# Black Holes and information

## Main Question or Discussion Point

I have a question about black holes. So imagine that we start putting energy in some region of space, until we reach Schwarzschild radius. We also reach the maximum amount of information we can store in that region of space. Still, the energy continues to collapse in smaller and smaller space, along the path to singularity. Does this not violates the amount of information we can store into a certain region of space?

## Answers and Replies

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Matterwave
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When you say "the maximum amount of information we can store in that region of space" what principle are you referring to and what is that maximum?

PeterDonis
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imagine that we start putting energy in some region of space, until we reach Schwarzschild radius.
How do you propose to do this? The energy can't just appear out of nowhere; it has to come from somewhere.

Grinkle
Gold Member
Does this not violates the amount of information we can store into a certain region of space?
I will offer a B level response that may not be correct or very well stated. As you add more information, you are increasing the event horizon radius. That is where the maximum information density comes from - the density stays constant because the EH radius increases.

From your post, I think you are picturing the actual "singularity" as containing the information. I believe that in GR, it is the EH that contains the information, and the EH is what must be used to describe the information density. There is no theory to describe / quantify the distribution of anything (mass, information, whatever) inside the EH.

When you say "the maximum amount of information we can store in that region of space" what principle are you referring to and what is that maximum?
I mean, the information you need to describe everything happening in a particular region of space is proportional to the surface area of that space. In this case we are talking about the black hole horizon. All the information you need to describe the black hole is proportional to its surface area. Well, the GR says that there are infinite amount of tidal forces and infinite density in the center of BH. So in order to describe this situation we would need infinite amount of information. And obviously, the surface of the horizon can not have that amount of information. It's limited by quantum mechanics. That's why the horizon grows when you throw something in the black hole.

martinbn
Can you explain what you mean by 'amount of information' and why it is infinite? Take the function $y=\frac1{x^2}$, when $x\rightarrow 0$ the function goes to infinity. Do I need an infinite amount of information to describe that? Well, I just did it in one line.

Grinkle
Gold Member
Well, the GR says that there are infinite amount of tidal forces and infinite density in the center of BH.
The only interpretation of the math I have seen is that GR makes no description of what is happening at the center of a BH - the theory breaks and we don't have any model of what is actually happening there.

So in order to describe this situation we would need infinite amount of information.
Is that true in principle? I think (I may be wrong) that the entropy of a neutron star is lower than its entropy was before it collapsed. Of course there was heat transfer in that process and I am not at all attempting to define what closed system I am talking about and the entropy of the universe as a whole did increase during the stars collapse. I am saying that if there is some very dense, exotic matter at the center of a BH, it might not have an entropy that is also getting very high and it might not itself contain very much information.

PeterDonis
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I think (I may be wrong) that the entropy of a neutron star is lower than its entropy was before it collapsed.
Gravitational clumping increases entropy so I don't think this is true. In the presence of gravity, entropy works differently from the textbook examples that are given for systems like gases.

if there is some very dense, exotic matter at the center of a BH
There isn't. In any case, the entropy of a BH is the area of its horizon; it doesn't matter what's inside it.

If a mass actually collapses in singularity, it means that on the way it is compressed into smaller regions of space. These regions will therefore have a certain area and a certain radius. If we see Bekenstein bound formula for maximum entropy we shall see that as the radius of a star, start to get smaller and smaller the entropy decreases. In certain moment the formula will reduce to a well known Black hole entropy equation(the star is a Black hole). But still after that point the mass will continue to collapse according to GR. So and the entropy must decrease to.

2pi (Boltzmann's constant)(Total Energy)(Radius) / (h-bar)(speed of light) = S
4pi (Boltzmann's constant)(Mass-square) G / (h-bar)(speed of light) = S(BH)

We are assuming here, that the only variable is the radius and everything else is a constant!

PeterDonis
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If a mass actually collapses in singularity, it means that on the way it is compressed into smaller regions of space.
Not once the matter is inside a black hole; at least, not in the sense required for the Bekenstein bound. The formulas you are quoting only work outside a black hole; they don't work inside one.

If we see Bekenstein bound formula for maximum entropy we shall see that as the radius of a star, start to get smaller and smaller the entropy decreases.
No, we don't, because the entropy of the star is much, much smaller than the entropy of a black hole with a Schwarzschild radius equal to the star's radius would be. So even though, as the star collapses to a black hole, the entropy of a black hole with Schwarzschild radius equal to the star's radius decreases, it is quite possible that the entropy of the star itself increases.

Yes, you are right! It would be good first to do the calculation before I write anything. What are the formulas which are working inside of a black hole? So the entropy of a black hole, despite the fact that the mass continues to shrink, remains a constant (unless you throw something)?

To me, it has to do with the uncertainty principle of quantum mechanics and the law of energy conservation cannot be violated, but only for very short durations. The Universe is able to produce mass and energy out of nowhere. Maybe this mass and energy disappear again very quickly as a result of supposed blackhole evaporation. One particular way in which this strange phenomenon manifests itself goes by the name of vacuum fluctuation

PeterDonis
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What are the formulas which are working inside of a black hole?
If you mean entropy formulas like the Bekenstein bound, there aren't any. Nobody outside the black hole can observe anything inside the hole, so there's no basis on which to do thermodynamics. That's why a black hole's entropy is based on the area of its horizon: the horizon is the boundary of what outside observers can observe.

PeterDonis
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The Universe is able to produce mass and energy out of nowhere.
No, it isn't. Vacuum fluctuations don't produce mass and energy out of nowhere.

Maybe this mass and energy disappear again very quickly as a result of supposed blackhole evaporation.
The energy radiated by black hole evaporation comes from the mass of the hole; it doesn't come out of nowhere. Nor does it disappear once it's radiated.

Can you explain what you mean by 'amount of information' and why it is infinite? Take the function $y=\frac1{x^2}$, when $x\rightarrow 0$ the function goes to infinity. Do I need an infinite amount of information to describe that? Well, I just did it in one line.
Not sure if this is relevant here, and at the risk of adding noise to the discussion (pardon the pun), there is a distinction to be made between standard information (of the type defined by Claude Shannon, of the genre of Boltzmann's "negentropy" in physical terms), and algorithmic information theory (a continuation along the lines of Kurt Godel) developed independently by Ray Solomonoff, Andrey Kolmogorov, and Gregory Chaitin--but really refined, expanded, and championed by Chaitin. In the physical sense, perhaps algorithmic information theory could be compared to the information of the Carnot engine that is the mechanism to extract useful information from the thermodynamic gradient, while the Shannon information is the information that is to be extracted. As it were, a compact description of an algorithm versus a compact but complete description of something (as specific case) of what that algorithm actually produced.

In the example mentioned, the equation might be thought as the object of algorithmic information while the specific arithmetic at each location would be the Shannon information. As I understand, the surface (Shannon) information relates to the volume in a similar manner as a 2-dimensional curl field is equal to the line integral of a contour enclosing it. Concerning a black hole, this would certainly make for an evermore complex calculation in the limit of smaller magnitude ds (or rather da), and greater accuracy, relative to the global formula mentioned.

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I think at points throughout this thread, there's some confusion between a generalised information entropy and a more specific thermodynamic entropy

Gravitational clumping increases entropy.... In the presence of gravity, entropy works differently from the textbook examples that are given for systems like gases.
I agree. But I did not find any sources, how entropy works with gravity (except Verlinde gravity theory).
Can you please post some links or explanation ? Thanks.

PeterDonis
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I did not find any sources, how entropy works with gravity
It's not discussed much in most textbooks. But heuristically, consider a large expanse of gas contained in some large volume. In the absence of gravity, the most likely state of the gas is filling the volume at uniform density. Small perturbations in density will tend to disappear as the gas re-diffuses to uniform density.

In the presence of gravity, however, the most likely state of the gas is all clumped together. If the gas is spread out at uniform density, small perturbations in density will get amplified.

What we don't really have at this point is a microscopic description of gravity that allows us to sharpen up the heuristic argument above into a rigorous derivation of how the processes in each case (spreading out to uniform density in the absence of gravity, vs. clumping in the presence of gravity) increase entropy in terms of the counting of microscopic states.

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What we don't really have at this point is a microscopic description of gravity that allows us to sharpen up the heuristic argument above into a rigorous derivation of how the processes in each case (spreading out to uniform density in the absence of gravity, vs. clumping in the presence of gravity) increase entropy in terms of the counting of microscopic states.
Yes, this was exactly the piont of my question.
I can imagine a calculating of entropy by counting of microscopic states in the case of absence of gravity. It is a subject of some textbooks (well, maybe more likely heuristicly).
But the same method in the presence of gravity leads to wrong result. So the question is how to compute entropy in presence of gravity. There is missing something in the computing method.

PeterDonis
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So the question is how to compute entropy in presence of gravity.
Yes, and the answer, as I said, is that we don't know, because we don't have a quantum theory of gravity, which means we don't know how to count microstates of the gravitational field (or of spacetime geometry, which is the same thing) in order to compute entropy statistically the way we can for non-gravitational systems.