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I am reading Sean Carroll's pop-science book 'From Eternity to Here' and am having trouble connecting the links in his discussion of the Holographic Principle.

At the outset, I would ask that you try to answer in terms of Carroll's discussion and without moving into concepts much more advanced than what he presents in his book. I do have a BS in physics and am conversant with thermodynamics, basics of GR, etc, but I want to grasp the argument in the particular terms Carroll puts forth in the book. And I don't want to lose that argument in a sea of math that I don't yet comprehend.

In other words, I'm not so much asking about the Holographic Principle itself, but about this particular presentation of the Holographic Principle.

I'll paraphrase the argument up to the point where I lose the thread:

Carroll uses the model of a box of particles to illustrate concepts regarding entropy and the second law. He puts the question "How much entropy could we pump into this box of fixed size"? Without gravity, he says, there's no limit. But with gravity, there is a limit. This is because the entropy in a black hole scales with its area. So if we added more entropy, we would make the black hole bigger, i.e. it would not be of fixed size. So: "there is a maximum amount of entropy you can possibly fit into a region of some fixed size, which is achieved by a black hole of that size." And since entropy counts possible microstates associated with a macrostate, "that means there are only a finite number of possible states within that region."

Carroll then goes on to claim that this fact overthrows the assumption of locality, which he defines as "the idea that different places in the universe act more or less independently of one another." THIS is the step I really don't understand.

The outline of Carroll's argument seems to be this:

a) Take two systems. The entropy of the two systems together is just the sum of the entropies of the individual systems (entropy encoded as logarithm). This means the max entropy we can fit into a box is proportional to the box's volume. b) But we've already seen that the max entropy that can fit into a box (of fixed size) is proportional to an area, i.e. the area of the largest black hole that can fit in the box. So there has been an "oops." c) The "oops" was the assumption of locality - i.e. that the systems are independent - which was used implicitly in deriving that the maximum allowed entropy is proportional to volume in the first place. So we have to toss out locality.

My confusion is over how we can conclude, from the fact that the maximum allowed entropy is constrained by area (2 dimensions) rather than volume (three dimensions), that it must be the case that "what goes on over here is not completely independent from what goes on over there."

Is it really just as simple as "If causality was strictly local, then the entropy of a black hole would be constrained by its volume, not by its surface area?" What would that look like? Would it be just our "naive" idea of a black hole?

I have been trying to mull this over in terms of information. Carroll says "the real world ... allows for much less information to be squeezed into a region that we would naively have imagined if we weren't taking gravity into account." So is it correct to say that the information (number of possible microstates) is restricted by the area (rather than by the volume) PRECISELY BECAUSE non-locality itself implies that there just isn't as much information in the system? Since specifying something about one part of the system also tells us about another (non-local) part of the system?

Maybe another way of framing my question is this: Suppose the original, gravity-less setup of the box of particles was somehow non-local. How would this non-locality affect the actual calculation of the sum of the two entropies? What would be the area to which the total maximum entropy would be confined in that case? Would it just be the surface area of the box?

And what if the maximum amount of entropy were restricted by the size of some one-dimensional attribute of the system, rather than the area? What would THAT have to say about locality and causality? How exactly is locality related to the number of dimensions of the feature of the system that constrains the maximum entropy of the system?

Just looking to fill out and clarify these issues. Thanks for any thoughts you can share!

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  • $\begingroup$ Couple of problems with that model... first of all, the world is about as local as it comes, so the idea is not exactly covered by facts. Secondly, the volume is not constant but increasing, which is not accounted for by this static argument. Third, we don't know what the degrees of freedom inside a black hole really are. The classical theory predicts something which may or may not be correct, but we don't have a single measurement to back it up. More to the point: black holes are a state of matter and the classical theory was always wrong about the states of matter, why bet on it now? $\endgroup$ – CuriousOne Jun 13 '16 at 3:00
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I will answer, for now, the easy part, which is the first. If entropy is proportional to the volume, then it would be proportional to $L^3$, and if it is proportional to the area ir will be proportional to $L^2$ (where $L$ is the characteristic distance regardless of shape, such as the radius, if the shape is a circumference , or the side if it is a box). The difference is important, because $L^3$ grows faster than $L^2$, thus the surface is limiting the growth of the volume. In other words, entropy is limited by the growth of being proportional to the area as opposed to growing as proportional to the volume.

In addition to that, the holographic principle breaks the notion of locality, at least as assumed by bell. For a full discussion see this answer How does the holographic principle imply nonlocality?

There are ways other than holography to circumvent bell's theorem too. Bell theorem implicitly assumes that space is the continuum, and that distances can be defined by the difference between the spatial positions. However, Wolfram, in his book has argued that nonlocality can be an artifact of the growth of causal networks . In other words, some systems could in principle look like a local space at the macroscopic level, but different points in space can be linked locally in the network, but they could be far away in "space", where space is a macroscopic property, not a local one.

It is currently unclear if wolfram ideas are wrong or right, but they represent an actual alternative to the assumptions taken by bell. It is fair to say that neither wolfram, nor anybody else that I know, has developed a full account of quantum mechanics based on this idea, but the possibility still exists.

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  • $\begingroup$ Before we talk about theory, should we not talk about observation? True non-locality would imply that there has to be some effect in my lab which is completely void of any causal relationship. I am not aware that this has ever been observed on any scale accessible to us. So if there is non-locality at some scale, it must be very efficiently screened by something... or... there simply is no non-locality, which then begs the question why we need to develop a non-local theory, again? $\endgroup$ – CuriousOne Jun 13 '16 at 5:17
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    $\begingroup$ @CuriousOne I totally disagree with your observation. It can remain a fully undeveloped theory until further development, but still a possibility. $\endgroup$ – Wolphram jonny Jun 13 '16 at 5:20
  • $\begingroup$ A theory needs to describe something. Which effect that I can observe in my lab is it supposed to describe? I am really not looking for anything special here. Just tell me what I am supped to see in an experiment that is not explained by local theories? $\endgroup$ – CuriousOne Jun 13 '16 at 5:22
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    $\begingroup$ @CuriousOne well, if your opinion does take a philosophical position but tries to conceal it, it doesn't work, everybody will notice it. I am not interested in arguing with you if that is what you have to offer. $\endgroup$ – Wolphram jonny Jun 13 '16 at 5:32
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    $\begingroup$ non sense, please post an answer so we can all vote $\endgroup$ – Wolphram jonny Jun 13 '16 at 5:43

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