Is Verlinde's Gravity Theory Based on Relativistic Assumptions?

[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.