Is Verlinde's Gravity Theory Based on Relativistic Assumptions?

[S.Daedalus]

In the experiment as above the neutron is moving toward its equilibrium, not the equilibrium of the Earth. The equilibrium is when the entropy increases, I think.

I think it's more accurate to say towards the equilibrium of the Earth-neutron system.

1. The correct analog model in my opinion would be the one with the state of particle N is determined. In this case the number of microstates would be (N-1)! The entropy then would be the entropy of (N-1) particles + the entropy of particle N, providing it is in mixed state (if in pure state then the entropy is 0). this picture is analog to the one with screen at x and neutron at x+\delta x, because screen at x has no information about neutron. Do you agree with this or not?

My particles in the toy model are classical objects, like silver pearls or something. They can't be in a mixed state; nevertheless, the entropy of the whole system can't be changed due to the insertion of a screen, as, ex hypothesi, the screen does not change the physics (else, holographic and 'regular' descriptions would not be equivalent). Thus, if the entropy of the system of N particles is ~ ln(N!), after replacing N - 1 particles with a screen, the entropy of the whole system will still be at that value, but that does not mean that particle N suddenly has acquired an entropy of ~ ln(N) -- it can't!

3. Screen at x+\delta x and neutron+screen at x defines the same physical system and if you equate the entropies you will find that neutron have an x-dependent entropy. In your toy model this necessarily means that particle N is described by the mixed state.

Which it can't, thus showing that this line of reasoning yields a contradiction. For a system to have a certain entropy, it is not necessary for each of its components to carry a fraction of that entropy! If you grant me that, then it immediately follows that the entropy of the screen at x + \delta x/the entropy of the system Earth + neutron does not necessarily have to be decomposable into the entropy of the screen at x plus the entropy of the neutron.

Yes, the holographic+entropic formulation FUNDAMENTALLY differs from the standard theory. There is no limit which takes the entropic formulation of gravity into the standard potential formulation and vice versa.

Well, that's somewhat besides the point, but at least Bee Hossenfelder begs to differ, claiming that both formulations are actually fully equivalent. (http://arxiv.org/abs/1003.1015)

These are not just states of neutron (free neutron), but states of neutron in the gravitational field of Earth, that is to say, they actually describe neutron-Earth system.

They are states of the neutron in a classical potential, which happens to be the gravitational potential of the Earth.

 

[qsa]

you guys are making a fundamental mistake. the entropy of verlinde has nothing to do with the statistical multi-particle entropy physics. black hole and the particle entropies are of an unkown microstate origin with conjecture of their values which are related to the energies. even unruh temp is not the usual one it has a different interpretation.that is all

 

[Fra]

It strikes me that a big problem is the understanding on the holographic principle. I think the notion that information about a black box is encoded on the surface of the box IMO signs the wrong way of seeing it. This is a typical established ignorance of how information is encoded. Usually the microstate; which is the context of the information, also contains information and this context is encoded on the other side of the screen, not ON the screen. The screen just enodes the state of the communication channels, or maybe equivalently the measurement "operators". The information involved in supporting and selecting these are somehow lost in the analysis. Extremely annoying!

But I realize that there is no point in arguing over this here, a lot of work would have to be put down to explain this clear enough to make those who are subject to this criqitue see that it's wrong.

/Fredrik

 

[CHIKO-2010]

Dear S.Daedalus, I think we are going on a circle...You are ignoring the key points in my replies and instead arguing about inessential wordings. Here we go:

My particles in the toy model are classical objects, like silver pearls or something. They can't be in a mixed state; nevertheless, the entropy of the whole system can't be changed due to the insertion of a screen, as, ex hypothesi, the screen does not change the physics (else, holographic and 'regular' descriptions would not be equivalent). Thus, if the entropy of the system of N particles is ~ ln(N!), after replacing N - 1 particles with a screen, the entropy of the whole system will still be at that value, but that does not mean that particle N suddenly has acquired an entropy of ~ ln(N) -- it can't!

Which it can't, thus showing that this line of reasoning yields a contradiction. For a system to have a certain entropy, it is not necessary for each of its components to carry a fraction of that entropy! If you grant me that, then it immediately follows that the entropy of the screen at x + \delta x/the entropy of the system Earth + neutron does not necessarily have to be decomposable into the entropy of the screen at x plus the entropy of the neutron.

The whole point in my numerous replies concerning your toy model was to show that your toy model is inadequate as a counterargument. It looks like I am talking about apples and you are keep saying that the orange is orange.

yes I understand that your example was totally classical, I just wanted to argue that it is not correct. Again, if you consider your toy model with N-1 particles and particle N with determined position and momentum the entropy of such system would be proportional to ln(N-1)! NOT to ln(N!). That is total entropy of the system is an entropy of the particle N (which is in your classical case is 0) and the entropy of (N-1) particles.

Again, the system neutron at x+\delta x and the screen at x has an entropy which is a sum of neutron's entropy and the entropy of the screen. This is simply because screen at x has nothing to do with the neutron at x+\delta x. It seems you are PURPOSELY IGNORING this part of my reasoning. The rest is written in my previous posts.

Well, that's somewhat besides the point, but at least Bee Hossenfelder begs to differ, claiming that both formulations are actually fully equivalent. (http://arxiv.org/abs/1003.1015)

Look, you have raised the question in your previous post and I have answered. Now your are claiming that this is not the point. if you believe that Verlinde's description of gravity and the standard description are equivalent than we can stop our discussion here. Judging from Hossenfelde's paper, I can say that she has no clue what Verlinde's theory is about.

They are states of the neutron in a classical potential, which happens to be the gravitational potential of the Earth.

And so what? Are you suggesting to describe Earth as a quantum mechanical particle? Fortunately for all of us it is classical with very high accuracy.

 

[CHIKO-2010]

you guys are making a fundamental mistake. the entropy of verlinde has nothing to do with the statistical multi-particle entropy physics. black hole and the particle entropies are of an unkown microstate origin with conjecture of their values which are related to the energies. even unruh temp is not the usual one it has a different interpretation.that is all

I totally agree with you. We were just trying to argue from the perspectives of the standard statistical system. My point is that the toy model suggested by S.Daedalus is not adequate, it is nor analogous to the physics of Verlinde's gravity

 

[S.Daedalus]

Again, the system neutron at x+\delta x and the screen at x has an entropy which is a sum of neutron's entropy and the entropy of the screen. This is simply because screen at x has nothing to do with the neutron at x+\delta x. It seems you are PURPOSELY IGNORING this part of my reasoning.

But that's the very bit of reasoning all of my past posts have been about! To show that, in the toy model, you can apply that same bit of reasoning, and arrive at a wrong conclusion, and hence, it is not sound in any other context. What you're saying amounts to a claim that the presence of a holographic screen changes the physics, i.e. makes it impossible for the neutron to be in a pure state; however, this runs counter to the idea of holography, in which the holographic description is exactly equivalent to the ordinary 3D one.

That the entropy of a system needs to be evenly distributed over its parts just isn't so. An example is the expansion of a gas cloud, where all of the particles could be in an initially known state, and from there, evolve unitarily, while the entropy of the gas as a whole increases. Your reasoning would have all of the gas particles evolve non-unitarily to increase the entropy, i.e. increase 'microscopic' entropy in order for 'macroscopic' entropy to rise as well -- which is just a level confusion, and that same level confusion is at work when you claim that in order for the total entropy of the system neutron + screen at x to be greater than the entropy of just the screen at x, the neutron must have a non-zero entropy.

You might perhaps argue that the neutron microstates and the screen microstates constitute different 'species' in some sense, but the fact that you can replace the entire system by a screen at x + dx that equivalently describes the same physical situation and on which all the microstates are indistinguishable shows this not to be so. I mean, how is this supposed to work anyway -- in a non-holographic setting, you agree that the neutron may be in a pure state, right? Then, going to a holographic description (screen at x), suddenly the neutron is forced to be in a mixed state. However, in a different holographic description (screen at x + dx), the (holographic 'image' of) the neutron can again be in a pure state? This doesn't make sense, at least not to me.

Judging from Hossenfelde's paper, I can say that she has no clue what Verlinde's theory is about.

Just out of curiosity, what specifically do you disagree with? Apparently, she's been in contact with Verlinde, who helped her with some clarifications.

And so what? Are you suggesting to describe Earth as a quantum mechanical particle? Fortunately for all of us it is classical with very high accuracy.

Well, I certainly don't disagree with that, and share your delight in not having to worry about quantum effects in everyday life, but I found your claim that the states under discussion actually describe the neutron-Earth system to be rather bizarre. Certainly, \langle A \rangle = \texttt{Tr}(A \rho_N) gives you the expectation value of an observable A in a measurement performed on the neutron, no?

 

[CHIKO-2010]

But that's the very bit of reasoning all of my past posts have been about! To show that, in the toy model, you can apply that same bit of reasoning, and arrive at a wrong conclusion, and hence, it is not sound in any other context. What you're saying amounts to a claim that the presence of a holographic screen changes the physics, i.e. makes it impossible for the neutron to be in a pure state; however, this runs counter to the idea of holography, in which the holographic description is exactly equivalent to the ordinary 3D one.

Its not just a presence of the holographic screen that makes neutron in gravitational field of Earth to be in mixed state, it is so because the very origin of gravitational interactions is entropic! Your toy model with non-interacting indistinguishable particles does not actually models physics of neutron-earth system within the Verlinde approach. Therefore, conclision you draw from the wrong toy model CANNOT be considered as a counterargument. Please, if you want to continue this discussion tell which of the points (1,2,3) of my reasoning you disagree with and why.

That the entropy of a system needs to be evenly distributed over its parts just isn't so.

? I never said that. What I have said is that if you have two subsystems A and B with entropies S_{A} and S_{B} the entropy of a whole system is S_{A}+S_{B}

An example is the expansion of a gas cloud, where all of the particles could be in an initially known state, and from there, evolve unitarily, while the entropy of the gas as a whole increases.

Nonsense. A gas of particles where you determine states of all individual particles has entropy = 0. Then, if you can trace unitary evolution of all the individual particles, the states of each particles will be uniquely defined at each given moment of time. Therefore, the entropy of your system will stay 0.

Your reasoning would have all of the gas particles evolve non-unitarily to increase the entropy, i.e. increase 'microscopic' entropy in order for 'macroscopic' entropy to rise as well -- which is just a level confusion, and that same level confusion is at work when you claim that in order for the total entropy of the system neutron + screen at x to be greater than the entropy of just the screen at x, the neutron must have a non-zero entropy.

No, not correct again. My claim is that a neutron interacting with Earth in the Verlide's theory carries an entropy which changes with its position relative to earth. Free, non-interacting neutrons are in pure states, of course.

You might perhaps argue that the neutron microstates and the screen microstates constitute different 'species' in some sense, but the fact that you can replace the entire system by a screen at x + dx that equivalently describes the same physical situation and on which all the microstates are indistinguishable shows this not to be so. I mean, how is this supposed to work anyway -- in a non-holographic setting, you agree that the neutron may be in a pure state, right? Then, going to a holographic description (screen at x), suddenly the neutron is forced to be in a mixed state. However, in a different holographic description (screen at x + dx), the (holographic 'image' of) the neutron can again be in a pure state? This doesn't make sense, at least not to me.

Look, I gave you an example within you beloved toy model. If you have a gas of N-1 particles and an isolated particle N which position and momentum you know, is this particle N distinguishable from the rest N-1 particles? Of course it is! If you mix all the particles, that is you do not know the position and momentum of particle N, than yes, those particles are indistinguishable, the entropy in this case increases. I do not understand why it is so hard for you to digest this rather simple picture.

Coming back to Verlinde (I am reapiting this again): A system of neutron at x+\delta x and a screen at x has an entropy S_{neutron}+S_{screen}(x) because these are independent subsystems. The same entropy must have a screen at x+\delta x, S_{screen}(x+\delta x). Then it follows that S_{neutron} is proportional to the gradient of the screen entropy. This gradient is NOT zero because it is the source of gravitation!

In your toy model: the entropy of N (with defined momentum and position) is 0 and the entropy of a gas of identical particles is ~ln((N-1)!). hence the total entropy is a sum of S_N=0 and S_{N-1}~ln((N-1)!). If I do not measure the position and momentum of particle N, the entropy becomes ~ln(N!) NOt equal to the previous entropy. How this two physical examples can be analogous to each other? (BTW, the entropy in normal understanding has zero spatial gradient, the position of a whole system is irrelevant).

Just out of curiosity, what specifically do you disagree with? Apparently, she's been in contact with Verlinde, who helped her with some clarifications.

Bee's understanding of equivalence of two physical theories is just wrong. The main her argument seems is based on the fact that if you can read equations from left to right you will be able to read them from right to left. Yes, of course, but in physics we always have the basics, "fundamentals" of a theory, and theories are differ because of those basic assumptions/conjectures are different. If two theories differ on the 'fundamental' level then although they may simultaneously describe some of the phenomena, they will have different predictions concerning for others. So let me trace down the difference between the standard potential and entropic approach to gravitation:

1. The starting macroscopic law is the Newtonian gravitational force law

The standard approach:

2. the force is described by the gradient of a potential field, which is defined in space (a function of space coordinates that satisfy Laplace's equation with certain boundary conditions)

3. relativistic generalization takes this potential field to the tensorial field

4. quantum generalization takes tensorial field to a quantized field which gives the notion of spin-2 particles. This microscopic particles do propagate in space and time.

Conclusion: Thus, microscopic description of Newtonian force law (the Newtonian potential) is determined by an appropriate limit (non-relativistic limit) of the exchange of virtual spin-2 particles between gravitating objects.

The entropic approach:

2. There force is described by the gradient of an entropy of some holographic screens. The space is not defined as the fundamental object, x is just a macroscopic parameter characterizing states on the holographic screen.

3. relativistic generalization seems to be possible, but it is in no (obvious) way is related to the microscopic description of the theory. you can formally define the gravitational potential but it is not a primary construct but rather is defined through the temperature and entropy of the screen.

4. the full microscopic description is not known, however, the basic thing is that at microscopic level no notion of space exist. Therefore, certainly there is no notion of quantized field, gravitons etc.

Obviously, these theories are fundamentally different, and cannot be claimed to be physically equivalent, although for macroscopic bodies they both reproduce Newton's force law.

Well, I certainly don't disagree with that, and share your delight in not having to worry about quantum effects in everyday life, but I found your claim that the states under discussion actually describe the neutron-Earth system to be rather bizarre. Certainly, \langle A \rangle = \texttt{Tr}(A \rho_N) gives you the expectation value of an observable A in a measurement performed on the neutron, no?

Yes, measurements are performed on a neutron, but they describe not just a neutron (free states) but interacting neutron sates.

 

[S.Daedalus]

Please, if you want to continue this discussion tell which of the points (1,2,3) of my reasoning you disagree with and why.

Your argument essentially boils down to: the complete system (i.e. 'screen at x + dx', 'neutron + Earth', etc.) has a certain entropy S. One of its subsystems ('screen at x', Earth...) has entropy S' < S. Hence, the other subsystem must have entropy S - S'. This, in general, isn't right (or rather, is only right under certain assumptions), independently of the microscopic details of the model.

? I never said that. What I have said is that if you have two subsystems A and B with entropies S_{A} and S_{B} the entropy of a whole system is S_{A}+S_{B}

There's the problem. This is only true if the two systems are composed of microscopically different species, or are isolated from one another. Think of two systems with N! and M! microstates respectively; that their entropies add is only the case if the system formed from their combination has a total of N!*M! microstates, which is only right if either both systems are still isolated from one another, or they are composed of microscopically distinguishable objects. For instance, if I bring a second string of M pearls to the toy model and adjoin it to the first one, the combined system has now (N + M)! microstates; thus, the entropy of the combined system is greater than the sum of the entropies of both original subsystems.

That the entropy at x + dx is the sum of the entropies at x plus the neutron entropy then only follows if you assume that either the system at x and the neutron are isolated from one another (in which case, how could there be any interaction between them?), or that there are in some sense different 'species' of microstates present. Neither of those assumptions has any good reason to hold.

Perhaps it's easier if we for the moment forget all about screens, and just consider holography as providing a bound on the maximal amount of entropy that can be 'stored' within a given volume, which happens to be proportional to the surface area of its boundary. Verlinde's argumentation can be exactly replicated in this setting (it must, since it's really the same setting looked at differently), only the entropy considered here is 'really' the entropy of a given volume. In this case, all that happens is that the entropy of the volume bounded at x + dx increases due to the 'mixing' of microstates, without any non-unitarity anywhere (just as the entropy of two systems in contact increases without any non-unitarity). This doesn't (and can't) change whether or not you describe it with screens at x and a neutron, at x + dx, or without any screens at all -- holography in this view is just the condition that you can't bunch more than a specific number of microstates in a given volume which is bounded by black hole formation.

Nonsense. A gas of particles where you determine states of all individual particles has entropy = 0. Then, if you can trace unitary evolution of all the individual particles, the states of each particles will be uniquely defined at each given moment of time. Therefore, the entropy of your system will stay 0.

I didn't say anything about tracing the evolution, I said that underneath everything, the evolution is unitary, even though the entropy of a gas increases -- we may not know the microscopic evolution precisely, which is why, after having let the system evolve for some time, we'd have to resort to a statistical, mixed-state description of any of its constituents, but that doesn't preclude an underlying, deterministic, reversible microdynamics; however, your argument would suggest that the fundamental microdynamics actually are irreversible and non-unitary.

Yes, of course, but in physics we always have the basics, "fundamentals" of a theory, and theories are differ because of those basic assumptions/conjectures are different.

Hmm. I don't think I'd agree with that. If you have two theories, one phrased in terms of interactions of bloops, and one in terms of interactions of floops, and both yield the same physical predictions, i.e. they can't be distinguished by experiment, I'd consider those theories equivalent. That's the whole basis of dualities, after all -- in AdS/CFT the theories don't even agree on something as fundamental as the number of space-time dimensions, and yet, their physical content is the same. It's the difference between (naive) scientific realism -- broadly, the stance that the fundamental constituents of your theory are in one to one agreement with the fundamental constituents of reality -- and instrumentalism, which basically states that a scientific theory is good if it predicts observations with high accuracy, while not committing to any specific interpretation of its fundamental elements.

If two theories differ on the 'fundamental' level then although they may simultaneously describe some of the phenomena, they will have different predictions concerning for others.

Again, I think this is invalidated by any number of dualities, such as AdS/CFT, or the various string theory dualities: those theories differ on a fundamental level, but agree on any predictions.

Obviously, these theories are fundamentally different, and cannot be claimed to be physically equivalent, although for macroscopic bodies they both reproduce Newton's force law.

Well, in all fairness, I think Bee really only talked about the equivalence between Newtonian and entropic gravity, only lightly touching on the GR level. And on this level, the equivalence is exact: everything predicted by the Newtonian theory is equivalently predicted by the entropic one; that's essentially by design. If our world were fully classical, i.e. if there were no QM or GR, one theory could at all points be exchanged for the other, they'd be fully dual.

Yes, measurements are performed on a neutron, but they describe not just a neutron (free states) but interacting neutron sates.

Well, interacting neutron states are, to me, still neutron states; they're not states of the neutron-Earth system anymore than my state right now is a state of the me-Moon system. It wouldn't make sense to say, for example, 'the me-Moon system is wasting too much time on the internet', it'd be rather unfair to lay the blame for that on the Moon.

 

[CHIKO-2010]

Your argument essentially boils down to: the complete system (i.e. 'screen at x + dx', 'neutron + Earth', etc.) has a certain entropy S. One of its subsystems ('screen at x', Earth...) has entropy S' < S. Hence, the other subsystem must have entropy S - S'. This, in general, isn't right (or rather, is only right under certain assumptions), independently of the microscopic details of the model.

Your are wrong. Once you have identified subsystems and they have entropies S and S' total entropy of the system is equal S+S'.

There's the problem. This is only true if the two systems are composed of microscopically different species, or are isolated from one another. Think of two systems with N! and M! microstates respectively; that their entropies add is only the case if the system formed from their combination has a total of N!*M! microstates, which is only right if either both systems are still isolated from one another, or they are composed of microscopically distinguishable objects. For instance, if I bring a second string of M pearls to the toy model and adjoin it to the first one, the combined system has now (N + M)! microstates; thus, the entropy of the combined system is greater than the sum of the entropies of both original subsystems.

That the entropy at x + dx is the sum of the entropies at x plus the neutron entropy then only follows if you assume that either the system at x and the neutron are isolated from one another (in which case, how could there be any interaction between them?), or that there are in some sense different 'species' of microstates present. Neither of those assumptions has any good reason to hold.

Your seems read a half of my post regarding this. Didn't I argue all the time that neutron (x+\delta x) + screen at x are two subsystems? I gave you the similar to yours example within your toy model. Are you saying that you are not able to determine neutron states without measuring screen microstates?

Yes, neutron can be DISTINGUISHED from the screen that describes only earth, that is neutron can be recognized as being different from Earth I hope you do not doubt this

Perhaps it's easier if we for the moment forget all about screens, and just consider holography as providing a bound on the maximal amount of entropy that can be 'stored' within a given volume, which happens to be proportional to the surface area of its boundary. Verlinde's argumentation can be exactly replicated in this setting (it must, since it's really the same setting looked at differently), only the entropy considered here is 'really' the entropy of a given volume. In this case, all that happens is that the entropy of the volume bounded at x + dx increases due to the 'mixing' of microstates, without any non-unitarity anywhere (just as the entropy of two systems in contact increases without any non-unitarity). This doesn't (and can't) change whether or not you describe it with screens at x and a neutron, at x + dx, or without any screens at all -- holography in this view is just the condition that you can't bunch more than a specific number of microstates in a given volume which is bounded by black hole formation.

The whole my argumentation was based on holography. A system of neutron at x+deltax and screen at x are identifiable subsystems exactly because screen at x has nothing to do with the neutron. Your analogy with the black hole is also wrong: your certainly can consider a black hole and a bunch of particles far away from it -- the entropy of this system is a sum of black hole entropy and entropy of associated with particles. Further, precisely holography dictates to identify the entropy of the screen at x+\delta x (on which neutron is indistinguishable) and the entropy of neutron + screen.

Hmm. I don't think I'd agree with that. If you have two theories, one phrased in terms of interactions of bloops, and one in terms of interactions of floops, and both yield the same physical predictions, i.e. they can't be distinguished by experiment, I'd consider those theories equivalent. That's the whole basis of dualities, after all -- in AdS/CFT the theories don't even agree on something as fundamental as the number of space-time dimensions, and yet, their physical content is the same. It's the difference between (naive) scientific realism -- broadly, the stance that the fundamental constituents of your theory are in one to one agreement with the fundamental constituents of reality -- and instrumentalism, which basically states that a scientific theory is good if it predicts observations with high accuracy, while not committing to any specific interpretation of its fundamental elements.

Again, I think this is invalidated by any number of dualities, such as AdS/CFT, or the various string theory dualities: those theories differ on a fundamental level, but agree on any predictions.

Sadly, but it seems you can't see what is more fundamental a house or a brick. The fundamental theoretical basis for ADS/CFT and various dualities is string theory. Yes, currently you have different formulations of string theory, however dualities are exactly pointing towards a unified description (M-theory?).

If you can find such a duality between microscopic description of Verlinde's gravity and standard approach I will agree that this two theories are just different formulations. However, I believe at microscopic level this theories are different (the paper we are discussing shows exactly this), and no such a duality is possible.

Well, in all fairness, I think Bee really only talked about the equivalence between Newtonian and entropic gravity, only lightly touching on the GR level. And on this level, the equivalence is exact: everything predicted by the Newtonian theory is equivalently predicted by the entropic one; that's essentially by design. If our world were fully classical, i.e. if there were no QM or GR, one theory could at all points be exchanged for the other, they'd be fully dual.

Bee simply reverted Verlinde's equations, this is not a proof of equivalence. Verlinde at least tries to justify why something he denotes by letter S must be called entropy, and something denoted by letter T is called the temperature. Can you tell me what the quantities ad hoc defined in Eqs (1) and (2) of Bee's paper have to do with entropy and temperature, other than that they are denoted by letters S and T? Yes they are constructed in such a way to
reproduce black hole entropy and temperature, but what black holes has to do with Newton's law? I do not think you can show that the gravity is indeed an entropic force without all ingredients provided by Verlinde.

Moreover, I believe that not everything predicted by usual approach to Newton's gravity is reproduced by entropic gravity -- predictions for neutron interference and neutron bound states within entropic gravity are in contradiction with observations.

Well, interacting neutron states are, to me, still neutron states; they're not states of the neutron-Earth system anymore than my state right now is a state of the me-Moon system. It wouldn't make sense to say, for example, 'the me-Moon system is wasting too much time on the internet', it'd be rather unfair to lay the blame for that on the Moon.

Ok, I think you do understand what I meant. Hopefully your state is still under the influence of Earth gravitational field (moon and other celestial objects you can safely ignore)

 

[S.Daedalus]

Your seems read a half of my post regarding this. Didn't I argue all the time that neutron (x+\delta x) + screen at x are two subsystems? I gave you the similar to yours example within your toy model. Are you saying that you are not able to determine neutron states without measuring screen microstates?

No. I think at bottom all I'm really saying is that there are many ways to break down the system 'screen at x + dx' into 'screen at x + neutron', which are all physically equivalent, corresponding to the many ways you can break down my N-particle system into one K-particle subsystem and one N - K particle subsystem: Let's associate the K particle system with the neutron, and the N - K particle system with the screen at x. Now, the screen has (N - K)! microstates; going by your argument, the neutron then must have N!/(N - K)! microstates, in order for the entropies to add up properly. However, the 'neutron' really only has K! microstates -- that's because there are \tbinom{N} {K} ways of choosing a K particle subsystem. With this missing factor, we get for the number of microstates on the screen at x + dx \tbinom{N} {K} K!(N - K)! = N!, which is the right answer.

The whole my argumentation was based on holography. A system of neutron at x+deltax and screen at x are identifiable subsystems exactly because screen at x has nothing to do with the neutron. Your analogy with the black hole is also wrong: your certainly can consider a black hole and a bunch of particles far away from it -- the entropy of this system is a sum of black hole entropy and entropy of associated with particles. Further, precisely holography dictates to identify the entropy of the screen at x+\delta x (on which neutron is indistinguishable) and the entropy of neutron + screen.

I think there's been a misunderstanding here. An area-entropy bound and holography aren't two different things -- holography just says that there's a maximum amount of entropy you can cram into a given volume (proportional to its boundary area), and if that bound is saturated, you end up with a black hole, whose entropy you can't increase and have the system stay the same size; it will invariably grow. Sure you can consider a black hole plus some particles, but this system will be larger than just the black hole alone, and in fact, the size of the black hole after you have thrown in the particles is the minimum size for this system -- this is just the generalized second law.

Sadly, but it seems you can't see what is more fundamental a house or a brick. The fundamental theoretical basis for ADS/CFT and various dualities is string theory. Yes, currently you have different formulations of string theory, however dualities are exactly pointing towards a unified description (M-theory?).

You're right, if it exists, M-theory provides such a description for the various string theories, but AdS/CFT is a somewhat different beast; the conformal field theory on the boundary is no string theory, but just an ordinary QFT. Yet, both describe the same physics equally well, despite differing in their fundamental constituents. (I hope you forgive me if I snip the discussion here; these posts are getting too sprawling to handle, and I think we're getting carried too far afield...)

 

[CHIKO-2010]

No. I think at bottom all I'm really saying is that there are many ways to break down the system 'screen at x + dx' into 'screen at x + neutron', which are all physically equivalent, corresponding to the many ways you can break down my N-particle system into one K-particle subsystem and one N - K particle subsystem: Let's associate the K particle system with the neutron, and the N - K particle system with the screen at x. Now, the screen has (N - K)! microstates; going by your argument, the neutron then must have N!/(N - K)! microstates, in order for the entropies to add up properly. However, the 'neutron' really only has K! microstates -- that's because there are \tbinom{N} {K} ways of choosing a K particle subsystem. With this missing factor, we get for the number of microstates on the screen at x + dx \tbinom{N} {K} K!(N - K)! = N!, which is the right answer.

You are again using incorrect analogies. Instead, I ask you again, please answer which of the statements below is incorrect and why:

1. Consider a particle at x+\delta x outside of Verlinde's screen at x. Because the screen at x cannot account for microstates of a particle the total entropy will be: S_particle(x+\delta x) + S_screen(x)

2. The above total entropy is equal to the entropy of the screen at x+\delta x

I think there's been a misunderstanding here. An area-entropy bound and holography aren't two different things -- holography just says that there's a maximum amount of entropy you can cram into a given volume (proportional to its boundary area), and if that bound is saturated, you end up with a black hole, whose entropy you can't increase and have the system stay the same size; it will invariably grow. Sure you can consider a black hole plus some particles, but this system will be larger than just the black hole alone, and in fact, the size of the black hole after you have thrown in the particles is the minimum size for this system -- this is just the generalized second law.

I do agree with the above statements. The only thing is that i do not understand why did you bother writing all this. You certainly misunderstood my previous msg. The only thing I wanted to say is that the black hole entropy accounts for the microstates that are 'hidden' behind the horizon, and similarly, screen at x accounts for microstates that is fitted in the volume surrounded by the screen. Similar to black hole physics, where the asymptotic observer sees thermal (dirty) radiation away from black hole, neutron away from the screen is described by 'dirty' (mixed) state.

You're right, if it exists, M-theory provides such a description for the various string theories, but AdS/CFT is a somewhat different beast; the conformal field theory on the boundary is no string theory, but just an ordinary QFT. Yet, both describe the same physics equally well, despite differing in their fundamental constituents. (I hope you forgive me if I snip the discussion here; these posts are getting too sprawling to handle, and I think we're getting carried too far afield...)

Dear S.Daedalus, I would not argue anymore with you on this. You seems do not want to understand what I am saying. I said that if you can find duality/equivalence on the fundamental level, yes i will agree with you.

A reliable theoretical evidence for the ADS/CFT correspondence is known only within string theory. this correspondence indicates that there is equivalencee between string theory on ADS background and boundary CFT. Similarly to prove equivalence of Verlinde's entropic gravity and standard approach, you have to find correspondence between Verlinde's space-less microscopic description and say quantized linearized theory of spin-2 field. Yes, both of these theories reproduce Newton's force law, but they disagree on other things.

 

[S.Daedalus]

You are again using incorrect analogies. Instead, I ask you again, please answer which of the statements below is incorrect and why:

1. Consider a particle at x+\delta x outside of Verlinde's screen at x. Because the screen at x cannot account for microstates of a particle the total entropy will be: S_particle(x+\delta x) + S_screen(x)

2. The above total entropy is equal to the entropy of the screen at x+\delta x

I really don't know why you want me to answer this again and again. But fine, once more: the entropy of a system is not necessarily the sum of the entropy of its subsystems, so it doesn't follow that simply because you can partition the system 'screen at x + dx' into the subsystems 'screen at x' and 'particle at x + dx', that the entropy of the screen at x + dx is given by the sum of the subsystem-entropies. If that were the case, then, for instance, in an expanding cloud of gas, the evolution of its microscopic constituents would necessarily be non-unitary, and thus, not be described by ordinary quantum mechanics.

I have illustrated this using my toy model, where it is very easy to see that you can partition it into subsystems, the sum of whose entropies is far less than the entropy of the total system. The reason for this is that there is an ambiguity in how to partition the total system, leading to a class of partitions that yield identical physics. Take, as a relevant example, partitions of a system of mass M into subsystems of mass m and (M - m). In the toy model, this corresponds to the example I gave in my last post, and hence, to \tbinom {N} {K} physically indistinguishable situations.

That same ambiguity may exist in partitioning the screen at x + dx into the subsystems 'screen at x' and 'particle at x + dx'; i.e. there is a number of ways to divide up the microstates of the screen that result in identical physical situations, in which case, S_{screen}(x) + S_{particle}(x + \delta x) \neq S_{screen} (x + \delta x). Your argument doesn't exclude, or even address, this issue, and hence, isn't sound.

 

[CHIKO-2010]

That same ambiguity may exist in partitioning the screen at x + dx into the subsystems 'screen at x' and 'particle at x + dx'; i.e. there is a number of ways to divide up the microstates of the screen that result in identical physical situations, in which case, S_{screen}(x) + S_{particle}(x + \delta x) \neq S_{screen} (x + \delta x). Your argument doesn't exclude, or even address, this issue, and hence, isn't sound.

Good to know that you agree with my statement # 1. That is, the entropy of particle + screen is S_particle(x+\deltax)+S_screen. The statement #2 follows from the fact that that screen at x+\delta x and particle + screen systems describe the same physical situation, namely, neutron in the gravitational field of Earth. That is, S_screen(x+\deltax)=S_particle(x+\deltax)+S_screen.

Note I do not disagree with your toy examples, I am just arguing that they are not adequately describe the physics we are discussing.

 

[S.Daedalus]

That is, the entropy of particle + screen is S_particle(x+\deltax)+S_screen.

Well, I certainly agree that the entropy of the particle plus the entropy of the screen is indeed the entropy of the particle plus the entropy of the screen. But it isn't (or not necessarily) the entropy of the whole system.

The statement #2 follows from the fact that that screen at x+\delta x and particle + screen systems describe the same physical situation, namely, neutron in the gravitational field of Earth.

That's true, but there is no reason that it should uniquely describe that physical situation, i.e. that there is just one way to partition the screen microstates into neutron and Earth microstates, as in my example where you get the same physical situation in a multitude of different ways.

Note I do not disagree with your toy examples, I am just arguing that they are not adequately describe the physics we are discussing.

They are not meant to describe the physics, they are meant as counterexamples to your logic, which continues to be: if you have a system with entropy S, and divide it into two subsystems, one of which has entropy S', the other must have entropy S - S'. Since one can show that this is not true for one specific system, it is not true in general.

 

[CHIKO-2010]

That's true, but there is no reason that it should uniquely describe that physical situation, i.e. that there is just one way to partition the screen microstates into neutron and Earth microstates, as in my example where you get the same physical situation in a multitude of different ways.

I think you must agree with #2 as well. If two systems (screen and screen+neutron) describe the same physics they have the same entropy (as well as other observables)!

I think I'll stop here. Was nice to discuss with you Verlinde's gravity. Thanks.