Dear Dr. B: Why do physicists worry so much about the black hole information paradox?

“Dear Dr. B,

Why do physicists worry so much about the black hole information paradox, since it looks like there are several, more mundane processes that are also not reversible? One obvious example is the increase of the entropy in an isolated system and another one is performing a measurement according to quantum mechanics.

Regards, Petteri”

Dear Petteri,

This is a very good question. Confusion orbits the information paradox like accretion disks orbit supermassive black holes. A few weeks ago, I figured even my husband doesn’t really know what the problem is, and he doesn’t only have a PhD in physics, he has also endured me rambling about the topic for more than 15 years!

So, I’m happy to elaborate on why theorists worry so much about black hole information. There are two aspects to this worry: one scientific and one sociological. Let me start with the scientific aspect. I’ll comment on the sociology below.

In classical general relativity, black holes aren’t much trouble. Yes, they contain a singularity where curvature becomes infinitely large – and that’s deemed unphysical – but the singularity is hidden behind the horizon and does no harm.

As Stephen Hawking pointed out, however, if you take into account that the universe – even vacuum – is filled with quantum fields of matter, you can calculate that black holes emit particles, now called “Hawking radiation.” This combination of unquantized gravity with quantum fields of matter is known as “semi-classical” gravity, and it should be a good approximation as long as quantum effects of gravity can be neglected, which means as long as you’re not close by the singularity.

Illustration of black hole with jet and accretion disk.
Image credits: NASA.

Hawking radiation consists of pairs of entangled particles. Of each pair, one particle falls into the black hole while the other one escapes. This leads to a net loss of mass of the black hole, ie the black hole shrinks. It loses mass until entirely evaporated and all that’s left are the particles of the Hawking radiation which escaped.

Problem is, the surviving particles don’t contain any information about what formed the black hole. And not only that, information of the particles’ partners that went into the black hole is also lost. If you investigate the end-products of black hole evaporation, you therefore can’t tell what the initial state was; the only quantities you can extract are the total mass, charge, and angular momentum- the three “hairs” of black holes (plus one qubit). Black hole evaporation is therefore irreversible.

Irreversible processes however don’t exist in quantum field theory. In technical jargon, black holes can turn pure states into mixed states, something that shouldn’t ever happen. Black hole evaporation thus gives rise to an internal contradiction, or “inconsistency”: You combine quantum field theory with general relativity, but the result isn’t compatible with quantum field theory.

To address your questions: Entropy increase usually does not imply a fundamental irreversibility, but merely a practical one. Entropy increases because the probability to observe the reverse process is small. But fundamentally, any process is reversible: Unbreaking eggs, unmixing dough, unburning books – mathematically, all of this can be described just fine. We merely never see this happening because such processes would require exquisitely finetuned initial conditions. A large entropy increase makes a process irreversible in practice, but not irreversible in principle.

That is true for all processes except black hole evaporation. No amount of finetuning will bring back the information that was lost in a black hole. It’s the only known case of a fundamental irreversibility. We know it’s wrong, but we don’t know exactly what’s wrong. That’s why we worry about it.

The irreversibility in quantum mechanics, which you are referring to, comes from the measurement process, but black hole evaporation is irreversible already before a measurement was made. You could argue then, why should it bother us if everything we can possibly observe requires a measurement anyway? Indeed, that’s an argument which can and has been made. But in and by itself it doesn’t remove the inconsistency. You still have to demonstrate just how to reconcile the two mathematical frameworks.

This problem has attracted so much attention because the mathematics is so clear-cut and the implications are so deep. Hawking evaporation relies on the quantum properties of matter fields, but it does not take into account the quantum properties of space and time. It is hence widely believed that quantizing space-time is necessary to remove the inconsistency. Figuring out just what it would take to prevent information loss would teach us something about the still unknown theory of quantum gravity. Black hole information loss, therefore, is a lovely logical puzzle with large potential pay-off – that’s what makes it so addictive.

Now some words on the sociology. It will not have escaped your attention that the problem isn’t exactly new. Indeed, its origin predates my birth. Thousands of papers have been written about it during my lifetime, and hundreds of solutions have been proposed, but theorists just can’t agree on one. The reason is that they don’t have to: For the black holes which we observe (eg at the center of our galaxy), the temperature of the Hawking radiation is so tiny there’s no chance of measuring any of the emitted particles. And so, black hole evaporation is the perfect playground for mathematical speculation.

[Lots of Papers. Img: 123RF]

There is an obvious solution to the black hole information loss problem which was pointed out already in early days. The reason that black holes destroy information is that whatever falls through the horizon ends up in the singularity where it is ultimately destroyed. The singularity, however, is believed to be a mathematical artifact that should no longer be present in a theory of quantum gravity. Remove the singularity and you remove the problem.

Indeed, Hawking’s calculation breaks down when the black hole has lost almost all of its mass and has become so small that quantum gravity is important. This would mean the information would just come out in the very late, quantum gravitational, phase and no contradiction ever occurs.

This obvious solution, however, is also inconvenient because it means that nothing can be calculated if one doesn’t know what happens nearby the singularity and in strong curvature regimes which would require quantum gravity. It is, therefore, not a fruitful idea. Not many papers can be written about it and not many have been written about it. It’s much more fruitful to assume that something else must go wrong with Hawking’s calculation.

Sadly, if you dig into the literature and try to find out on which grounds the idea that information comes out in the strong curvature phase was discarded, you’ll find it’s mostly sociology and not scientific reasoning.

If the information is kept by the black hole until late, this means that small black holes must be able to keep many different combinations of information inside. There are a few papers which have claimed that these black holes then must emit their information slowly, which means small black holes would behave like a technically infinite number of particles. In this case, so the claim, they should be produced in infinite amounts even in weak background fields (say, nearby Earth), which is clearly incompatible with observation.

Unfortunately, these arguments are based on an unwarranted assumption, namely that the interior of small black holes has a small volume. In GR, however, there isn’t any obvious relation between surface area and volume because space can be curved. The assumption that such small black holes, for which quantum gravity is strong, can be effectively described as particles is equally shaky. (For details and references, please see this paper I wrote with Lee some years ago.)

What happened, to make a long story short, is that Lenny Susskind wrote a dismissive paper about the idea that information is kept in black holes until late. This dismissal gave everybody else the opportunity to claim that the obvious solution doesn’t work and to henceforth produce endless amounts of papers on other speculations.

Excuse the cynicism, but that’s my take on the situation. I’ll even admit having contributed to the paper pile because that’s how academia works. I too have to make a living somehow.

So that’s the other reason why physicists worry so much about the black hole information loss problem: Because it’s speculation unconstrained by data, it’s easy to write papers about it, and there are so many people working on it that citations aren’t hard to come by either.

Thanks for an interesting question, and sorry for the overly honest answer.