whcih involves virtual particle/anti-particles becoming real particles when one or the other disappears behind the event horizon, won't hold water since one does not even have the notion of particles in such a theory.

Bottom line: This stuff is so far out that no one can really speak authoritatively and everything is speculation. While Hawking conceded his bet with Preskill regarding the so-called "black hole information problem", so far as I know Kip Thorne has not accepted Hawking's rationale. Hawking's reasoning involves making use of the AdS/CFT correspondence from string theory, and even that is nothing more than a conjecture of Maldecena. A lot of smoke has been published on this subject including Susskind's The Black Hole War which is just full of rank speculation being presented as absolute fact.

Beware of anything that you read on this subject.

16. Originally Posted by Markus Hanke
Surprisingly, I have found a paper on arXiv which appears to make the exact same point as myself :

http://arxiv.org/pdf/1302.2849.pdf

The paper is even quite new, dated Feb 2013. Allow me to quote ( page 4 of the paper ) :

"But it is the notion of black hole
entropy (or more generally of horizon entropy [16]) itself that challenges in a very serious way the usual
continuum picture of spacetime. A black hole horizon is nothing but a particular region of spacetime, so it
is spacetime itself that has entropy. And something that has entropy has a microstructure, whose number
of degrees of freedom that entropy counts. If spacetime was a continuum, that entropy would be infinite.
So the finite value for the horizon entropy is a clear indication that there exist a discrete microstructure for
spacetime [17].
"

How cool is that
Hm, I'm not quite sure I can call it a particular region of space time yet. And I do not agree with the quoted argument. After all my cup out water is also just a particular region of spacetime. That doesn't make it quantized.
In other words, the same argument would apply everywhere.

I will however take a closer look. But bear in mind i will be looking for a need of explanation.

17. Originally Posted by Markus Hanke
Surprisingly, I have found a paper on arXiv which appears to make the exact same point as myself :

http://arxiv.org/pdf/1302.2849.pdf

The paper is even quite new, dated Feb 2013. Allow me to quote ( page 4 of the paper ) :

"But it is the notion of black hole
entropy (or more generally of horizon entropy [16]) itself that challenges in a very serious way the usual
continuum picture of spacetime. A black hole horizon is nothing but a particular region of spacetime, so it
is spacetime itself that has entropy. And something that has entropy has a microstructure, whose number
of degrees of freedom that entropy counts. If spacetime was a continuum, that entropy would be infinite.
So the finite value for the horizon entropy is a clear indication that there exist a discrete microstructure for
spacetime [17].
"

How cool is that
Nothing surprising at all. One can find almost anything in ArXiv. Remember that the archive is a repository for electronic preprints, many (most) of which will not be published in any reputable refereed journal.

I have not read, and don't intend to read, the entire paper in detail, particularly after seeing the phrase "background independent" used early on as a requirement for a quantum theory of gravity. This assertion is rather common among the loop quantum gravity crowd, but they are not able to even define what it means, let alone produce any coherent theory that one might agree to have such a property.

It may well be that in some future theory that replaces the current quantum field theories and general relativity that the concept of the spacetime manifold is replaced by something else. I have no doubt that any unification of quantum theories with theories of gravitation will require fundamentally new ideas, and replacement of the manifold with something else as the setting for the dynamics of the theory may well prove to be one of the necessary radical modifications. But at this stage of the game, no one has made and specific serious proposal as to what that something else might be. Rather all that one has in that regard is smoke and mirrors from the LQG crowd and vociferous denials by the string theory crowd that there is any merit at all in LQG.

There is a great deal of hand waving in the literature, but not a great deal that can be supported by rigorous mathematics, and to date not even a hint of a testable new prediction from any of the various approaches to quantum gravity.

I personally will have to wait until one or the other, or more likely a third party, produces a theory that makes actual testable predictions and then is able to fortify their predictions with supportive experimental data. In the meantime the pressure in academia to publish, whether the publications be meritorious or complete nonsense, will continue and I anticipate no shortage of papers in the archive on this and other subjects. It is fortunate that such papers can be handled completely electronically and do not require the sacrifice of a tree just so that some academic can have his name attached to some ink spots. Numbers of papers do not make a scientific reputation. What counts is the useful content of a "few but ripe" papers of real merit.

18. All points taken, thanks for the feedback guys.
Just to clarify this, I do not assert that there is anything to the idea - how could I, since I am as per yet missing major junks of the physics and maths involved. I am just finding the idea interesting, and was a little surprised to find the same argument on arXiv.

Remember that the archive is a repository for electronic preprints, many (most) of which will not be published in any reputable refereed journal.
So then, what is the actual purpose of arXiv ? What do the authors gain by publishing their preprints there ?

19. ArXiv provides a means of rapidly disseminating an idea for two purposes: 1) to rapidly obtain comments from the community and 2) to establish priority for an idea. It is a very useful tool and of great benefit to the scientific community.

There are also some crackpots who put things in the archive because they cannot get them out in any other way. While it necessary to have sponsorship from some recognized institution to put things on ArXiv, the standards are not very high. A lot of stuff that is put there will never see publication in any reputable journal. On the other hand there some papers published there that are of great importance.

Even papers that are never published MAY be usefully placed in the archive. For instance in the todo last year over the apparent discovery of supeluminal neutrinos, the preliminary paper discussing the experiment and some theoretical arguments against the validity of the experiment appeared quickly in ArXiv. This was legitimate science at work, even though the problem turned out to be a bad connection in the instrumentation.

On the other subject of discrete models of space and/or spacetime it is worth noting that several attempts have been made to construct a lattice model and there are still some researchers pursuing that idea. But to date no one has been able to make a lattice model work.

20. So how would one best keep up with latest developments in, say, GR and quantum gravity ? Is there a big, well known peer review journal for that area, or is there a repository of individual journals ?

21. Originally Posted by Markus Hanke
So how would one best keep up with latest developments in, say, GR and quantum gravity ? Is there a big, well known peer review journal for that area, or is there a repository of individual journals ?
GR is mature area so you can find it well presented in several textbooks -- for instance see the list for GR here http://www.thephysicsforum.com/misce...oks.html#post7

There are different approaches to quantum gravity, none of which have produced any testable hypotheses and all of which involve a lot of speculation and very advanced mathematics. The most well-known and most widely pursued approach is string theory. I think that if any of the current avenues is likely to producd fruit then it is probably string theory, and I am not overy optimistic that it will do so without major revisions to the general approach and a lot more time. It is virtually impossible for anyone but a specialist to "keep up" with the latest developments.

There is a lot of junk publilshed in the name of string theory. So if you want to read relatively cuttiing-edge stuff and still avoid going out into lala land you might try reading articles in ArXiv that are published by people in the area with solid reputations. Some (I don't claim to be able to produce a list of all) such solid researchers are: Edward Witten, Juan Maldacena, Joseph Polchinski, John Schwarz and Michael Green. You will find research papers in the area virtually impossible to read unless you are a specialist in the area.

There are a few textbooks written on string theory. They are also rather difficult to read, in no small part because 1) they are highly mathematical and 2) the mathemtical foundations of string theory are very shaky (for instance there is no good definition for what strinig theory actually is). If you are feeling brave here are some titles:

Superstring Theory (2 vols) -- M.B. Green, J.H. Schwarz, E. Witten

String Theory (2 vols) -- Joseph Polchinski

String Theory and M-Theory -- Becker, Becker and Schwarz

A First Course in String Theory -- Zwiebach (this is the most elementary of the books)

Quantum Fields and Strings: A Course for Mathematicians (2 vols) - Deligne, Kazhdan, Etingof, Morgan, Freed, Morrison, Jeffrey, Witten (ranges from difficult to indecipherable depending on which section and which author)

Keeping up with journal articles basically requires deep expertise and access to a large research library. This is a full-time job. An appointment to the Institute for Advanced Studies would be a big help.

Good luck.

22. You will find research papers in the area virtually impossible to read unless you are a specialist in the area.
I wouldn't even attempt to read anything in the area of String Theory. It is way beyond me at this stage. I am more interested in General Relativity, where I can at least follow the general ideas, if not always all the mathematical details...any journal recommendations ?

23. Originally Posted by Markus Hanke
I wouldn't even attempt to read anything in the area of String Theory. It is way beyond me at this stage. I am more interested in General Relativity, where I can at least follow the general ideas, if not always all the mathematical details...any journal recommendations ?
Journals are devoted to cutting-edge research. There is very little such cutting edge research in general relativity as it is a rather mature subject. The cutting edge stuff generally lies at the boundary between general relativity and high-energy physics, and that is basically string theory and the other approaches to quantum gravity.So, you won't find a lot of accessible journal articles on general relativity.

That said any of the general journals on physics might from time to time have something on the subject -- various APs journals , American Journal of Physics, and others . You might also search ArXiv for articles on relativity, with the usual caveat that you will not have the benefit of peer review and that relativity does tend to attrack the wackos who are out to show that Einsteiin was wrong.

You should also be aware that journals are written for specialists. Journal page charges are quite high, and siince space is at a premium articles tend to assume that the reader is very familiar with the discipline and generally don't spend a lot of time with background (survey articles are something of an exception but don't expect to be spoon fed even there). So, to "keep up the area" you probably already need to be pretty close to up to date just to follow the articles.

For general relativity and cosmology you would probably be well-served with the books by Wheeler et al, Weinberg, Hawking and Ellis, Peebles, Peacock in the list of recommended texts. Books tend to be a bit more leisurely and usually provide something in the way of background and introduction, and references to other texts that can provide even more background and introduction.

I would be very leery of articles in the popular press unless they are written by well-known and respected researchers -- people like Kip Thorne or Steven Weinberg.

24. Originally Posted by Markus Hanke
So then, what is the actual purpose of arXiv ? What do the authors gain by publishing their preprints there ?
arXiv is a great dissemination tool and is a useful copyright dodge with respect to some of the more draconian journals (as once the paper is published it belongs to the journal and not the author etc). But as DrRocket has mentioned, the barrier to submit is not very high and so if you are not an expert in the field in question you have to keep your eyes open.

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