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Relation between Verlinde's Entropic gravity and AdS/CFT

  1. Jul 13, 2011 #1
    Hi there PF

    I have been thinking about Verlinde's entropic gravity in relation to AdS/CFT. In Verlinde's gravity, gravity is caused by changes in entropy on holographic screens. Now if we look at the dimensionality of this picture, we have gravity in 3+1 dimensions (regular space, like where Newtons law og gravitation works) and then we have the holographic screens, embedded in the 3+1 dimensions, as 2+1-dimensional objects. Now this looks to me, as we have:

    • Gravity in 3+1-dimensions
    • Information on holographic-screens (2+1-dimensions)

    This looks to me, like we have situation, which is described in the AdS/CFT-conjecture. In AdS/CFT we have the following

    • Gravity in n+1-dimensions
    • A conformal field theory in (n-1)+1-dimensions

    Here, can the duality then be seen as a relation between gravity and a CFT, where one causes the other? If so, then it looks viable with Verlinde's gravity.

    Also, this thought leads to two questions from my side. 1) Is it possible to describe the information on the holographics screens (in Verlinde's gravity) as a conformal field? And 2) Can Verlinde's gravity (in this relation) be seen as a proof of the AdS/CFT-conjecture in the low-energy/infrared regime?

  2. jcsd
  3. Jul 13, 2011 #2
    Schreiber, from what I understand, holographic screen are present in the bulk, so we have the bulk theory dual to the screen, not the CFT on the boundary.
  4. Jul 13, 2011 #3


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    also the technique and the motivation are very different
  5. Jul 14, 2011 #4
    But what is the information on holographic screen then? Can't it be considered a CFT?
  6. Jul 14, 2011 #5
    When Erik Verlinde talks about "entropy" and "temperature", you should put those words in mental quotation marks. "Entropy" really means "phase space volume", and "temperature" means "density of energy levels". It's just that the relationships between these quantities resemble the relationships in thermodynamics. Only on or behind a gravitational horizon, such as inside a black hole, does it truly become thermodynamics.

    His idea is best understood in terms of the matrix realization of string theory. The diagonal matrix elements represent the position in space of D-branes, and the off-diagonal matrix elements represent the open strings between the D-branes.

    Suppose you have some D-branes which make up a black hole; they will be described by some part of the matrix diagonal, and there will be a sub-matrix of off-diagonal matrix entries connecting them. In this case, the off-diagonal degrees of freedom will be about the same size as the on-diagonal degrees of freedom. This is the thermodynamic equilibrium, equilibrium between off-diagonal and on-diagonal.

    Outside a black hole, it's different. The off-diagonal elements are close to zero. This holds both for off-diagonal elements between two D-branes in space, and for off-diagonal elements between a D-brane inside a black hole and another D-brane outside it.

    His claim is that gravity is an effective force which you see happening on the matrix diagonal - acting on the positions of matter in space - which is just an approximation to the full matrix dynamics. The full matrix is evolving according to some matrix-model Hamiltonian through the abstract space of matrix values. Because it is Hamiltonian dynamics, there is conservation of phase-space volume. But if you just focus on the on-diagonal degrees of freedom, and construct an approximate dynamics by integrating out the off-diagonal elements, you will find some effective forces acting on the on-diagonal degrees of freedom (positions of matter in space). Technically, they are adiabatic reaction forces. Adiabatic processes are slow, and the on-diagonal degrees of freedom change slowly compared to the off-diagonal degrees of freedom. The conservation of phase space volume in the full dynamical system becomes the presence of a reaction force acting on the slow variables in the approximation; this is a general principle that was explored in some depth by the physicist Michael Berry. Verlinde is applying this to matrix theory.

    Gravity is the leading effect, but all the other forces are supposed to be reaction forces too, higher-order corrections to the reduced dynamics of the on-diagonal degrees of freedom.

    There's no paper spelling this out yet, but it's in his talk at Strings 2011.

    In matrix theory, you aren't working in space-time any more. You have a matrix which evolves in time, and events in ordinary space correspond to what happens on the matrix diagonal. In AdS/CFT, on the other hand, you still have space and time on the boundary, just with fewer dimensions than in the bulk. So that is the first bridge to be crossed.

    Now what about the holographic screens? In Verlinde's idea, the entropy is not on the screens. All the actual entropy (the matrix "entropy", anyway) is behind the gravitational horizons. "Entropy" (volume in matrix phase space) is an adiabatic invariant for the reduced dynamics of the matrix diagonal, and an adiabatic invariant is a quantity that doesn't change. Matter in space attracts other matter because of the underlying matrix dynamics. Distance from a screen (such as the equipotential surfaces he considers in his 2010 paper) is a relationship between on-diagonal degrees of freedom, and it's connected to these pseudo-thermodynamic quantities "entropy" and "temperature" (see my first paragraph), but matrix "entropy" isn't located outside the black holes and isn't changing outside the black holes.

    The holographic screens really apply at a level of description above the matrix level employed by Verlinde - Bousso's http://www.ccsem.infn.it/issp2011/docs/bousso.pdf" [Broken]. This is a statement that the entropy of matter along lightlike surfaces is bounded by A/4G. This must have something to do with black holes forming if you get too much matter together - Bousso's bound is the entropy for a black hole with surface area A - but he says the origin of the general bound is obscure and should arise from a principle of quantum gravity, such as the holographic principle.

    As I said, in AdS/CFT you still have space and time, you just have less space dimensions on the boundary. At the end of his talk linked above, Bousso discusses holographic screens in the AdS bulk. The bound applies in AdS space, but here you can understand it in terms of the boundary theory; the entropy in the bulk corresponds to the entropy in the boundary CFT, and introducing a screen in the bulk corresponds to introducing a cutoff on the boundary CFT. Moving the screen out to the boundary at infinity corresponds to removing the cutoff in the CFT. So in AdS space, there is a CFT corresponding to a holographic screen: a cutoff version of the full boundary CFT.

    In other spaces (that are not anti de Sitter), we don't have the gauge/gravity duality worked out - we don't know the boundary CFT - and so we don't know the cutoff appropriate for a screen in the middle of the bulk space, either. But it seems likely that when we do have the counterpart to AdS/CFT for other spaces (flat space, de Sitter space,...) figured out, it will be possible to define regularizations of the underlying CFT which correspond to specific holographic screens in the bulk.

    To fully realize Erik Verlinde's argument in the context of AdS/CFT, the boundary CFT would need to be derived from a matrix model. I'm not sure of the status of that line of research, but the other Verlinde brother, Herman, has http://arxiv.org/abs/1104.2605" [Broken] constructing a twistor matrix model for N=4 Yang-Mills, the boundary CFT for Type IIB strings in AdS5 x S5. So it might be explored in that context.

    For the real world, we want dS/CFT, with time holographically emergent from a Euclidean CFT at past infinity. Some of us are waiting anxiously for a paper by Hartman and Anninos, promised at Strings 2011, which will finally provide a detailed example of dS/CFT, ten years after Andrew Strominger first described how it should work in principle. One of the issues awaiting resolution is that in de Sitter space, you have, not just black-hole event horizons, but observer-dependent cosmological horizons. "Behind the horizon" - which in Verlinde's scheme is where the on-diagonal/off-diagonal thermodynamic equilibrium is realized - should correspond to beyond the cosmological horizon. But that space should be just like space here - unlike the space inside a black hole. So there are major unanswered questions. My guess is that cosmological horizons in de Sitter space are more like mid-bulk holographic screens and that the matrix "entropy" isn't there, it's at the boundary at past infinity. I also guess that this has something to do with the state of the universe before inflation being something like the state of the universe inside a black hole - it's a place where off-diagonal and on-diagonal degrees of freedom are in equilibrium, and the usual description of space does not apply. But those are just guesses.
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  7. Jul 14, 2011 #6
    It is just odd, because it seems like he is treating the situation as being space and time in his paper http://arxiv.org/abs/1001.0785, not like it is derived from matrix models.. Ohh well, it is probably my ignorance and lack of physics yet :wink:
  8. Jul 14, 2011 #7
    Btw I just had a thought on this. Now if we look at the Verlinde picture with the holographic screen, what would actually happen if we moved the screen to boundary of space (i.e. to infinity), would this setup then resemble the AdS/CFT, because the screen is now on the boundary of the bulk?

    And btw Mitchell: You say that in Verlinde's picture, all the other forces are supposed to be emergent aswell? This would, in my view, make a good correspondance to strings/M-theory, since the fundamental object here is the string (or membrane) and the forces (which we call fundamental forces) are just emergent phenomena comming from strings/membranes (i.e. fundamental forces are not fundamental at all, since they are emergent from the fundmental objects: strings/membranes). Does this make sense to you ? :biggrin:
  9. Jul 15, 2011 #8


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    Here are the slides of Verlinde's talk:

    http://www-conference.slu.se/strings2011/presentations/5%20Friday/1220_Verlinde.pdf [Broken]

    He is thinking about MOND and probably dark fluid, since he cited H.Zhao. Here's some papers:


    Refining MOND interpolating function and TeVeS Lagrangian

    HongSheng Zhao, Benoit Famaey
    (Submitted on 16 Dec 2005 (v1), last revised 5 Jan 2006 (this version, v3))
    The phenomena customly called Dark Matter or Modified Newtonian Dynamics (MOND) have been argued by Bekenstein (2004) to be the consequences of a covariant scalar field, controlled by a free function (related to the MOND interpolating function) in its Lagrangian density. In the context of this relativistic MOND theory (TeVeS), we examine critically the interpolating function in the transition zone between weak and strong gravity. Bekenstein's toy model produces too gradually varying functions and fits rotation curves less well than the standard MOND interpolating function. However, the latter varies too sharply and implies an implausible external field effect (EFE). These constraints on opposite sides have not yet excluded TeVeS, but made the zone of acceptable interpolating functions narrower. An acceptable "toy" Lagrangian density function with simple analytical properties is singled out for future studies of TeVeS in galaxies. We also suggest how to extend the model to solar system dynamics and cosmology, and compare with strong lensing data (see also astro-ph/0509590).


    Dark Fluid: Towards a unification of empirical theories of galaxy rotation, Inflation and Dark Energy

    HongSheng Zhao (SUPA, St Andrews) Baojiu Li (DAMTP, Cambridge)
    (Submitted on 10 Apr 2008)
    Empirical theories of Dark Matter like MOND gravity and of Dark Energy like f(R) gravity were motivated by astronomical data. But could these theories be branches rooted from a more general hence natural framework? Here we propose the natural Lagrangian of such a framework based on simple dimensional analysis and co-variant symmetry requirements, and explore various outcomes in a top-down fashion. Our framework preserves the co-variant formulation of GR, but allows the expanding physical metric be bent by a single new species of Dark Fluid flowing in space-time. Its non-uniform stress tensor and current vector are simply functions of a vector field of variable norm, resembling the 4-vector electromagnetic potential description for the photon fluid, but is dark (e.g., by very early decoupling from the baryon-radiation fluid). The Dark Fluid framework naturally branches into a continuous spectrum of theories with Dark Energy and Dark Matter effects, including the $f(R)$ gravity, TeVeS-like theories, Einstein-Aether and $\nu\Lambda$ theories as limiting cases. When the vector field degenerates into a pure Higgs-like scalar field, we obtain the physics for inflaton and quintessence. In this broad setting we emphasize the non-constant dynamical field behind the cosmological constant effect, and highlight plausible corrections beyond the classical MOND predictions. Choices of parameters can be made to pass BBN, PPN, and causality constraints. The Dark Fluid is inspired to unify/simplify the astronomically successful ingredients of previous constructions: the desired effects of inflaton plus quintessence plus Cold DM particle fields or MOND-like scalar field(s) are shown largely achievable by one vector field only.

    And mainly this one, where Zhao makes the same representation of the octopus of DM/DE/Matter, but with an iceberg:


    An ecological approach to problems of Dark Energy, Dark Matter, MOND and Neutrinos

    HongSheng Zhao
    (Submitted on 21 Nov 2008)
    Modern astronomical data on galaxy and cosmological scales have revealed powerfully the existence of certain dark sectors of fundamental physics, i.e., existence of particles and fields outside the standard models and inaccessible by current experiments. Various approaches are taken to modify/extend the standard models. Generic theories introduce multiple de-coupled fields A, B, C, each responsible for the effects of DM (cold supersymmetric particles), DE (Dark Energy) effect, and MG (Modified Gravity) effect respectively. Some theories use adopt vanilla combinations like AB, BC, or CA, and assume A, B, C belong to decoupled sectors of physics. MOND-like MG and Cold DM are often taken as opposite frameworks, e.g. in the debate around the Bullet Cluster. Here we argue that these ad hoc divisions of sectors miss important clues from the data. The data actually suggest that the physics of all dark sectors is likely linked together by a self-interacting oscillating field, which governs a chameleon-like dark fluid, appearing as DM, DE and MG in different settings. It is timely to consider an interdisciplinary approach across all semantic boundaries of dark sectors, treating the dark stress as one identity, hence accounts for several "coincidences" naturally.
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  10. Jul 16, 2011 #9
    Anyone? :)
  11. Jul 17, 2011 #10
    The idea has developed further. See his recent talks at Perimeter Institute and at Strings 2011. The slides from Strings 2011 don't display some characters properly, but you can see them in the Perimeter PDF.

    I'm trying to use http://arxiv.org/abs/0709.2136" [Broken] as a model for thinking about his claims, and will have more to say [strike]soon[/strike] eventually.
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  12. Jul 17, 2011 #11


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    Mitchel, what is your opinion about the iceberg paper?
  13. Jul 17, 2011 #12
    But what would happen if we moved Verinde's holographic screens to the boundary of the bulk? Would it then resemble AdS/CFT, if the bulk theory was indeed AdS?
  14. Jul 19, 2011 #13


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    Mitchell has given you the explaination of both. there is no direct connection like I have said before, but like all the cases of physics theories they are somehow deeply connected. this paper which is a find by MTD2 is one attempt to unravel the connection. however, it is of general idea and also the emphasis is on information, entropy and such.


    On the origin of the holographic principle

    Jae-Weon Lee
    (Submitted on 18 Jul 2011)
    Recently, it has been suggested that quantum mechanics and gravity are not fundamental but emerge from information loss at causal horizons. Based on the formalism it is shown that the holographic principle also aries naturally from the information loss of bulk fields observed by an outside observer. As an application, Witten's prescription for the Rindler spacetime is derived.
  15. Jul 19, 2011 #14
    Thank you so much :) It has been a most helpful discussion with you guys, and the clarification has been terrific :) I am sorry for my misunderstanding of the meaning of the papers, because I could not see the connection between the two research areas from the papers.
  16. Jul 19, 2011 #15
    It's just that I would prefer to be precise (and correct) in what I say, and I can't form an opinion about the relationships between all these ideas right away. But I can say a few more things.

    First, one of the inspirations for Verlinde's idea was the study of http://arxiv.org/abs/0907.2695" [Broken]. Almost from the beginning of AdS/CFT, it was known that a black hole in AdS is dual to a thermal state (plasma) on the boundary. But what about the formation of a black hole, what is the boundary dual description of that process? So Verlinde and coauthors had to develop a boundary description of something like a neutron star in the bulk, with lots of bulk fermions held apart by the exclusion principle. The principle of the AdS/CFT correspondence is that a bulk field (that is, a type of particle in the bulk) corresponds to a type of combination of boundary field operators. You could think of the boundary fields as "holographic preons" that combine to make the bulk fields. So a bulk fermion is a holographic combination of boundary preons, and when the AdS neutron star collapses into a black hole, the combinations of boundary preons come apart and you have a preon plasma on the boundary. This corresponds - in Verlinde's new picture - to equipartition of energy, inside the black hole, between D-branes and the open strings between them; whereas, for objects outside black holes, the energy is mostly in their constituent D-branes, not in the strings.

    I'm still confused about some details. For example, in "entropic gravity 2.0", all the actual gravitational entropy is inside horizons (e.g. inside black holes), and it's only at the horizon that the gravitational force becomes a truly entropic force. Away from horizons, gravity is supposed to be an adiabatic reaction force. Precisely because there's no gravitational entropy outside horizons, the gravitational entropy can't change out there, but it's "because" the entropy doesn't change that gravity outside the horizon exists; it's an emergent force whose existence and form is dictated by the phase space volume associated with the degrees of freedom (the open strings) which are being integrated out in the description which focuses only on ordinary, "commutative" space-time coordinates. But then how do we interpret the "entropic" force associated with an equipotential surface in space surrounding a mass that is not a black hole? In this case there's no event horizon inside the holographic screen. There's matter inside the screen, so maybe we can still say something about open strings which connect a D-brane inside the surface with a D-brane outside the surface, but I don't see how the argument works.

    So I find this new version of entropic gravity quite plausible, but the details are still fuzzy and they may make a big difference to how it works in the end. I did notice, at 45 minutes into the Perimeter Institute talk, that Verlinde was using a formula from CFTs (Cardy formula) in order to estimate the "entropy" (phase space volume) associated with a particular equipotential surface (of the sort that appeared in his 2010 paper). So it's sort of consistent with the idea that CFT-based holography could provide a rigorous justification for his argument. (In response to the original post, that would mean that AdS/CFT was proving entropic gravity, rather than the other way around.) But many of the arguments are still heuristic, they rely on guesses and plausibility, and even Verlinde says there are unanswered questions about how it should work.

    As for HongSheng Zhao's papers, it seems very likely that Verlinde is drawing some inspiration from his model unifying dark energy, cold dark matter, and hot dark matter, except that Verlinde wants visible matter to be part of this one continuum. About Jae-Weon Lee's work, I can't comment yet.
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  17. Jul 19, 2011 #16
    One thing I didn't get in verlindes illustration in terms of matrix models is how can he actually disentangle between what's on the diagonal and what's not? To my understanding, matrix model lagrangians usually exhibit a symmetry acting on the matrices as a rotation in matrix space of some sort, e.g. U(N). But for example, a diagonal hermitian matrix will not stay diagonal upon some U(N) rotation of matrix coordinates. Still, if such a symmetry exists at the level of the lagrangian, physics shouldn't depend on this. So how does he tell apart what he calls diagonal and off-diagonal? Obviously I have some misconception of how these matrix models work.
  18. Jul 22, 2011 #17
    Readers of this thread may be interested in http://motls.blogspot.com/2010/01/erik-verlinde-why-gravity-cant-be.html" [Broken] about the viability of Verlinde's idea in its current form (see the "slow comments" beneath the article, for July 2011, rather than the "fast comments" that pop up). I expressed Erik's ideas (as I understand them) in the language of Matrix theory (the matrix model of M theory), and Luboš's final statement was that it's obvious that no version of these ideas can work in Matrix theory, if you know the technical details. I don't, so I intend to keep thinking about entropic gravity, and to use it as a spur to learn enough about Matrix theory so I can eventually judge for myself; but meanwhile, there's the considered opinion of a physicist who was a co-inventor of matrix string theory (and who is Erik Verlinde's harshest critic on the web).
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  19. Jul 22, 2011 #18


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    Erik Verlinde is also a co-inventor of matrix theory:


    "This Matrix string theory was first proposed by Luboš Motl in 1997 [1] and later independently in a more complete paper by Robbert Dijkgraaf, Erik Verlinde, and Herman Verlinde [2]. Another matrix string theory equivalent to Type IIB string theory was constructed in 1996 by Ishibashi, Kawai, Kitazawa and Tsuchiya [3]. This version is known as the IKKT matrix model."
  20. Jul 26, 2011 #19
    It's really a third topic, but I also want to mention in this thread Stephen Adler's "trace dynamics", which is supposed to be a subquantum matrix mechanics which reduces to quantum mechanics in a thermodynamic limit. Adler is one of those eminent physicists who, late in his career, has taken on the challenge of deriving quantum mechanics from something deeper. According to chapter 8 of his http://arxiv.org/abs/hep-ph/0505177" [Broken], this arose from his work on quaternionic quantum mechanics, and the difficulties he had in extending this to quaternionic quantum field theory. He couldn't get the usual framework of quantum operator equations to work, and instead ended up inventing his new matrix mechanics.

    Here I might quote a comment of mine from the discussion with Lubos:

    "We can describe the universe by a matrix with D-branes on the main diagonal and open-strings off the diagonal. We can put the state of the universe into approximately block-diagonal form. Black holes are blocks in which diagonal and off-diagonal entries are in equilibrium. When matter outside a black hole falls in, the matrix elements connecting the matter block to the black hole also enter equilibrium, and the black hole grows. Finally, for the gravitational interaction between two objects outside a black hole, we integrate out the matrix elements connecting their blocks (the open strings)."

    Lubos agreed that this describes, in a quite standard way, how one recovers the space-time description of the world from Matrix theory. The key insight is that the matrix entries on the diagonal can take the values of space-time coordinates, so if a matrix element on the diagonal had, as its expectation value, some vector "x", you could interpret that as a position vector, and hence as the existence of an object at position "x". Off-diagonal entries between two such diagonal elements then correspond to something connecting those two space-time points, and in the black hole, where the off-diagonal elements become as "heavy" as the on-diagonal elements, space is effectively becoming noncommutative; you need the full matrix, and not just the diagonal, to see what's going on.

    Lubos, of course, dissents from Erik Verlinde's idea, which I further summarized as follows:

    "I then understand Erik to be saying two things. First, the growth of a black hole is an entropic process, occurring in a phase space of volume exp(A/4G). Second, the ordinary gravitational interaction is not an entropic force, but the matter degrees of freedom on the diagonal do still "feel" the volume of the off-diagonal phase space, through something like a Berry phase, and this determines the effective form of the interaction. (A/4G)v is Erik's guess as to the log of this volume. (v must be dimensionless, i.e. it's v/c.) A horizon forms when v=c, and the diagonal degrees of freedom go into equilibrium with the off-diagonal degrees of freedom."

    Lubos wrote:

    "You can't "feel" the entropy of states that have nothing to do with states in which you are in, and even if you interpreted this bizarre statement in some way - like counting intermediate states that may be exchanged - the force wouldn't be linear in entropy or its derivatives in any sense. It could be proportional to the number of microstates, but there would also be couplings etc.

    "... try to formulate any actual formula that would link the force in a matrix model to the entropy of anything or its derivative, and it will take a few seconds to show that this formula is wrong."

    I can't yet judge for myself, but there is a good chance that this is true, because - if you listen to Verlinde's discussion with his audience, especially at Perimeter Institute - he says that while he's using Matrix theory to communicate his ideas, he's skeptical that it is the full story. So Verlinde may also be sensing these technical difficulties for his concept, but he interprets these difficulties oppositely to Lubos - as a sign that there's a deeper theory.

    So I'm wondering if some version of Verlinde's idea can be realized in Adler's trace dynamics.

    I will mention one other reason why I'm interested in trace dynamics - and this is even further afield from the topic of this thread - and that is its origin in the study of a higher division algebras. There is a long and ongoing history of attempts to link quaternions and octonions (and even sedenions, a 16-dimensional nondistributive algebra) to physics, in various ways. For example, there was an attempt to explain quark confinement by attaching the quarks to the nonassociative part of the state in an octonionic quantum mechanics, which would be unobservable, there were attempts to realize "hadronic supersymmetry" using octonions, and so on. It's one of those mathematical tangents which isn't quite mainstream but which at least has some mathematical substance to it. It would be quite interesting if some form of octonionic trace dynamics explained quantum mechanics, the standard model, and the dark forces all at once. :-)
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