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!