# Holographic Principle discussion in Sean Carroll's book

• I
• mindarson
In summary, Sean 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,

#### mindarson

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.

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 (possibly for reasons other than gravity). 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? How would we know that? (Maybe there's no obvious physical mechanism analogous to those by which the area-dependence was deduced in the case of Black Holes.)

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!

mindarson said:
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.

Unfortunately, a pop science book is not really a good source, even when it's written by a reputable scientist and expert in the field like Carroll. We should really be looking at textbooks or peer-reviewed papers--for this topic, probably the latter, since it's new enough not to be covered in most textbooks.

mindarson said:
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.

If you really want to limit the discussion this way, then this thread should be a "B". Also, I'm not sure your questions can be answered at that level, other than to say "it's complicated, and the complications are beyond the level you have picked for this discussion".

PeterDonis said:
Unfortunately, a pop science book is not really a good source, even when it's written by a reputable scientist and expert in the field like Carroll. We should really be looking at textbooks or peer-reviewed papers--for this topic, probably the latter, since it's new enough not to be covered in most textbooks.

Wow, thanks for trying!

In the first place, nothing in my post implies that I am NOT looking at textbook-level sources. And indeed, I am. But parallel to that steeper learning curve, there is absolutely nothing wrong with trying to get the general lay of the land from popular books. That's like saying you shouldn't read the introduction to a textbook because it isn't as in-depth as the body. Nonsense. Everyone has to start somewhere.

PeterDonis said:
If you really want to limit the discussion this way, then this thread should be a "B". Also, I'm not sure your questions can be answered at that level, other than to say "it's complicated, and the complications are beyond the level you have picked for this discussion".

Why did you even bother to type anything at all if you can't answer the questions? Do you honestly believe that every effort to explain physics concepts that doesn't involve postdoctoral mathematics is in vain simply because there is always more to say about any topic at a higher level of knowledge? Get real. I've explained special relativity to people who didn't even know algebra. Is their understanding complete? Obviously not. But it wasn't in vain, because they walked away knowing more than they did before I tried. Which was all they wanted.

I sincerely hope you're not a teacher of any kind.

mindarson said:
nothing in my post implies that I am NOT looking at textbook-level sources

Yes, it does, because you didn't include any, and PF rules on acceptable sources say you should. If you are looking at such sources, those are the ones you should be referencing for an "I" level discussion, not a pop science article.

mindarson said:
there is absolutely nothing wrong with trying to get the general lay of the land from popular books.

Yes, there is, because pop science books and articles, even the best of them, can't be relied on to give you "the general lay of the land". Sorry, but the PF rules on acceptable sources are there from bitter experience; we've seen what happens when people try to use pop science sources to do what you suggest.

mindarson said:
That's like saying you shouldn't read the introduction to a textbook because it isn't as in-depth as the body. Nonsense.

What is nonsense is your statement. A textbook is not a pop science source, and it has to undergo a level of review and scrutiny that no pop science source does. So your analogy fails.

mindarson said:
Why did you even bother to type anything at all if you can't answer the questions?

Because the way you posed your question made it unanswerable as you posed it, for the reasons I have given. See further comments below.

mindarson said:
Do you honestly believe that every effort to explain physics concepts that doesn't involve postdoctoral mathematics is in vain simply because there is always more to say about any topic at a higher level of knowledge?

I said nothing of the kind. You labeled this thread as an "I". That indicates an undergraduate level discussion. You can't have that level of discussion based on a pop science source.

I strongly suggest that you dial back the attitude and consider what I'm telling you. If you want an "I" level discussion of this topic, you will need to provide acceptable sources. Since you say you are consulting such sources, that should be easy, and you could have already posted links or references to them instead of complaining. If you really want just a very basic discussion based on what's in a single pop science source, then this thread should be a "B" thread and you will get very rudimentary answers. This is an advanced topic and even an "I" level discussion might not do it justice.

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PeterDonis said:
Yes, it does, because you didn't include any, and PF rules on acceptable sources say you should. If you are looking at such sources, those are the ones you should be referencing for an "I" level discussion, not a pop science article.

Okay. But it's not like my grandma wrote this stuff, and it's not like it isn't well-established physics. The solution to your objection is for me to point to page 421 of the first edition of Carroll's textbook Spacetime Geometry: An Introduction to General Relativity, published by Addison-Wesley, where he makes the same argument, only in a condensed form.

You're acting like I asked if quantum mechanics really says cats don't exist because Joe down at the local bar told me so. I'm sorry that you apparently run into so many boneheads that you have to start out assuming every new person you meet must be one and condescend accordingly.

That said, in the unlikely event that I come back here for help with anything, I'll review the rules so as to minimize the chance of this sort of exchange.

mindarson said:
it's not like it isn't well-established physics.

Actually, it's not. It's speculation at this point, since we have no experimental evidence in this regime and no well-established theory of the microstates of a black hole.

mindarson said:
page 421 of the first edition of Carroll's textbook Spacetime Geometry: An Introduction to General Relativity, published by Addison-Wesley, where he makes the same argument, only in a condensed form.

Thank you for the reference. I will take a look when I have a chance. Also, others might be able to comment on this reference.

The rest of your post merits a warning, not a response, and further posts in the same vein will receive one.

Dale
I think you are trying to reason backwards. The statement is that if entropy is proportional to area then we have non-locality, not the other way around, which is what you are trying to figure out. And for this statement (entropy prorportional to area implies non-locality), you've already written his argument. If you had two boxes that are completely indipendent, then the total number of states is the product, so the the entropy (log of that) is the sum of entropies. But when you put two things together area doesn't add, so the assumption(that they had nothing to do with each other) must be wrong.

Nugatory and haushofer
Funny, I have never looked upon it that way, but as I understand your post the holographic principle indicates because of the "entropy scales like surface instead of volume" non-locality?

haushofer said:
I understand your post the holographic principle indicates because of the "entropy scales like surface instead of volume" non-locality?

The term "locality" can have different meanings, but as it's being used here it means, heuristically, that the microstates of two disjoint regions of space are statistically independent. That implies that the entropy of two disjoint regions of space, taken as a single system, is the sum of the entropies of the two regions taken individually. But that can only be true for all possible sets of disjoint regions of space if entropy is proportional to volume. However, the holographic principle shows that entropy is proportional to surface area, not volume.