Large black holes and information

[Elroch]

Hawking and others suggest that it is a very special deal that macroscopic black holes lose quantum information, but I only partially agree.

It is clearly important that they do not preserve most quantum numbers - black holes may be a way to break a lot of conservation laws of particle physics. But as for other information about particles (eg dynamical variables) they don't seem very special at all.

To explain what I mean, compare two situations. In the first case you have some matter about which you know a great deal and drop it into a black hole. Soon it is characterised by a few quantum numbers. Lots of information lost.

Second case you have a lot of gas molecules and know a lot about their state. You vent the gas into a sealed box. After a very short while uncertainty means you have no predictive power about their special state. All the information that you have left is a temperature and a few other numbers.

Isn't this really rather similar?

Of course the second case is not so dramatic: you still have molecules. But if you ignore that detail there is a strong analogy.

The argument that black holes destroy information in a more radical way seems to come from the idea that if you know a wave function you have perfect predictive power about the wave function in the future. This is a red herring, because in the real world we never know the value of a wave function (except in some sense for a bound state), we just have some observations which give us partial uncertain knowledge about the quantum state. These observations allow us to make uncertain predictions which get increasingly inaccurate (without further observations).

[Note: the issue of virtual quantum black holes losing information (which Hawking wrote about) sounds intriguing and may be more significant to our understanding of what we can know and predict about the universe]

 

[joris_pixie]

_small_ detail ?

I'm not very familiar with the subject. But I do believe almost all information is lost except mass and total angular moment.

I think you should read up a little more about quantum mechanics and relativity ;)

 

[Matterwave]

I think the difference between the two types of "loss of information" you describe is that for the case of the gas molecules information is only lost practically, while for the case of the black hole, information is lost even in principle.

What I mean is, for the gas molecule case, if you could track all the gas molecules that your sample interacts with, then you wouldn't lose information, you would just get one really big wave function. You only "lose information" when you trace out the environment.

In the black hole case, I think (i'm no expert) you lose information even if you don't do any "trace out" procedures.

 

[Elroch]

I'm not very familiar with the subject. But I do believe almost all information is lost except mass and total angular moment.

You are essentially right, although to be complete it would have been worth mentioning electrical charge as the third and final quantity preserved by black holes.

I think you should read up a little more about quantum mechanics and relativity ;)

Good advice to absolutely anyone interested in physics, doubtless including yourself

The point was that for the gas in a box too almost all information about the state was lost except two numbers (any two of temperature, pressure and mass will do, if you know the dimensions of the box and the type of molecule).

To make the analogy a little closer, a black hole is characterised by the precise particles it is emitting as black body radiation. But without more observations we know the temperature but not the precise random distribution of emission. For the black hole, a few numbers tell us everything predictable about the distribution of these emitted particles in the future; for the gas a few numbers tell us everything about the distribution of the molecules and the emitted radiation in the future.

Searching for flaws in this analogy, with the gas in a box, one can in principle take measurements which give more precise information about the state of the molecules for a tiny fraction of a second (after which this information is lost again). For the black hole, this cannot be done unless there are some sorts of random fluctuations in the state of the black hole which could serve this purpose (I believe there are). Also you could probe into the nature of the molecules using more powerful observations (Perhaps the random variations in a black hole are susceptible to probing?). But these issues don't stop the fact that in both cases information about a system disappears.

On reflection, I think it is loose wording in what I have read that is the issue. The mystery was that classical black holes were supposed to lose the ability to represent information, rather than merely to lose that information itself (as was often written). And the answer to this mystery is that while a black hole is macroscopically described by three parameters its state is described by a very large number of quantum numbers associated with its event horizon, in the same way that the gas is macroscopically described by a few numbers but its state is given by the very large number of quantum numbers for the wave function of trillions of molecules. These microstates presumably contain information which affects discrepancies in the behaviour of the black hole from a classical one, both in the emitted radiation and the very fine detail of how external particles would interact with the black hole.

 

[Elroch]

I think the difference between the two types of "loss of information" you describe is that for the case of the gas molecules information is only lost practically, while for the case of the black hole, information is lost even in principle.

What I mean is, for the gas molecule case, if you could track all the gas molecules that your sample interacts with, then you wouldn't lose information, you would just get one really big wave function. You only "lose information" when you trace out the environment.

In the black hole case, I think (i'm no expert) you lose information even if you don't do any "trace out" procedures.

This is probably the "red herring" I referred to earlier. If you knew the wave function for all the molecules in the gas, this would provide you with the wave function for all time. But the wave function is merely a useful tool. In reality you have real observations, not a wave function. The predictive power of such observations vanishes entirely in a surprisingly small fraction of a second due to the uncertainty in the behaviour of individual molecules when they collide. All the information you are left with is the basic thermodynamic parameters.

 

[Matterwave]

What if we looked at a simple case where one particle moves into an environment of simply 2 particles? Certainly, we could (at least in principle) keep track of that wave-function, even if they interacted with each other. But we cannot keep track of that wave-function if that 1 particle went into a black hole (created by 2 particles).

It seems to me that that is an essential difference and not a red herring. In one case, information is lost due to a practical matter (we can't perform all those measurements, nor can we keep track of all those variables), while in the black hole case, information is lost in principle (the no hair theorem says we can't know the full state of the inside of a black hole from the outside).

 

[Elroch]

What if we looked at a simple case where one particle moves into an environment of simply 2 particles? Certainly, we could (at least in principle) keep track of that wave-function, even if they interacted with each other.

The wave function is not something we know about the universe, it is something which encapsulates our state of knowledge about a system. Most physicists agree that it is merely a way to predict future observations from past observations.

But suppose we start with a box with a small number of particles in. Actually, let's make it two, because that illustrates the point fine.

Suppose initially we know everything we can know as much as quantum mechanics permits us about the particles.

Perhaps we have a pretty good idea of the momentums, and some idea of the positions. Our knowledge of the energy (derived from knowledge of the momentums) is preserved indefinitely without getting degraded (think temperature). But the fact that we can't know both position and momentum in each direction means that we lose all information about the position of the particles very quickly. The probabilistic density function becomes constant in the box very quickly.
The fact that there is more than one particle means we lose a lot of information each time they collide - a little error in position or momentum makes a big difference after several bounces off the cushion on a snooker table. In particular, uncertainty about these collisions loses all direction information very rapidly.

Some maths leads to the fact that equally quickly all we know about the momentum of the particles is that they have the classical Maxwell–Boltzmann distribution of momentum for each particle (with some anticorrelation between the two because we have only two particles and we have a pretty good idea of the total energy). In particular we know nothing about their directions (except anticorrelations we can derive from conservation of energy).

So even with 2 particles in a box, our knowledge of their state becomes limited to their temperature as a gas in a fraction of a second. Nothing more.

But we cannot keep track of that wave-function if that 1 particle went into a black hole (created by 2 particles).

It seems to me that that is an essential difference and not a red herring. In one case, information is lost due to a practical matter (we can't perform all those measurements, nor can we keep track of all those variables), while in the black hole case, information is lost in principle (the no hair theorem says we can't know the full state of the inside of a black hole from the outside).

The state of a black hole is the state of its event horizon. Bear in mind that past this point, the radius becomes time like, so points inside the black hole can be considered the future of things that fall in. Moreover these points are infinitely far in the future relative to any point outside of the black hole with any choice of frame, because of the infinite time dilation at the event horizon. They are certainly not "now" in any sense. The classical "no hair" theorem has since been qualified by the realisation that the microstates of the black hole's event horizon provide a sort of quantum hair, so is not strictly correct for real black holes.

I've just noticed that this thread is highly relevant. The theorems of black hole thermodynamics are almost unbelievably perfect. Worth mentioning is that contrary to losing the ability to store information, a black hole has the maximum entropy possible for its energy (mass), so it actually increases the information capacity.