Is Verlinde's Gravity Theory Based on Relativistic Assumptions?

[S.Daedalus]

Don't you agree with this?

Well, I don't disagree, I'm just puzzled: if the neutron's entropy at height z is different from its entropy at height z + dz, and it propagates from z to z + dz, that must mean that it undergoes non-unitary evolution, it seems to me. Right?

But then, the solution to the modified quantum bouncer derived from this assumption, as shown in the paper, leads to real energy eigenvalues, and thus, unitarity. Those two statements seem to be at odds with each other, and I'm not sure how to resolve that tension.

As I've pointed out in my previous your analogy with the system 1 atom + bulk gas is wrong, simply because a subsystem consisting of 1 atom is not statistical, it does not make sense to talk about entropy of 1 atom.

Yes, pointing that out was the purpose of the analogy; the argument being, that in the system neutron + screen, it may make just as little sense to talk about the entropy (gain) of the neutron, which both Motl (in discussing the number of microstates available to the neutron rising) and the paper (in talking about the non-unitariness of the z-translation operator, or alternatively the different entropy of the neutron at different z's) seem to be doing. And if there's no entropy gain in the neutron, there doesn't seem to be a problem for Verlinde's reasoning -- there's no decoherence due to rising number of microstates, nor is there any need to modify the quantum bouncer.

 

[CHIKO-2010]

Yes, pointing that out was the purpose of the analogy; the argument being, that in the system neutron + screen, it may make just as little sense to talk about the entropy (gain) of the neutron, which both Motl (in discussing the number of microstates available to the neutron rising) and the paper (in talking about the non-unitariness of the z-translation operator, or alternatively the different entropy of the neutron at different z's) seem to be doing. And if there's no entropy gain in the neutron, there doesn't seem to be a problem for Verlinde's reasoning -- there's no decoherence due to rising number of microstates, nor is there any need to modify the quantum bouncer.

if you do not assume that the neutron carries entropy in quantum bouncer you will arrive at a wrong classical limit, which is approached as number of bounds states is large n (large z_n). In that limit neutron behaves as a classical particle which according to Verlinde must carry entropy (again see eq.3.14).

 

[S.Daedalus]

if you do not assume that the neutron carries entropy in quantum bouncer you will arrive at a wrong classical limit, which is approached as number of bounds states is large n (large z_n). In that limit neutron behaves as a classical particle which according to Verlinde must carry entropy (again see eq.3.14).

3.14 just describes the entropy gained by the screen as the particle merges with it; besides, it should be fine if the neutron carries some fixed entropy, it's just problematic if its entropy increases.

 

[czes]

The particle increases the entropy of 1 bit when it approaches at 1 Compton wavelength to the sphere.
Does it mean the entropy of the whole spherer = sum of the number of the wavelengthes of all particles inside the sphere ?
S=n R/l(c)

 

[CHIKO-2010]

3.14 just describes the entropy gained by the screen as the particle merges with it; besides, it should be fine if the neutron carries some fixed entropy, it's just problematic if its entropy increases.

Yes, but please also note that n bits in 3.14 specifically describes a particle of mass m on the screen (see, e.g., the discussion just above 3.14 and just below 3.15).

I think the fact that the entropy of neutron must be x-dependent is pretty clear from Verlinde's paper. Again, consider a holographic screen that suurounds a mass M, say at distance x from M. This microstates on this holographic screen carry information (entropy) concerning the object M. let us put now a test particle of mass m at a distance x+\delta x from M. The total entropy of test particle + screen is

S_{\particle}(x+\delta x) + S_{screen}(x)

which can be equated with the entropy os a screen at distance x+\delta x, that is screen with a test particle on it, S_{screen}(x+\delta x). Since \delta S_{sceen} is proportional to \delta x, S_{\particle} CAN NOT be x-independent. I think this is trivial.

The equation I have highlighted, assumes that neutron states and the microstates on the screen are uncorrelated which is perfectly OK, since the creen contains only the information in the surrounded volume. If you assume that neutron states are entangled with microstates on the screen, than you will get even in bigger troubles -- to describe a neutron in quantum bouncer you have to sum up screen microstates at each x. you will certainly get decohered picture.

 

[S.Daedalus]

I think this is trivial.

I may just be dense, but I don't see it. I agree that the entropy of the system screen + particle must be greater the closer the particle gets to the screen, but this only translates to an entropy increase in the particle if you assume irreversible, non-unitary dynamics, which I think is neither necessary, nor appropriate, if you want the particle's evolution to be describable by ordinary quantum mechanics.

The equation I have highlighted, assumes that neutron states and the microstates on the screen are uncorrelated which is perfectly OK, since the creen contains only the information in the surrounded volume.

What do you make of this quote from Verlinde's paper: "Eventually the particle merge [sic] with the microscopic degrees of freedom on the screen, but before it does so, it already influences the amount of information stored on the screen."?

Besides, assuming that the screen microstates and the neutron state are uncorrelated seems at odds with your proposal to replace the neutron by a screen containing it: for then, microstates on neither screen (not the Earth's nor the neutron's) would change with their respective position, and hence, the total entropy would be independent of location -- leading to no entropic gravity at all.

 

[czes]

I agree with CHIKO-2010 here.
Due to Holographic principle the screen contains only the information in the surrounded volume. When the particle is far away from the surface of the massive body you have 2 Horizon Events. One is of the massive body with a radius R where is higher entropy and another is of the particle where is the radius (R+x).

The maximum entropy is when the body collapses into a Black Hole. The lower entropy is for the body of radius R and the lowest is for the system with a distant particle (R+x).

The gravity as entropic force acts toward the higher entropy (the future Black Hole).

 

[CHIKO-2010]

What do you make of this quote from Verlinde's paper: "Eventually the particle merge [sic] with the microscopic degrees of freedom on the screen, but before it does so, it already influences the amount of information stored on the screen."?

Besides, assuming that the screen microstates and the neutron state are uncorrelated seems at odds with your proposal to replace the neutron by a screen containing it: for then, microstates on neither screen (not the Earth's nor the neutron's) would change with their respective position, and hence, the total entropy would be independent of location -- leading to no entropic gravity at all.

I do not see contradictions here -- holographic description of both masses m and M, being at "positions" x+\delta x and x=0, requires a screen at x+\delta x, so this screen has an entropy S(x+\delta x). Note that, space has not yet emerge for region < x+\delta x, so the position of m and M are encoded in microstates on the screen.

You perhaps did not noticed but there is NO total dependence of entropy S on x. The explicit dependence on x (gradient of S) is compensated by the change in the energy due to the work of an entropic force, that is total derivative of S wtr to x IS indeed 0. That is how the entropic force is defined in the first place!

 

[Fra]

The kind of analysis I would like to see to move forward is to define in terms of an inference abstraction, the notion that Verlinde thinks is "independent".

"Starting from first principles, using only space independent concepts like energy, entropy and temperature, it is shown..."

Energy, entropy and temperature are all different measures and their definition in terms of constructable measures are far from clear enough IMHO. In particular does it seem quite obvious that each of these measures are observer dependent, and the nature of a possible observer-covariant view is not clear either.

But of course you have to start somewhere and apply the admitted heuristic arguments, so did Verlinde.

In the end he notes

"This brings us to a somewhat subtle and not yet fully understood aspect. Namely, the
role of h-bar"

I think this is connected to how the measures are really somehow discrete.

In my opinion the weakest part of the whole argument isn't that the idea is all misguided, it's that it's heuristic and MIXING baggage notions the we well understand in classical setting, but not so in the general setting, with the holograpic conjecture which is also a bit unclear. I think he is not radical enough.

Can we reconsider how the measures energy, entropy and temperature are supposedly to be rationally constructed without relying on classical concepts, or fictious ensembles etc and instead only use the state of the observer as constructing tools and see how space and gravity is emergent along with the construction?

Since all horizons are observer dependent, that also seems to hint the duality that there may be two descriptions of the same thing, one with gravity one w/o.

Even in GR we have that. The free falling oberver does not see gravitation, it's just doing a random walk. So it seems clear that gravity seems simpler from the inside perspective. Only to an outside observer, does the mysterious gravity reveal itself. To the free fall observer it's just a random walk.

So it seems what we need to understand is why two observers, both doing a random walk - attract, right?

That is pretty close to asking, why two observers that are communicating, will have a tendency to negotiate agreements. And if space; is simply a measure of disagreement, then the connection is clear.

This is the obvious rational I see behind verlindes idea, but to make it clear, the notions of entropy, energy and temperature and the spacetime structure and the distance metric etc needs to be reconstructed.

/Fredrik

 

[czes]

So it seems what we need to understand is why two observers, both doing a random walk - attract, right?

That is pretty close to asking, why two observers that are communicating, will have a tendency to negotiate agreements. And if space; is simply a measure of disagreement, then the connection is clear.

This is the obvious rational I see behind verlindes idea, but to make it clear, the notions of entropy, energy and temperature and the spacetime structure and the distance metric etc needs to be reconstructed.
/Fredrik

I would like to refer everything from holographic point of view.
The ordinary hologram is made of the interfered waves of the coherent light rays. If we assume each that interference encodes a constant time dilation we get the space-time as in General Relativity. Each object will follow the curvature of that space. An inner observer is doing random walk in his space.

There are naturally more interferences and time dilations close to a massive object. The object absorbs more interferences toward the higher density closer to massive object and accelerates. it is the Unruh effect.

Therefore gravity may be shown as an entropic force (object moves toward the Event Horizon with higher entropy)
or
also as the result of the computer program where each point of interference has encoded a constant time dilation.

 

[Fra]

I would like to refer everything from holographic point of view.

From my perspective, a version of the holographic principle is seen as an equilibrium condition, and thus I can't accept it as a starting point for the reconstruction.

This doesn't mean I think the holographic connections is baloney. On the contrary, there is interesting logic there, but it's not a starting point for me, the understanding on that is deeply entangled with general theory scaling, and theory interactions. I think it's at that level we should take the stance.

In my perspective the holographic abstraction is best understood in terms of two interacting theories. When these two theories have establishd a stable communication channel, then each theory can describe the other theory via this channel, in the sense that they are "consistent". But when there is no communication channel, they are not consistent or can be said to encode each other. It's clear here that one can USE the holographic idea as a contraints to a process where communication channels are emergent, but the problem is that it's just an expectation, the generally can collapse, resulting in a revision.

/Fredrik

 

[czes]

Holographic principle is a new approach and has to be investigated carrefully and exact.
I noticed that Compton wave length and time have special meaning in calculation.
Verlinde wrote about Ccompton wavelength just close to event horizon. I think we can use it for another calculatios. May you have seen them on my website: www.hologram.glt.pl

 

[czes]

Holographic principle is a new approach and has to be investigated carrefully and exact.
I noticed that Compton wave length and time have special meaning in calculation.
Verlinde wrote about Ccompton wavelength just close to event horizon. I think we can use it for another calculatios. May you have seen them on my website: www.hologram.glt.pl

 

[S.Daedalus]

I do not see contradictions here -- holographic description of both masses m and M, being at "positions" x+\delta x and x=0, requires a screen at x+\delta x, so this screen has an entropy S(x+\delta x). Note that, space has not yet emerge for region < x+\delta x, so the position of m and M are encoded in microstates on the screen.

What does x = 0 mean if space has not emerged beyond x + dx?

Anyway, I think we're getting a bit tangled up here, and probably talk past each other a little. Perhaps we should go back to basics: does the entropy of a neutron falling in a gravity field change with position, or doesn't it? (If it does, why?)

 

[CHIKO-2010]

What does x = 0 mean if space has not emerged beyond x + dx?

Anyway, I think we're getting a bit tangled up here, and probably talk past each other a little. Perhaps we should go back to basics: does the entropy of a neutron falling in a gravity field change with position, or doesn't it? (If it does, why?)

The starting point is a screen with an entropy S(x), where x is some macroscopic parameter describing microstates on the screen. Another such a parameter is an energy. An object with mass M is described through the microstates on the screen around it. Then take a test particle (neutron) of mass m at x+\delta x, the entropy of the screen becomes S(x+\delta x). On the other hand, this is an entropy of the screen "placed" at x+\delta x, where the test particle is also described by some microstates on the screen. Integrating out those microstates gives back S(x). Now since the entropy is an additive quantity, S(x+\delta x)= S_{without neutron} (x+\delta x)+S_{neutron}(x+\delta x) = S(x)+S_{neutron}(x+\delta x). Therefore, S_{\neutron}=\delta S~\delta x and hence the neutron entropy depends on the distance from M. So, yes, the entropy of neutron falling in the gravitational field changes with the position of neutron.

 

[S.Daedalus]

The starting point is a screen with an entropy S(x), where x is some macroscopic parameter describing microstates on the screen. Another such a parameter is an energy. An object with mass M is described through the microstates on the screen around it. Then take a test particle (neutron) of mass m at x+\delta x, the entropy of the screen becomes S(x+\delta x). On the other hand, this is an entropy of the screen "placed" at x+\delta x, where the test particle is also described by some microstates on the screen. Integrating out those microstates gives back S(x). Now since the entropy is an additive quantity, S(x+\delta x)= S_{without neutron} (x+\delta x)+S_{neutron}(x+\delta x) = S(x)+S_{neutron}(x+\delta x). Therefore, S_{\neutron}=\delta S~\delta x and hence the neutron entropy depends on the distance from M. So, yes, the entropy of neutron falling in the gravitational field changes with the position of neutron.

The thing is, you can mirror this reasoning exactly for the example of the expanding gas cloud. The cloud consisting of N - 1 particles has a certain number of microstates, and hence, a certain entropy. Let's mentally draw a line around those N - 1 particles, leaving particle N out -- perhaps it's just a little bit further from the cloud's center than all of the others. The gas cloud including the vanguard particle N then has a higher entropy, corresponding to the additional microstates, i.e. the additional permutations of the N particles that lead to 'the same' gas cloud. One could similarly 'integrate out' the additional microstates conferred by adding the Nth particle, and get the entropy of the original gas cloud back. But that doesn't mean that the additional microstates are somehow intrinsic to particle N! (Though that is a logical possibility: one could add some object to the gas cloud that's distinct from the gas particles, and has 'internal' microstates corresponding to the difference between the microstates of the N and the (N - 1)-particle gas clouds.)

Thus, that the screen at x + dx has a greater number of microstates than the screen at x does not (necessarily) mean that these microstates are intrinsic to the neutron. Rather, it just means that there are a number of distinct screens that can be coarse-grained to obtain the screen at x -- that there are a number of distinct screens that describe the same physical situation, that of the neutron being at point x + dx. That's, I think, where the additional microstates reside.

 

[czes]

The most general interpretation of entropy is as a measure of our uncertainty about a system. The equilibrium state of a system maximizes the entropy because we have lost all information about the initial conditions except for the conserved variables; maximizing the entropy maximizes our ignorance about the details of the system.
http://en.wikipedia.org/wiki/Entropy

In a cloud of gas the motion of a particle is good defined when it is outside of the cloud (low entropy).
When it is inside the motion is not well defined and entropy is high.

 

[CHIKO-2010]

The thing is, you can mirror this reasoning exactly for the example of the expanding gas cloud. The cloud consisting of N - 1 particles has a certain number of microstates, and hence, a certain entropy. Let's mentally draw a line around those N - 1 particles, leaving particle N out -- perhaps it's just a little bit further from the cloud's center than all of the others. The gas cloud including the vanguard particle N then has a higher entropy, corresponding to the additional microstates, i.e. the additional permutations of the N particles that lead to 'the same' gas cloud. One could similarly 'integrate out' the additional microstates conferred by adding the Nth particle, and get the entropy of the original gas cloud back. But that doesn't mean that the additional microstates are somehow intrinsic to particle N! (Though that is a logical possibility: one could add some object to the gas cloud that's distinct from the gas particles, and has 'internal' microstates corresponding to the difference between the microstates of the N and the (N - 1)-particle gas clouds.)

Dear S.Daedalus, You have again failed to produce a correct analogy. A system of N-1 identical particles + one isolated particle cannot possibly have an entropy higher than a system of N identical particles. This is just wrong. I agree with czes on this. Besides, I do not quite understand how your previous post negates my argument.

 

[CHIKO-2010]

Thus, that the screen at x + dx has a greater number of microstates than the screen at x does not (necessarily) mean that these microstates are intrinsic to the neutron. Rather, it just means that there are a number of distinct screens that can be coarse-grained to obtain the screen at x -- that there are a number of distinct screens that describe the same physical situation, that of the neutron being at point x + dx. That's, I think, where the additional microstates reside.

No, it does mean precisely that certain microstates are intrinsic to the neutron, since if you remove neutron (take to infinity) the entropy of screens at ANY x will be the same. I do relevant to the problem idealization here, assuming that we have two-body problem at hand, neutron-Earth.

 

[S.Daedalus]

Dear S.Daedalus, You have again failed to produce a correct analogy. A system of N-1 identical particles + one isolated particle cannot possibly have an entropy higher than a system of N identical particles. This is just wrong. I agree with czes on this. Besides, I do not quite understand how your previous post negates my argument.

Huh? Where do you think I said this? I merely said that an N particle system has a greater entropy than the N - 1 particle system, but that this entropy increase does not come from additional entropy contained in the Nth particle, but rather, from new microstates opened up to the system as a whole.

 

[CHIKO-2010]

Huh? Where do you think I said this? I merely said that an N particle system has a greater entropy than the N - 1 particle system, but that this entropy increase does not come from additional entropy contained in the Nth particle, but rather, from new microstates opened up to the system as a whole.

Sorry, I indeed misunderstood your previous post on this point. I DO understand that there is no entropy associated with an isolated particle

However, the analogy your have drawn is still not adequate:

1. Whatever populates a holographic screen at x with an entropy S(x) it cannot describe the neutron at x+\delta x, since the entropy of the screen is a maximal entropy which can be "fitted" in a volume surrounded by the screen. This is in accord with the holographic principle -- e.g. black hole entropy is given by "tracing" microstates inside the black hole horizon.

Therefore your analogy with the gas of identical particles where one of the isolated particles are associated with the neutron is NOT correct. Microstates at the screen at x with entropy S(x) DO NOT now anything about the neutron at x+\delta x. Do you agree with this?

2. The neutron is described by the screen with an entropy S(x+\delta x), which can be viewed as the one placed at x+\delta x. Yes, on this screen neutron looses its individuality and is described by the microstates on the screen.

If you agree with the above, than it is easy to convince yourself that neutron does carry position dependent entropy, see one of the previous posts of mine.

 

[S.Daedalus]

Sorry, I indeed misunderstood your previous post on this point.

No harm done.

1. Whatever populates a holographic screen at x with an entropy S(x) it cannot describe the neutron at x+\delta x, since the entropy of the screen is a maximal entropy which can be "fitted" in a volume surrounded by the screen. This is in accord with the holographic principle -- e.g. black hole entropy is given by "tracing" microstates inside the black hole horizon.

Therefore your analogy with the gas of identical particles where one of the isolated particles are associated with the neutron is NOT correct. Microstates at the screen at x with entropy S(x) DO NOT now anything about the neutron at x+\delta x. Do you agree with this?

I do. I was (and to a certain extent still am) puzzled by some comments of Verlinde (like the one I quoted earlier) regarding this issue, but the matter is largely separate from the point I was trying to make in my last few posts.

2. The neutron is described by the screen with an entropy S(x+\delta x), which can be viewed as the one placed at x+\delta x. Yes, on this screen neutron looses its individuality and is described by the microstates on the screen.

This is, I think, where I disagree. The screen at x + dx does not merely describe the neutron, but the whole system (neutron + gravitating body, i.e. the screen at x). That this screen has additional microstates/entropy due to the presence of the neutron does not necessarily imply that these additional microstates are indeed microstates of the neutron.

Perhaps to make things a bit more clear, let's look at a toy model a bit more explicitly. Take N - 1 (N at this point would be cleaner, but I wish to keep consistency with previous posts) particles arranged on a one dimensional lattice, i.e. something like pearls on a string. This system has (N - 1)! microstates, corresponding to the number of permutations of the pearls. If a is the lattice spacing, we can 'replace' the system by a 'screen' at point x = (N - 1)a -- the screen here being pretty much a purely rhetorical device which we only need to make the parallel to the neutron + Earth case more obvious.

Now let's add an Nth particle at location Na. Clearly, the system formed by the N particles now has N! microstates. We then replace this system again by a 'screen' at Na, and are then, I think, in a position to exactly replicate your previous argumentation: We can 'coarse-grain' S_{Na} to obtain S_{(N - 1)a}, and hence, conclude that S_{particle N} = S_{Na} - S_{(N - 1)a} -- and in particular, that particle N has N 'internal' microstates, which it, of course, doesn't! Those microstates are only there because of the combination of particle N with the N - 1 others. Similarly, the screen at x + dx has its higher entropy not because of the entropy of the neutron, but because of the combination of the neutron and the Earth (i.e. the screen at x).

To be sure, it is possible to construct a system obeying these entropy relations in such a way -- instead of particle N being a pearl like all of the others, it could, for instance, be some object exhibiting an N-fold symmetry, such that all (N - 1)! permutations combined with N's N symmetry transformations yield again a physically indistinguishable situation; this is the possibility your argument stipulates. But it's not the only possibility, and, if you want quantum mechanics to be unitary, also not the favoured one.

So again, I can't see a reason for, in order to have the total entropy increase, the entropy of the neutron to increase.

(czes, by the way, I'm not ignoring you on purpose, however, I have a hard time figuring out what exactly you're arguing for/against. Maybe if you could clarify I can figure out what to reply to, and how...)

 

[jonnke123]

his theories are based on findings of surroundings not of sceinces not yet known

 

[jonnke123]

so the entropy of n is not co-herent with spatial constant as say {d=^n+4^} as to wit space and gravity have no constant except when in an osmostatic state

 

[CHIKO-2010]

I do. I was (and to a certain extent still am) puzzled by some comments of Verlinde (like the one I quoted earlier) regarding this issue, but the matter is largely separate from the point I was trying to make in my last few posts.

Ok, if you do agree than you must also agree that the entropy of a system screen at x and +
neutron at x+delta x must be S_screen(x)+S_neutron(x+\delta x). Is not it so?

This is not a separate point. when you equate the above entropy with the entropy of a screen at x+\delta x, S_screen(x+_{\delta x}), you will obtain that the neutron entropy depends on the position. Do you agree with this?

This is, I think, where I disagree. The screen at x + dx does not merely describe the neutron, but the whole system (neutron + gravitating body, i.e. the screen at x). That this screen has additional microstates/entropy due to the presence of the neutron does not necessarily imply that these additional microstates are indeed microstates of the neutron.

Perhaps to make things a bit more clear, let's look at a toy model a bit more explicitly. Take N - 1 (N at this point would be cleaner, but I wish to keep consistency with previous posts) particles arranged on a one dimensional lattice, i.e. something like pearls on a string. This system has (N - 1)! microstates, corresponding to the number of permutations of the pearls. If a is the lattice spacing, we can 'replace' the system by a 'screen' at point x = (N - 1)a -- the screen here being pretty much a purely rhetorical device which we only need to make the parallel to the neutron + Earth case more obvious.

Now let's add an Nth particle at location Na. Clearly, the system formed by the N particles now has N! microstates. We then replace this system again by a 'screen' at Na, and are then, I think, in a position to exactly replicate your previous argumentation: We can 'coarse-grain' S_{Na} to obtain S_{(N - 1)a}, and hence, conclude that S_{particle N} = S_{Na} - S_{(N - 1)a} -- and in particular, that particle N has N 'internal' microstates, which it, of course, doesn't! Those microstates are only there because of the combination of particle N with the N - 1 others. Similarly, the screen at x + dx has its higher entropy not because of the entropy of the neutron, but because of the combination of the neutron and the Earth (i.e. the screen at x).

To be sure, it is possible to construct a system obeying these entropy relations in such a way -- instead of particle N being a pearl like all of the others, it could, for instance, be some object exhibiting an N-fold symmetry, such that all (N - 1)! permutations combined with N's N symmetry transformations yield again a physically indistinguishable situation; this is the possibility your argument stipulates. But it's not the only possibility, and, if you want quantum mechanics to be unitary, also not the favoured one.

I 100% agree with your statements concerning your 'toy' model. What I am trying to say is that this model describes physically different situation and cannot be considered as the analog of Verlinde's theory. in your example your explicitly assume that particles on the screen and the one added to it are necessarily indistinguishable. Again, I DO understand that individual particle cannot carry any entropy, and the increase of entropy in your example is related with the increase of possible microstates in the whole system. BTW, note that if you 'measure, identify' the state of an added particle (position, energy etc) than you won't have any increase of entropy in your model.

The situation is indeed different in Verlinde's theory. It is true that on the screen at x+\delta x, that describes neutron and Earth together, neutron has no 'individuality', since all the microstates have the same energy (equipartition -> maximal entropy). However, if you look at the screen at x, that describes only Earth, then the entropy of the neutron-Earth system is the sum of neutron's and screen's entropies, that is, neutron's states are distinguishable. After all, neutron states are those which are measured in experiments!

 

[S.Daedalus]

However, if you look at the screen at x, that describes only Earth, then the entropy of the neutron-Earth system is the sum of neutron's and screen's entropies, that is, neutron's states are distinguishable.

You could conclude the same thing from the toy model, after having inserted the screen at (N - 1)a. But whether or not the screen is there doesn't change the physics -- you don't 'see' the screen from the outside. It would still look like -- i.e. be indistinguishable by experiment from -- there now being N particles. The same goes for the Earth-neutron system: if we agree that the neutron in the ordinary, non-entropic gravity setting doesn't have a position-dependent entropy, introducing the screen in place of the Earth doesn't change anything. (Else, you couldn't later replace the neutron-Earth system by another screen and expect the physics to remain equivalent, either.) The microstates are those of the system, whether it be represented by a screen at x + dx, by a neutron and a screen at x, or just by a neutron and the Earth in the usual setting. Else, you'd expect the physics to change depending on where you introduce the screen: in the usual, no-screen setting, the neutron has no entropy, and is in a pure state. Replace the Earth by a screen, and suddenly, the neutron acquires entropy, and is transformed into a mixed state. Replace neutron and Earth by a screen, and again the neutron has no distinguishable microstates. I don't see how this could possibly square with the idea of holography, i.e. that 2D screen and 3D bulk descriptions are dual, and fully equal to one another.

 

[CHIKO-2010]

You could conclude the same thing from the toy model, after having inserted the screen at (N - 1)a. But whether or not the screen is there doesn't change the physics -- you don't 'see' the screen from the outside. It would still look like -- i.e. be indistinguishable by experiment from -- there now being N particles.

No, I think you are wrong on this. Measuring just microstates on the screen at x gives NO information whatsoever about states of Nth particle at x+\delta x. Therefore the entropy on the screen is S_screen(x) and the total entropy S_screen(x)+S_neutron(x+\deltax). On the other hand, this is equal to S_screen(x+\deltax).

The same goes for the Earth-neutron system: if we agree that the neutron in the ordinary, non-entropic gravity setting doesn't have a position-dependent entropy, introducing the screen in place of the Earth doesn't change anything. (Else, you couldn't later replace the neutron-Earth system by another screen and expect the physics to remain equivalent, either.) The microstates are those of the system, whether it be represented by a screen at x + dx, by a neutron and a screen at x, or just by a neutron and the Earth in the usual setting. Else, you'd expect the physics to change depending on where you introduce the screen: in the usual, no-screen setting, the neutron has no entropy, and is in a pure state. Replace the Earth by a screen, and suddenly, the neutron acquires entropy, and is transformed into a mixed state. Replace neutron and Earth by a screen, and again the neutron has no distinguishable microstates. I don't see how this could possibly square with the idea of holography, i.e. that 2D screen and 3D bulk descriptions are dual, and fully equal to one another.

I think you are totally confused here. In non-entropic set-up to describe gravitation in the neutron-Earth system you do not need entropy at all -- gravity happens because of x-dependent gravitational potential. Verlinde said that this potential is a fiction, the key point is x-dependent entropy associated to the neutron-Earth system. Moreover, if you do not have such an x-dependent entropy you do not have even the notion of space.

The next question then is where does this entropy come from? Verlinde's answer is that it is associated with some (yet unspecified) microstates that live on holographic screens. next, you ask what do these screens has to do with the gravitating bodies? Verlinde's answer is that microstates on each screen describes objects the screen is surrounding, according to the holographic conjecture.

Now, if you remove any of the above ingredients the whole construction collapses. That is, no screens, no x-dependent entropy, no gravity!


Coming back to the problem of quantum bouncer. It is usually solved in the reference frame where Earth is in rest. Now according to Verlinde, neutron-earth system gravitates because the change in position of neutron (relative to Earth) changes the entropy of the system. I think that this basic fact about verlinde's theory is enough to derive the result of 1009.5414. Indeed look at the perform active spatial translation on neutron, this operation changes relative earth-neutron distance, and hence changes entropy. Therefore, the operator of spatial translations are non-unitary, and the results of 1009.5414 follows. It is in fact not even necessary to argue whether the entropy change is associated with neutron or not. Anyway, the quantum states of neutron are the quantum states in the presence of gravitational field (interacting states), and these states will be influenced by the entropy change in the system.

 

[S.Daedalus]

No, I think you are wrong on this. Measuring just microstates on the screen at x gives NO information whatsoever about states of Nth particle at x+\delta x.

I didn't say it does. Let's start with the system of N - 1 particles. Then, replace those particles by a screen. From the 'outside', both those systems should look the same -- that's what holography is all about. In particular, both systems will have (N - 1)! microstates. OK so far?

Then, add particle N. Added to the N - 1 particle system, it's plain that now the system has N! microstates, while the Nth particle on its own does not bring any new microstates to the table. However, added to the system in which the N - 1 particles are replaced by a screen, the situation should be identical; the Nth particle still does not have any 'internal' microstates, and the total number of microstates still increases to N!.

You claim that, in the case of the neutron falling in a gravitational potential, this should be different. That the description of the cases 'N particles' and 'N - 1 particles replaced by a screen + Nth particle' should be different. I don't think there's a good reason to assume this; and it's flat wrong in the toy model. The entropy is not S_{screen}(x = (N-1)a) + S_{particle N}, at least not in any meaningful way, because the microstates of the N particle system are not the microstates of the N - 1 particle system times the microstates of the Nth particle, either in the case in which there 'actually are' N - 1 particles or in the 'holographic' case where those particles have been replaced by a screen.

From the fact that the entropy of a screen placed at x + \delta x is higher than the entropy of the screen at x, you conclude that this increase in entropy is due to the additional entropy of the neutron at x + \delta x. The toy model shows that this need not be so. In this model, the screen at x + \delta x is equivalent to a screen at Na, i.e. a screen replacing the entire N particle system with its holographic description. The entropy of this screen is greater than the entropy of a screen at x = (N -1)a: S_{screen}(x + \delta x) = S_{screen}(Na) &gt; S_{screen}((N-1)a) = S_{screen}(x). But S_{screen}(Na) \neq S_{screen}((N-1)a) + S_{particle N}, because S_{particle N} = 0!

I think you are totally confused here. In non-entropic set-up to describe gravitation in the neutron-Earth system you do not need entropy at all -- gravity happens because of x-dependent gravitational potential. Verlinde said that this potential is a fiction, the key point is x-dependent entropy associated to the neutron-Earth system. Moreover, if you do not have such an x-dependent entropy you do not have even the notion of space.

I didn't say anything in conflict with this. I merely contrasted the cases of non-entropic, classical gravity -- in which a neutron's evolution is unitary, and hence, its entropy is constant, and possibly 0 -- with what you claim about entropic gravity, which results in the idea that the holographic formulation differs from the classical one in the neutron suddenly necessarily having non-vanishing entropy, which I don't think can be right.

Now, if you remove any of the above ingredients the whole construction collapses. That is, no screens, no x-dependent entropy, no gravity!

I agree completely. But it's the entropy of the entire system that depends on x, not just of the neutron.

It is in fact not even necessary to argue whether the entropy change is associated with neutron or not. Anyway, the quantum states of neutron are the quantum states in the presence of gravitational field (interacting states), and these states will be influenced by the entropy change in the system.

The translation operator defined in the paper acts on the states of the neutron, and takes them to higher entropy states, which is where its non-unitarity stems from. If the neutron states were of constant entropy, that operator would not have to be non-unitary.

 

[czes]

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.

 

[CHIKO-2010]

I didn't say it does. Let's start with the system of N - 1 particles. Then, replace those particles by a screen. From the 'outside', both those systems should look the same -- that's what holography is all about. In particular, both systems will have (N - 1)! microstates. OK so far?

Then, add particle N. Added to the N - 1 particle system, it's plain that now the system has N! microstates, while the Nth particle on its own does not bring any new microstates to the table. However, added to the system in which the N - 1 particles are replaced by a screen, the situation should be identical; the Nth particle still does not have any 'internal' microstates, and the total number of microstates still increases to N!.

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?

2. Now, I can also consider the screen at x+\delta x. in this case, yes, the entropy is analogous of N indistinguishable particles, the number of microstates is N! Do you agree with this or not?

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. If you agree with 1,2, then you must agree with 3

I didn't say anything in conflict with this. I merely contrasted the cases of non-entropic, classical gravity -- in which a neutron's evolution is unitary, and hence, its entropy is constant, and possibly 0 -- with what you claim about entropic gravity, which results in the idea that the holographic formulation differs from the classical one in the neutron suddenly necessarily having non-vanishing entropy, which I don't think can be right.

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. Why do you expect some kind of continuity? Again, there cannot be any deformation (gradual, continuous or whatever) that can approach the standard theory.

I agree completely. But it's the entropy of the entire system that depends on x, not just of the neutron.

The translation operator defined in the paper acts on the states of the neutron, and takes them to higher entropy states, which is where its non-unitarity stems from. If the neutron states were of constant entropy, that operator would not have to be non-unitary.

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.