# Why are physicists surprised that the information of a black hole is proportional to its boundary? #2

Intuitively, one of two things must happen to information that enters a black hole: either it is destroyed when the object enters the black hole, or it is transferred to the black hole. If we assume that the former does not happen, then the information must be encoded in the black hole's properties. By the no-hair theorem, black holes have only three properties: mass, angular momentum, and electric charge. There can obviously be black holes without angular momentum or electric charge, but there can be no black holes without mass, so if the information is encoded anywhere at all, it must be encoded in the mass. By extension, since mass is proportional to the area of the event horizon, the information of a black hole must be proportional to its boundary.

What is wrong with the above argument? If there is nothing wrong, then why are physicists surprised that the information of a black hole is proportional to its boundary?

• If the information is encoded anywhere at all, it must be encoded in the mass. The mass is one number. How would you code all the info into one number? Dec 1, 2022 at 4:17
• There can obviously be black holes without angular momentum or electric charge, but there can be no black holes without mass, so if the information is encoded anywhere at all, it must be encoded in the mass. Why does the existence of black holes without angular momentum mean the information cannot be encoded in angular momentum? (Same question for electric charge, obviously.) Dec 1, 2022 at 15:48
• Mass isn't proportional to the area of the event horizon. The area is quadratic in the mass. Dec 1, 2022 at 19:37
• @Ghoster presumably it would provide some range of information - for example, if I know you are a human, then I have some idea what your age is (1 < age <120), your height, that you have two eyes and two arms, etc. Dec 2, 2022 at 1:34
• The no hair theorem is a statement about classical G.R. It does not apply once you bring in quantum field theory. Dec 3, 2022 at 0:25

In my view/understanding, answering your question requires understanding the information loss paradox. This requires that we introduce Hawking radiation, and try to imagine how we would actually go about quantizing fields inside a black hole.

By the no-hair theorem, black holes have only three properties: mass, angular momentum, and electric charge.

The problem is that there is tension between the no-hair theorem and quantum mechanical unitary evolution.

On the one hand, the no-hair theorem says that an outside observer can only really measure the three properties you mentioned of a black hole.

On the other hand, there is still extra information associated with a black hole. It is the information inside the event horizon. For example, if you throw an encyclopedia into a black hole, then the information on the pages of the encyclopedia is inside the event horizon.

That isn't a problem so long as the information stays inside the event horizon, inaccessible to the observer. However, once we take into account Hawking radiation, then black holes will shrink in size, and, it is widely thought, evaporate. (Although one logical possibility is that you're left with a "remnant" that still has all the information inside.)

Now combine the black hole evaporation with quantum mechanical unitary evolution. At least mathematically, it should be possible to take the final state of the system, after the black hole has evaporated, and evolve it backward to infer what happened inside the black hole before it evaporated. In other words, you should be able to reconstruct the words in the encyclopedia (at least in principle -- in practice you would need to measure an enormous amount of subtle information about the Hawking radiation and do a truly mind-bogglingly universe-bendingly epic calculation; the information loss paradox is at its heart an "in principle only" thought experiment).

In other words, there is more information available to the outside observer, than the no-hair theorem suggests, if they are able to wait for the black hole to evaporate.

Furthermore, detailed calculations suggest that the Hawking radiation has an entropy proportional to the area of the black hole.

Where does this information live, while the black hole has not evaporated?

Well, in a normal quantum field theory, you would expect information to be localized in small volumes of space. To take a concrete example, imagine quantizing a free field theory on a cubic lattice with $$N$$ grid points on each side. Then you are effectively quantizing $$N^3$$ coupled harmonic oscillators, so the size of the Hilbert space grows with the volume, in an ordinary field theory.

In order that the number of degrees of freedom contained in the black hole, be consistent with the entropy of the Hawking radiation that can be used to reconstruct what is going on inside, it must be that the size of the Hilbert space describing the interior of the black hole grows as the area of the black hole, and not the volume. This is surprising, because it is not how an ordinary QFT behaves. It seems to suggest there must be something fundamentally non-local about quantum gravity.

Now, of course, most of these arguments are quite speculative. There is no widely agreed upon solution of the information loss paradox, and there is no solid proof that the holographic principle is correct. However, this is the kind of logic that people follow, and the end result of this chain of logic is surprising, if it is correct.

• +1, interesting, but I do not think this answer should be accepted. 1. The size of Hilbert space grows multiplicatively, not additively with the addition of degrees of freedom. 2. Less critically, the size of Hilbert space is already infinite for a single harmonic oscillator. 3. More critically, the amount of information "contained" in a volume already goes like the area in "regular" QFT by Srednicki's 1993 calculation arxiv.org/abs/hep-th/9303048. Thus the distinction between "ordinary QFT" and quantum gravity raised in this answer appears to be spurious. Dec 1, 2022 at 17:54
• does the no-hair theorem imply a measurement from a long distance? I was under the impression that black holes could still have local temporary fluctuations in color charge for example. It's necessary if you assume that a proton doesn't cross the event horizon all at once. Dec 1, 2022 at 19:03

Historically, the above argument contains multiple surprising (or non-obvious) remarks, regardless of whether or not they are true.

1. Assuming information is not destroyed.
2. Assuming that information must be encoded in mass, angular momentum or electric charge and not something else.
3. Assuming that something proportional to mass should be proportional to the boundary.

Speaking to point #3 in particular, the Bekenstein bound is that the entropy is proportional to the surface area

$$S \propto R^{2} \propto M^{2}$$. This is not linear with mass--the Bekenstein bound is not obvious from your statement "since mass is proportional to the area of the event horizon"

I am not an expert on this topic.

Questions on it appear here often enough, even though I think it is quite speculative and not really a solid part of "established mainstream physics".

If we tolerate exchange of ideas on this topic on this site at all, we should also allow questioning the very assumptions behind the claims, so people don't get the wrong idea that the claims are all solid and verified.

... the information must be encoded in the black hole's properties

This is often assumed, but I don't understand why. From an outside observer's point of view, in the "normal" frame far from the black hole, the information-carrying body (which is not itself a black hole) falling towards the black hole never gets inside it. After some time, it gets so close to the horizon that it effectively freezes somewhere above it, with all its information. It does not go through the horizon to the "interior", it does not spread uniformly on the horizon. So it should not be counted as part of the black hole. Certainly its mass above the event horizon does not contribute to the mass $$M$$ defining the event horizon.

By the no-hair theorem, black holes have only three properties: mass, angular momentum, and electric charge. ...if the information is encoded anywhere at all, it must be encoded in the mass. By extension, since mass is proportional to the area of the event horizon, the information of a black hole must be proportional to its boundary.

Technical correction: black hole's mass is proportional to the Schwarzschild radius, thus to square root of the black hole's surface area.

But the no-hair theorem is about the ideal model with all mass "inside" that defines the black hole mass. It does not capture a mass that may be hovering above or on the event horizon. Such hovering mass of the fallen bodies would not typically be symmetrically distributed, has different gravity effect than mass localized in the singularity and thus would seem to break assumptions of the no-hair theorem. Black hole with some mass above the horizon seems to be a different thing than black hole without such mass.

... why are physicists surprised that the information of a black hole is proportional to its boundary?

I suspect most physicists do not care or believe that "information of a black hole is proportional to its boundary". Even the concept of "information of the black hole" is quite dubious. A hard drive falling towards the black hole won't get lost from the space outside the hole at some finite time. It will remain in that space indefinitely, even though very close to the horizon.

Unless, the hard drive at some point of the fall somehow becomes a black hole itself, with its own horizon, and the two horizons then merge. But this seems like a wild speculation.

• @safesphere > "If the mass of a BH were inside, it would have no effect outside, so we would measure it as zero." Do you consider this a controversial statement? Can you provide a reference? Dec 5, 2022 at 22:44
• > "Similarly, as far as we are concerned, the inside of the black hole has not happened yet...and can have no effect on us in any regard, including gravity." But black hole does have gravity. And the corresponding mass that generates this gravity can be only on the horizon or inside. Just because we can calculate ADM mass using field far from the black hole does not mean mass is located in space far from the hole. Dec 6, 2022 at 15:24
• Why do you assume $mc^2 + K$ is conserved? In standard uses of this expression, $m$ is constant in time, and it is called rest mass or invariant mass. This can change in time only if something changes in the internal state of the falling body. It does not change when it only changes height in gravity field. Dec 6, 2022 at 21:55
• I agree that a falling body can't go through the horizon, but this does not mean there is no mass inside. Mass $M$ used in the Schwarzschild solution is not hovering above the horizon, it can only be inside. Dec 6, 2022 at 22:02
• > "The total energy is conserved when measured in the same coordinates before and after. " Why do you believe this? It is not the case already in nonrelativistic mechanics, where we have to introduce potential energy of the body pair to maintain conservation of kinetic + potential energy. In GR, there is no general concept of potential energy, and I don't see how mass of the falling body could supplant it. The falling body can accelerate without its mass changing in any way. Where do you think the Schwarzschild solution mass is located? I think it should be in the center singularity. Dec 7, 2022 at 13:07