B Definition of black hole, according to Maldacena

[javisot]

TL;DR Summary

Is a black hole equivalent to a quantum system with S degrees of freedom, where S is proportional to the area of the BH?

I would like to discuss a topic of definitions.

In natural language, a black hole is defined as a region of spacetime from which (classical) information cannot escape. This definition is very good and satisfactory in the context of relativity. I'd like to discuss more modern definitions, especially the one used by Maldacena,

"A black hole is equivalent to a quantum system with S degrees of freedom, where S is proportional to the area of the BH"



I have a simple question, the universe is also a quantum system with S degrees of freedom and it's not a black hole. I suppose the difference is that in a BH, S is proportional to the area of the BH, and in the case of the universe, S is not proportional to the area of the universe, right?

(I find Maldacena's definition ambiguous since everything can be treated as a quantum system with S degrees of freedom)

 

[.Scott]

For a BH, S is proportional to the bounding area at the event horizon - and it is at the maximum possible value for that area. For all other objects, S has not reached the capacity of the bounding area.

 

[javisot]

For a BH, S is proportional to the bounding area at the event horizon - and it is at the maximum possible value for that area. For all other objects, S has not reached the capacity of the bounding area.

From your words I understand that for a black hole, entropy (S) is proportional to the area of its event horizon and saturates the Bekenstein bound. For all other objects, entropy remains below this maximum limit for the given bounding area.

I would like to know more about some words from Maldacena at the beginning, saying that this definition only applies in very specific cases.

 

[.Scott]

From your words I understand that for a black hole, entropy (S) is proportional to the area of its event horizon and saturates the Bekenstein bound. For all other objects, entropy remains below this maximum limit for the given bounding area.

I would like to know more about some words from Maldacena at the beginning, saying that this definition only applies in very specific cases.

I just listened to the first 4 minutes. could you give me a video time-tag for what you are asking about.

 

[javisot]

(I assume we all know that when we say "the area of a BH" we mean the area of its event horizon)

I just listened to the first 4 minutes. could you give me a video time-tag for what you are asking about.

Between minute 1:00 and 2:00 he comments on it, but they are just quick comments without clarification.

 

[.Scott]

The sound quality isn't great. But what he says is:
"In general the black holes can be replaced with a quantum system"
"by quantum system we mean some system of qubits ...{garbled}... acting on it"
"And this is suppose to hold for black holes in nature, in principle."
"We, of course. haven't tested this idea for black holes in nature"

He is referring to the available observational data from black holes.
Personally, I would even put the event horizon into the category of "Untested Theories". Most put the presumed singularity into that category (perhaps permanently). And most everything related to resolving the black hole QM issues is untested.

So, if you build your black hole with no spin, no charge, and with General Relativity, you get a Schwarzschild black hole. As you add more theory, the characteristics of the black hole changes. Perhaps with enough thought we will successfully predict what humanity might someday be able to verify.

 

[PeterDonis]

the universe is also a quantum system with S degrees of freedom

What makes you think so?

 

[PeterDonis]

everything can be treated as a quantum system with S degrees of freedom

What makes you think so?

 

[PeterDonis]

I would even put the event horizon into the category of "Untested Theories".

There is at least one sense in which this has to be true: the definition of the event horizon is global--it is the boundary of the region (the black hole) that is not in the causal past of future null infinity. But to know where that boundary is, you have to know the entire future of the spacetime. And of course we don't and we can't. So we can never truly know where any event horizon actually is, or even whether there is an actual event horizon, as opposed to just an apparent horizon (a locally trapped surface where light locally can't move outward).

 

[javisot]

The sound quality isn't great. But what he says is:
"In general the black holes can be replaced with a quantum system"
"by quantum system we mean some system of qubits ...{garbled}... acting on it"
"And this is suppose to hold for black holes in nature, in principle."
"We, of course. haven't tested this idea for black holes in nature"

He is referring to the available observational data from black holes.
Personally, I would even put the event horizon into the category of "Untested Theories". Most put the presumed singularity into that category (perhaps permanently). And most everything related to resolving the black hole QM issues is untested.

So, if you build your black hole with no spin, no charge, and with General Relativity, you get a Schwarzschild black hole. As you add more theory, the characteristics of the black hole changes. Perhaps with enough thought we will successfully predict what humanity might someday be able to verify.

00:00:59- "we have examples where certain black hole can be represented...well, we think that in general black holes can be replaced by a quantum system, described by quantum mechanic, and by quantum system we mean some system of qubits with some hamiltonian acting on it. This is suppose to hold for black holes in nature, in principle. We, of course, haven't tested this idea for black holes in nature. 00:01:30

00:01:30- "but, we have some toys models or simpler situations where we think/we know that a black hole correspond to a quantum system" 00:01:48


He thinks that in general this is true for any black hole, but we only have toy models that confirm it. I don't understand what he mean by toy model or simpler situation which demonstrate that this definition is correct. In the following minutes, it is true that he comments on important things about it.

 

[javisot]

What makes you think so?

The correspondence principle and all the Maldacena talks I've watched. Susskind also often speaks in those terms.

 

[PeterDonis]

The correspondence principle and all the Maldacena talks I've watched. Susskind also often speaks in those terms.

You're going to have to give more specific references. @.Scott has already pointed out the obvious reason why no object except a black hole satisfies the condition you have stated. Note that "S", as your own OP says, is not "the entropy of the object", it's "the area of the black hole horizon corresponding to the objects mass", i.e., in natural units in which $G = c = 1$, it's $16 \pi M^2$, where $M$ is the object's mass. (Technically S is proportional to this, but that's a minor detail.)

(It's also not the case, btw, that an object's entropy is the same as the number of degrees of freedom the object has, as you seem to be implying. But I think that's another minor detail compared to the key issue described above.)

 

[Demystifier]

I would put this definition in the following form. Any object of finite size has a (quantum) entropy $S$ and a boundary with area $A$. Such an object is called black hole iff $S=A/4$ (in natural units).

 

[javisot]

Any object of finite size has a (quantum) entropy $S$ and a boundary with area $A$.

The universe too?

 

[Demystifier]

The universe too?

Would you say that Universe has a finite size?

 

[javisot]

Would you say that Universe has a finite size?

The observable universe?, sure, but I mean,
I don't see it as strange to say that the universe (or a black hole) is a quantum system if you assume the correspondence principle and, as in the case of Maldacena, the holographic principle are the foundations of your theory. I can find any references that prove a principle, because principles are not proven, you either assume them or you don't.

I understand that the answer to my question is what you (Peter and Scott) are saying, only in the case of the black hole S is proportional to the area of the event horizon (the universe doesn't have an event horizon like a black hole, obviously)

Maldacena's definition of a black hole is not really ambiguous.

 

[Demystifier]

The observable universe?, sure, but I mean,
I don't see it as strange to say that the universe is a quantum system if you assume the correspondence principle and, as in the case of Maldacena, the holographic principle are the foundations of your theory. I can find any references that prove a principle, because principles are not proven, you either assume them or you don't.

I see your question. The observable Universe is defined by the horizon, with the area $A$. If we would assume that the entropy of this Universe is $S=A/4$, then the Universe would satisfy the definition of black hole, and yet it looks silly to claim that the Universe is a black hole. The resolution of this conundrum is that it is wrong to assume that the entropy of the Universe is $S=A/4$, it must be less than that.

Indeed, note that the original motivation for the Bekenstein proposal (that black hole entropy is proportional to the area) originated from the classical laws of black holes mechanics, that look analogous to the laws of thermodynamics. The classical cosmological horizon does not satisfy any such thermodynamic-like laws, hence there is no good reason to think of Universe as a thermodynamic system, so it is not so plausible that entropy of the Universe is $A/4$.

 

[javisot]

I see your question. The observable Universe is defined by the horizon, with the area $A$. If we would assume that the entropy of this Universe is $S=A/4$, then the Universe would satisfy the definition of black hole, and yet it looks silly to claim that the Universe is a black hole. The resolution of this conundrum is that it is wrong to assume that the entropy of the Universe is $S=A/4$, it must be less than that.

Indeed, note that the original motivation for the Bekenstein proposal (that black hole entropy is proportional to the area) originated from the classical laws of black holes mechanics, that look analogous to the laws of thermodynamics. The classical cosmological horizon does not satisfy any such thermodynamic-like laws, hence there is no good reason to think of Universe as a thermodynamic system, so it is not so plausible that entropy of the Universe is $A/4$.

I found this, but it's orders of magnitude more complex than I can understand, https://arxiv.org/abs/2504.05763

 

[Demystifier]

I found this, but it's orders of magnitude more complex than I can understand, https://arxiv.org/abs/2504.05763

This paper is only a couple of weeks old, I'm sure Maldacena in his talk was not aware of it. It will be interesting to see what other experts in the field will say about this paper.