Thursday, January 15, 2009


Cookie MonsterThese days, everybody is talking about entropy. In fact, there is so much talk about entropy I am waiting for a Hollywood starlet to name her daughter after it. To help that case, today a contribution about the entropy of black holes.

To begin with let us recall what entropy is. It's a measure for the number of micro-states compatible with a given macro-state. The macro-state could for example be given by one billion particles with a total energy E in a bag of size V. You then have plenty of possibilities to place the particles in the bag and to assign a velocity to them. Each of these possibilities is a micro-state. The entropy then is the logarithm of that number. Don't worry if you don't know what a logarithm is, it's not so relevant for the following. The one thing you should know about the total entropy of a system is that it can't decrease in time. That's the second law of thermodynamics.

It is generally believed that black holes carry entropy. The need for that isn't hard to understand: if you throw something into a black hole, its entropy shouldn't just vanish since this would violate the second law. So an entropy must be assigned to the black hole. More precisely, the entropy is proportional to the surface area of the black holes, since this can be shown to be a quantity which only increases if black holes join, and this is also in agreement with the entropy one derives for a black hole from Hawking radiation. So, black holes have an entropy. But what does that mean? What are the microstates of the black hole? Or where are they? And why doesn't the entropy depend on what was thrown into the black hole?

While virtually nobody in his right mind doubts black hole have an entropy, the interpretation of that entropy is less clear. There are two camps: On the one side those who believe the black hole entropy counts indeed the number of micro-states inside the black hole. I guess you will find most string theorists on this side, since this point of view is supported by their approach. On the other side are those who believe the black hole entropy counts the number of states that can interact with the surrounding. And since the defining feature of black holes is that the interior is causally disconnected from the exterior, these are thus the states that are assigned to the horizon itself. These both interpretations of the black hole entropy are known as the volume- and surface-interpretations respectively. You find a discussion of these both points of view in Ted Jacobson's paper "On the nature of black hole entropy" [gr-qc/9908031] and in the trialogue "Black hole entropy: inside or out?" [hep-th/0501103].

A recent contribution to this issue comes from Steve Hsu and David Reeb in their paper

Steve is a neighbor here on blogspot over at Information Processing. In their paper Steve and David examine the question how much matter one can stuff into a volume bounded by a given surface, and how much entropy this matter can carry. In flat space-time the relation between the volume of an area and its surface is trivial, it's just Euclidean geometry. But not so if space-time is strongly curved!

To see this, consider the often made analogy of a curved space to a rubber sheet. Draw a circle on it. That's your surface. But it's a rubber sheet, meaning you can deform the sheet inside the circle arbitrarily. You could for example form it to a bag and stuff a lot of gold into it.
Bag of Gold
This pictorial terminology is sadly not my invention: these kind of solutions have been known to be possible in General Relativity for a long time, and have been dubbed “bags of gold” by Wheeler already in the early 70s. Their defining property is that they have a potentially arbitrarily large interior volume, but a small surface area.

Steve and David in their paper now construct a weird kind of solution they dub “monsters,” which exemplifies what one can do with these bags. To understand what a monster is, consider some stuff (eg coins of gold) dispersed in space-time, such that the background is to good approximation flat. Now pick up these coins and put them closely together - so close that they almost, but not entirely, form a black hole. What you achieve in this way is that you get a strong gravitational field and a deviation of the volume-surface relation from flat space. That process of picking up and redistributing the coins should not be thought of as a process that is actually dynamically happening, but just as a way to create the initial conditions*. If you create these initial conditions carefully you can achieve most importantly two things:
  1. You can get the asymptotic mass a far away observer would measure (ADM mass) to be arbitrarily small, no matter how many coins you have had. The reason for this is that the strong gravitational field contributes with a negative binding energy.

  2. You can similarly get an arbitrarily large entropy inside a sphere with fixed surface area, think of the coins as the particles forming a particular micro-state. The reason is that the volume can get arbitrarily large, and you can stuff all the coins in, even though the surface area and the asymptotic mass might remain small.

The authors also show in their paper that if you create the monster state and let it evolve in time, it inevitably forms a black hole. Since it can have been arbitrarily close to being a black hole, it is plausible to expect that almost all of this entropy goes into the black hole. If the volume interpretation of the black hole entropy was correct, this would be in conflict with it. Weirder than that, the monster solution must have come out of a white hole in the past. This solution is thus very similar to an expanding and re-collapsing closed FRW universe embedded in empty space.

Despite these monster solutions existing in GR, there remains the question however whether they do exist in reality, since they are somewhat pathological and constructed. Though it might be possible to argue these states will never be formed from any sensible initial condition, in a quantum theory the situation is more tricky since everything that can happen does happen - even though it might be very improbable. That means the monsters could be spontaneously formed through tunneling processes. That might however in practice not happen even once during the lifetime of the universe.

Steve was visiting PI in November and gave a very clear talk about the monsters, that is recommendable if you want to know more details. You can find it at PIRSA 08110026 and the slides are here.

* You shouldn't take the picture too literally though, much like in the often used example with the marble on the rubber-sheet it is slightly misleading as there isn't actually something "on" the spacetime (the sheet) that extends into an additional dimension.


  1. This is so well explained, very interesting post. Bravo and thanks a lot!


  2. Thanks for the post Bee - dumb question: what is FRW?

  3. The monsters idea is weird, yet more to feed my fascination and appreciation of how weird our universe is. That definition of entropy sounds rigorous, but it is unambiguous even in a case of including the state of collections of matter which include radiative nuclei etc? Also, considering continua of possible values of velocity (the example given) or etc, how can there be a "number of possibilities"?

    Another problem about entropy is "the arrow of time." The AoT is often believed to be relative, or defined only because of entropy and not inherent absolute character of time flow. But consider the issue of interfering in a “time-reversed world” and how the changes in such a world undermine the credibility of arguing that the AoT is just relative. As I proposed earlier, what if I e.g. deflect a backwards-happening bullet so it “then” misses the barrel it “came out of” (but not yet in the view of the intruding other world relative to which the intrusion is benchmarked.) If that interference happens, the bullet maybe goes past the gun it should have come out of, runs into a tree etc. and then we have a ridiculous “past” that continues to get more wrecked as time (?) goes on.

    In principle, our world could be such a time-reversed world if there’s no true physical distinction (or one that matters to showing “legitimacy”, versus some distinctions about nuclear decay etc. that have equal “standing” regarding genuineness.) Yet now many of us can believe that an intervention from another world etc., regardless of what time flow they were in relative to us, could actually change our own past? Such questions are part of the foundational framing and can’t be brushed off from not being more directly operational expressions of what we already know.

  4. Hi Changcho:

    FRW = Friedmann-Robertson-Walker, sorry about that. It's one of the standard textbook examples for spherical collapse (see eg MTW = Misner, Thorne, Wheeler). Best,


  5. I find entropy to be a bit of a false lead, after all successive observations from a distribution should statistically pick out the most likely values.

    The real question is why are successive observations always time ordered?

    Perhaps the forward time shift operator acts on a left (right) bounded measurable set (that is time has a definite beginning) so that in the forward direction it is unitary (preserves inner products of states) but the inverse operator (reverse time shift) has a non-trivial kernel, and is thus not unitary. You can only pull this sort of magic off in infinite Hilbert spaces.

    Actually that is a pretty easy theorem to prove: Shift operators away from a finite left (right) boundary can be symmetries of a Hermitian operator, while the inverse shift operator, towards a finite left (right) boundary can never be a symmetry of a Hermitian operator; precisely because the kernel is not trivial. So traveling back in time towards a finite temporal boundary can never be a symmetry of a real observable. Thus the arrow of time always points away from a finite boundary in the past.

    Should I write that up and publish it?


  6. Hi Bee,

    "On the one side those who believe the black hole entropy counts indeed the number of micro-states inside the black hole. I guess you will find most string theorists on this side, since this point of view is supported by their approach."

    I don't think that terms like inside or outside of black holes apply for the microscopic description of the black hole in string theory. There the microscopic degrees of freedom of the black hole are basically open strings in D-branes configurations. So there is no notion of the inside of a black hole. The low energy description of these D-branes configurations though correspond to solutions of the supergravity theory (p-branes). If then we compactify the derived metric down to our 4 dimensions we'll get eventually the metric of the black hole, where notions as the horizon for example have a meaning. From there we can calculate the entropy from the area. This matches of course the entropy calculated by counting states in the microscopic picture.

    Unless you meant something else.


  7. Hi Giotis:

    Thanks for the clarification. I wasn't referring to inside vs outside but to surface versus volume. I think the corresponding question would be after you compactify down, would the degrees of freedom be found merely on the horizon? Best,


  8. Well, I meant "radioactive nuclei" and am still interested in how entropy applies to such entities, but oddly enough "radiative nuclei" turns up quite a few apparently legitimate non-typo references - I never heard of that, perhaps nuclei that emit gamma rays in response to excitation etc. But like I was saying, look at a muon which before it decays is structureless and just like a stable particle in terms of properties, but when it decays there is then more complexity in the universe than before etc - how can entropy be coherently defined for such entities, absent a mechanism we can describe inside the way we can for ordinary "heat engines" etc?

  9. In string theory, things with large volumes and small areas tend to be very unstable, see eg

    Having said that, I do think that it's a great pity that Prof Hsu's work has not received the attention it deserves. So your posting is very welcome. This is what physics blogs should be.

  10. I think from outside of a black hole (Earth) if you can measure its entropy you're not going to get information from inside it because you're causally disconnected from that region of spacetime. And to get an estimation of the entropy from inside the black hole I suppose you should perform a change of reference system and see what happens with the entropy after that. But I suppose that, being causally disconnected and one spacelike dimension inside interchanged with the timelike one, the change of reference would not be mathematically sound. Is that correct?

  11. Hi Bee,

    Have you physicists no mercy, for I haven’t had reason to believe in monsters since childhood and now I’m presented with this. Seriously though, it’s a great post which serves to remind that the second law is more of what’s the monster as far as physics is concerned. It will be interesting to see that in future if this means you must further refine the models of black holes or rather find reason to repeal the law. Anyway, I don’t see this gives me reason enough to once again sleep with the light on:-)



  12. Black hole has maximal entropy.

  13. Neil,

    Do monsters have a geometrical structure, or are these things contrived by mortal men whose fear has overtaken them?:) I' think such abstraction are mental capacities that allow wo/men to advance into the realm of the mental constructs. Interesting to see such real "allotrope geometrical values in the real world, from such abstractions.

    You might think the loss of geometry like the loss of, say, Latin would pass virtually unnoticed. This is the thing about geometry: we no more notice it than we notice the curve of the earth. To most people, geometry is a grade school memory of fumbling with protractors and memorizing the Pythagorean theorem. Yet geometry is everywhere. Coxeter sees it in honeycombs, sunflowers, froth and sponges. It's in the molecules of our food (the spearmint molecule is the exact geometric reaction of the caraway molecule), and in the computer-designed curves of a Mercedes-Benz. Its loss would be immeasurable, especially to the cognoscenti at the Budapest conference, who forfeit the summer sun for the somnolent glow of an overhead projector. They credit Coxeter with rescuing an art form as important as poetry or opera. Without Coxeter's geometry as without Mozart's symphonies or Shakespeare's plays our culture, our understanding of the universe,would be incomplete. This quote taken from article and posted in this Blog entry. I give a more in depth explanation at Moshe's article on Maldacena.

    Will have to see if comment takes there or not.

    Faster then light in a medium Neil. Ice or the earth serve as the backdrop for evidence of particle collisions. We know they happen in the cosmos, and, we know they happen at the LHC with Gran Sasso.

    How is this translated....microstate blackholes?


  14. Thanks for this well explained posting!

    When talking about entropy in relation to the physics of Black Holes (BH), it is typically conjectured that statistical thermodynamics and QFT apply in a way that is familiar to our laboratory settings. But since BH are non-trivial space-time structures emerging near a gravitational singularity, what grounds does one have to make this tacit assumption? For example, is there experimental confirmation of the Beckenstein area law? Can one precisely define what a "surface" is near the singularity, where space-time is likely to evolve into an unfamilar topology?



  15. Hi Bee,

    I wonder if it’s been considered what’s required to conquer a monster is a demon? In this case perhaps one inspired by Maxwell :-)



  16. I don't think that string theorists would uniformly say that the degrees of freedom of a black hole must be "inside". They are equally well associated with the horizon. The very question "where are they?" is largely unphysical, analogously to the question Where does the gauge theory live? that was answered by Moshe Rozali.

    What matters is whether one can predict physics. The details of the microstates influence the evolution of the exterior microscopically - it is imprinted in Hawking radiation - but it doesn't influence the exterior macroscopically - the radiation is thermal if we only care about the macro description.

    Does it mean the information is inside or the surface? The question has no privileged answer. The people inside will surely think that at least a part of the entropy is carried by them who are inside. The people outside may find the whole interior unphysical because they can't ever see it, so they will associate the entropy with the surface that simplifies calculations.

    There is no contradiction here. In fact, black hole complementarity implies something much stronger: the degrees of freedom inside are not independent from those outside, despite the spacelike separation.

  17. Hi Ervin,

    BH are non-trivial space-time structures emerging near a gravitational singularity,

    You should try to find out exactly what you mean with that. First, what do you mean with a black hole? Presumably the presence of a horizon, since that is what makes the hole black. Then, in what sense is that 'emerging near the gravitational singularity'? Well, it isn't. The horizon can be arbitrarily far away from where classically the singularity would be. The horizon can be in a regime with arbitrarily small curvature. I will repeat that once again because it is a very common confusion: the horizon, which is what makes the black hole black, is formed in a region with a background that can be arbitrarily close to being flat. It is to excellent precision described by semi-classical physics unless the black hole has reached Planckian size.

    Besides this however you seem to assume that the entropy of the black hole has something to do with a quantum field in its inside. That is most definitely not the case. Best,


    PS: You don't have to post as 'Anonymous'. Check option Name/URL under the comment window, a box will open where you can enter a name. You don't have to provide an URL.

  18. Ervin might referring to Sean Carroll's post?:)

    I may of been a bit incorrect in my explanation at Sean Carroll's. I'm trying. Of course his comment might not have to be correlated at all in this context and one can take Sean Carroll as his blog post is.

    See Blackhole Wars Ervin. And then, Moshe's newest post.


  19. Somewhat related to this topic, an anomaly in the German GEO600 gravity waves detection experiment could be the first demonstration of the most amazing discovery in fundamental physics in decades - the discreteness of space-time and possible confirmation of the holographic principle:

  20. The article name is:

    Our world may be a giant hologram

    and the article link is:


  21. Hi Tkk,

    Thanks that was indeed an interesting article you offered. I must admit however I’ve never been able to get my head around that all we refer to as reality is nothing more then a holographic projection. I have no problem with the horizon constituting to being what contains the hologram, it’s just that I can’t imagine what serves as the projector.

    If any of you out there ever talk to Susskind or 't Hooft you might ask them if you don't know already.



  22. Dear Bee,

    Thank you for your explanations and clarifications in terminology. I am not fully familiar with the physics of BH and this is the reason why I ask. I must confess, however, that your reply did not really answer my questions:
    a) what observational grounds do we have to state that the concept of "entropy" from statistical physics is fully applicable to BH?
    b)is there confirmation of Beckenstein law from astrophysical data?
    c)for BH that are reaching Planckian size, what motivates the assumption that the horizon has a conventional topology?

    Best regards,


  23. Hi Tkk,

    I've read this paper already some months ago and tried to figure out what the guy is talking about. I couldn't really make sense of it. Best,


  24. Hi Ervin,

    a) I don't even know what you mean with the question. Who applied where and when what concept to black holes that you doubt? Could you be somewhat more specific? What I wrote was: one identifies the black hole area with an entropy for the reason that the area has the right properties and it matches with Hawkings results according to which black holes have a temperature (the inverse of which you can integrate to get the entropy). If you read my post you will figure out that there is some debate about what this entropy de facto physically means.

    b) There is nothing known in the universe that violates the bound. That is of course not a confirmation. What would you consider a 'confirmation'?

    c) I don't even know whether it still has a horizon or whether that is a meaningful question then. Why do you think so?



  25. Hi Bee,

    1)Indeed, Hawking entropy as an integrated inverse temperature makes sense. The physical origin of BH entropy is not universally accepted and this is what prompted my question: can astrophysical data be used to settle the debate on where the BH entropy is coming from?

    2) By confirmation I mean data that reasonably matches the area law. Is there such confirmation?

    2)I don't really know if it is a sensible question to ask but it seems to me that concepts such as "surface" and "area" loose their conventional meaning near the Planck scale.



  26. Hi Ervin,

    1) no.

    2) see my above reply. the entropy bounds are so large, there is nothing known that breaks them. I wouldn't call that a confirmation though.

    3) that is correct



  27. "The one thing you should know about the total entropy of a system is that it can't decrease in time."

    Please don't say this!!! I know it's a small quibble, but I think it's important to remember that entropy does not have to increase over time... it's only likely to do so! In small systems, it's not even very likely to do so. The consequences of this are measurable.

  28. Hi Aaron,

    cool, thank you very much for the pointer ro the paper by Wang et al., "Experimental Demonstration of Violations of the Second Law of Thermodynamics for Small Systems and Short Time Scales".

    Actually, I've started reading Feyerabend's "Against Method" over our vacation, and stumbled about his bold statement (there are many bold statements in the book, though ;-) that It is well known [...] that statistical thermodynamics is inconsistent with the second law of the phenomenological theory (page 24 in the 1993 Verso edition at Google Books), which he later specifies (page 27) as It is now known that the Brownian particle is a perpetual motion machine of the second kind and that its existence refutes the phenomenological second law.

    I was/I am quite sure that this is not true without further qualifiers. Interestingly, Feyerabend didn't give any references to bolster this assertion, though usually his footnotes can exceed the main text on a page.

    As such, a Brownian particle doesn't do net work, as explained by Feynman's ratchet, and Brownian motors and other more complicated contrivances had not yet been known when Feyerabend wrote his book, I guess.

    So, the paper seems just to address this point - I will have a closer look.

    Interestingly, Feyerabend refers an old German paper by Reinhold F├╝rth, which is said to show that there is a kind of "uncertainty principle" which forbids to establish entropy decrease via the measurement of temperature/heat exchange. And, indeed, in the paper by Wang, they evaluate the work done by/done at the Brownian particle by really measuring mechanical work via integration of dv·F, where velocity and force are accessible by the experimental setup...

    Very interesting stuff!

  29. Bee:

    The experiment apparatus continues to exhibit tiny amount of vibrational noises no matter how 'perfect' they tried to eliminate all sources of noise. More detailed analysis of the noise shows it has characteristics curve closely matching 'noise' of space-time as we go near Planck's length. But with a difference - the noise shows itself many orders of magnitude at lengths larger then Plank. I.e. quantum fluctuation space-time could be showing itself at lengths much larger than Planck. How to explain?

    One possibility is the holographic principle - information (thus entropy) of surface = that of volume. Since volume >> surface, the only way this can happen is Planck size inside the volume is >> than surface. Two Plank sizes! When they calculate out the necessary Planck sizes according to HP, the Planck size in volume (i.e. within the universe) has a much larger length, which yields a quantum flux character approximately matching that of the said apparatus 'noise'.

    If this noise is truly irreducible and the characteristic pattern confirmed, then it is a strong confirmation of 1) large Planck length inside the universe, and the accepted Planck length at the surface of the universe, 2) discreteness of space-time 3) HP.

    I personally is intrigued by implications of discreteness on the time side. I.e. 2 Planck times!

  30. Hi Tkk,

    Yes, very intriguing indeed. Did you actually read the paper? I tried, and I also looked up the papers he is referring to (mostly his own) because I thought it would make for an interesting blog post. I couldn't make sense of it, and then I lost interest. Your equation volume >> surface is obviously nonsense, sorry to be so blunt. Best,


  31. This comment has been removed by the author.

  32. Hi Arun,

    This temporary violation of the second law is not a new idea in fact James Clerk Maxwell in 1878 wrote in a review in Nature the following (sourced from nature):

    “The truth of the second law is ... a statistical, not a mathematical, truth, for it depends on the fact that the bodies we deal with consist of millions of molecules...
    Hence the second law of thermodynamics is continually being violated, and that to a considerable extent, in any sufficiently small group of molecules belonging to a real body. “

    It doesn’t appear that Wang et al has extended thinking much past Maxwell’s. What I do like however is that experiments like these as far as I tell suggest we separate ourselves from notions that entropy might be responsible for the arrow of time. That is to say are we to suppose that for the brief few seconds where the experiment showed a decrease in entropy time ran backwards.



  33. It also appears that Penrose doesn’t consider the second law as being a sacred cow either, for in his “Road to Reality” page 692 he states:

    “In my view entropy has the status of a ‘convenience’, in present day theory rather then being ‘fundamental’---though there are indications that, in the deeper context where quantum-gravitational considerations become important (especially in relation to black hole entropy) there may be a more fundamental status for this kind of notion.”

    What Penrose considers significant about the black hole entropy question is that it seems to indicate that the lion’s share of entropy is manifest in and confined to these objects. One way to look at this would be to say that without them the background radiation temperature would be significantly higher. On the other hand if you are one of those that equate entropy with information would be to say that the overwhelming portion of the universes information is confined to them. Which ever way you slice it the current makeup and character of reality is reliant on their existence in being largely responsible for assuring that equilibrium if looked at from a potential standpoint is something that is not being reached in the straight forward manner which thermodynamics normally suggests.



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