Quantum evaporation of 2-d black holes

by John Baez

• Abhay Ashtekar, Quantum evaporation of 2-d black holes, 21 September 2010. PDF of the slides, and audio in either .wav (45MB) or .aif format (4MB).

Abhay Ashtekar has long been one of the leaders of loop quantum gravity. Einstein described gravity using a revolutionary theory called general relativity. In the mid-1980s, Ashtekar discovered a way to reformulate the equations of general relativity in a way that brings out their similarity to the equations describing the other forces of nature. Gravity has always been the odd man out, so this was very intriguing.

Shortly thereafter, Carlo Rovelli and Lee Smolin used this new formulation to tackle the problem of quantizing gravity: that is, combining general relativity with the insights from quantum mechanics. The result is called "loop quantum gravity" because in an early version it suggested that at tiny distance scales, the geometry of space was not smooth, but made of little knotted or intersecting loops.

Later work suggested a network-like structure, and still later time was brought into the game. The whole story is still very tentative and controversial, but it's quite a fascinating business. Maybe this movie will give you a rough idea of the images that flicker through people's minds when they think about this stuff:





... though personally I hear much cooler music in my head.

Now, one of the goals of any theory of quantum gravity must be to resolve certain puzzles that arise in naive attempts to blend general relativity and quantum mechanics. And one of the most famous is the so-called black hole information paradox. (I don't think it's actually a "paradox", but that's what people usually call it.)

The problem began when Hawking showed, by a theoretical calculation, that black holes aren't exactly black. In fact he showed how to compute the temperature of a black hole, and found that it's not zero. Anything whose temperature is above absolute zero will radiate light: visible light if it's hot enough, infrared if it's cooler, microwaves if it's even cooler, and so on. So, black holes must 'glow' slightly.

Very slightly. The black holes that astronomers have actually detected, formed by collapsing stars, would have a ridiculously low temperature: for example, about 0.00000002 degrees Kelvin for a black hole that's 3 times the mass of our Sun. So, nobody has actually seen the radiation from a black hole.

But Hawking's calculations say that the smaller a black hole is, the hotter it is! Its temperature is inversely proportional to its mass. So, in principle, if we wait long enough, and keep stuff from falling into our black hole, it will 'evaporate'. In other words: it will gradually radiate away energy, and thus lose mass (since E = mc²), and thus get hotter, and thus radiate more energy, and so on, in a vicious feedback loop. In the end, it will disappear in a big blast of gamma rays!

At least that's what Hawking's calculations say. These calculations were not based on a full-fledged theory of quantum gravity, so they're probably just approximately correct. This may be the way out of the "black hole information paradox".

But what's the paradox? Patience — I'm gradually leading up to it. First, you need to know that in all the usual physical processes we see, information is conserved. If you've studied physics you've probably heard that various important quantities don't change with time: they're "conserved". You've probably heard about conservation of energy, and momentum, and angular momentum and electric charge. But conservation of information is equally fundamental, or perhaps even more so: it says that if you know everything about what's going on now, you can figure out everything about what's going on later — and vice versa, too!

Actually, if you've studied physics a little but not too much, you may find my remarks puzzling. If so, don't feel bad! Conservation of information is usually not mentioned in the courses that introduce the other conservation laws. The concept of information is fundamental to thermodynamics, but it appears in disguised form: "entropy". There's a minus sign lurking around here: while information is a precise measure of how much you do know, "entropy" measures how much you don't know. And to add to the confusion, the first thing they tell you about entropy is that it's not conserved. Indeed, the Second Law of Thermodynamics says that the entropy of a closed system tends to increase!

But after a few years of hard thinking and heated late-night arguments with your physics pals, it starts to make sense. Entropy as considered in thermodynamics is a measure of how much information you lack about a system when you only know certain things about it — things that are easily measured. For example, if you have a box of gas, you might measure its volume and energy. You'd still be ignorant about the positions of all the molecules inside. The amount of information you lack is the entropy of the gas.

And as time passes, information tends to pass from easily measured forms to less easily measured forms, so people say entropy increases. But the information is still there in principle — it's just hard to access. So information is conserved.

There's a lot more to say here. For example: why does information tend to pass from easily measured forms to less easily measured forms, instead of the reverse? Does thermodynamics require a fundamental difference between future and past — a so-called "arrow of time"? Alas, I have to sidestep this question, because I'm supposed to be telling you about the black hole information paradox.

So: back to black holes!

Suppose you drop an encyclopedia into a black hole. The information in the encyclopedia seems to be gone. At the very least, it's extremely hard to access! So, people say the entropy has increased. But could the information still be there in hidden form?

Hawking's original calculations suggested the answer is no. Why? Because they said that as the black hole radiates and shrinks away, the radiation it emits contains no information about the encyclopedia you threw in — or at least, no information except a few basic things like its energy, momentum, angular momentum and electric charge. So no matter how clever you are, you can't examine this radiation and use it to reconstruct the encyclopedia article on, say, Aardvarks. This information is lost to the world forever!

So what's the black hole information paradox? Well, it's not exactly a "paradox". The problem is just that in every other process known to physicists, information is conserved — so it seems very unpalatable to allow any exception to this rule. But if you try to figure out a way to save information conservation in the case of black holes, it's tough. Tough enough, in fact, to have bothered many smart physicists for decades.

Indeed, Stephen Hawking and the physicist John Preskill made a famous bet about this puzzle in 1997. Hawking bet that information wasn't conserved; Preskill bet it was. In fact, they bet an encyclopedia!

In 2004 Hawking conceded the bet to Preskill, as shown above. It happened a conference in Dublin — I was there and blogged about it. Hawking conceded because he did some new calculations suggesting that information can gradually leak out of the black hole, thanks to the radiation. In other words: if you throw an encyclopedia in a black hole, a sufficiently clever physicist can indeed reconstruct the article on Aardvarks by carefully examining the radiation from the black hole. It would be incredibly hard, since the information would be highly scrambled. But it could be done in principle.

Unfortunately, Hawking's calculation is very hand-wavy at certain crucial steps — in fact, more hand-wavy than certain calculations that had already been done with the help of string theory (or more precisely, the AdS-CFT conjecture). And, neither approach makes it easy to see in detail how the information comes out in the radiation.

This finally brings us to Ashtekar's talk. Despite what you might guess from my warmup, his talk was not about loop quantum gravity. Certainly everyone working on loop quantum gravity would love to see this theory resolve the black hole information paradox. I'm sure Ashtekar is aiming in that direction. But his talk was about a warmup problem, a "toy model" involving black holes in 2d spacetime instead of our real-world 4-dimensional spacetime.

The advantage of 2d spacetime is that the math becomes a lot easier there. There's been a lot of work on black holes in 2d spacetime, and Ashtekar is presenting some new work on an existing model, the Callen-Giddings-Harvey-Strominger black hole. This new work is a mixture of analytical and numerical calculations done over the last 2 years by Ashtekar together with Frans Pretorius, Fethi Ramazanoglu, Victor Taveras and Madhavan Varadarajan.

I will not attempt to explain this work in detail! The main point is this: all the information that goes into the black hole leaks back out in the form of radiation as the black hole evaporates.

But the talks also covers many other interesting issues. For example, the final stages of black hole evaporation display interesting properties that are independent of the details of its initial state. Physicists call this sort of phenomenon "universality".

Furthermore, when the black hole finally shrinks to nothing, it sends out a pulse of gravitational radiation, but not enough to destroy the universe. It may seem very peculiar to imagine that the death spasms of a black hole could destroy the universe, but in fact some approximate "semiclassical" calculations of Hawking and Stewart suggested just that! They found that the dying black hole emitted a pulse of infinite spacetime curvature — dubbed a "thunderbolt" — which made it impossible to continue spacetime beyond that point. But they suggested that a more precise calculation, taking quantum gravity fully into account, would eliminate this effect. And this seems to be the case.

For more, listen to Ashtekar's talk while looking at the PDF file of his slides!