Now I wrote the word information. For the first time in this blog, actually. People working in the field of quantum gravity use this word a lot, but not always precisely. It has a precise meaning both in classical and in quantum physics. Let me convince you that serious problems may already exist with classical information when paired with black holes, so that I can talk about quantum information in another blog post.
Classical information is the shared entropy between two systems. It has never been anything else than that, and will never be. If you are talking about a set of states and their probability distribution, you are talking about entropy. If you think you have information but you don't know what it predicts, you don't have information, you have entropy. In particular, imagine I have 4 bits of information (which allow me to reduce the entropy of system X, say, by 4 bits). Suppose I encode these 4 bits in a string 4 million bits long and the channel scrambles 23 million, say, of these bits. If the receiver of this string can reconstruct the 4 bits of information (via decoding), no information was lost. She can also reduce the information of X by 4 bits, and thus make exactly the same prediction that I, the sender, was able to make. The stuff that was lost was entropy, not information. The 3,999,994 bits that were used to encode the signal aren't predicting anything. Information is about prediction, after all, nothing else.
But I'll assume you know all this already, and if not you can read a little bit about it in a review I wrote.
But I'll assume you know all this already, and if not you can read a little bit about it in a review I wrote.
Before I go on, there is one last thing you have to know about black holes (those that have no charge, and don't spin on their axis crazily). They can only be distinguished by their mass. That's it. No color, no smell, no weird shape. With this in mind, you may already carry out devilish thought experiments in your head. What if I throw two different things of equal mass into the black hole? After they are swallowed, can you tell me which one I threw in? The answer appears to be no and we would have to seriously think about accusing the black hole of treacherous villainy. But first things first. Let's burn some books first.
OK, burning books is generally frowned upon (particularly by me), but let's keep the analogy for a moment. Let's imagine I have two different books. They weigh exactly the same. But one is, say, Shakespeare's "Hamlet", and the other, oh, Darwin's "Origin". (Two books I like a lot, by the way).
Let's say I throw one or the other into the fire, and they have exactly the same mass, and the cover and pages are exactly the same, except the text (unlike in the pics above). And let's imagine that after they have burned up, the ashes are just ashes. Was information lost? In practice, yes. In principle, no. In terms of classical communication theory, we are dealing with a noisy channel. The receiver cannot access the book, but only watch the flames. And you may think that the flames cannot possibly tell us about the identity of the book I just incinerated. But indeed they could, in principle. When "Hamlet" burns up, the flames and smoke are just a tiny bit different from what happens when "Origin" burns up. It may be imperceptible to your eye because it is lost in the natural variability of fire, but it is there. It must be there, otherwise the laws of physics would be violated. You can imagine an ultra-sensitive measurement device that can distinguish the two, or you can take a page from the book written by Shannon, and make your life a ton easier. You see, it is really quite normal that noise in a channel overwhelms the signal. But if there is any signal at all, then it can be protected from the noise via a process known as "encoding". This process makes the signal state identifiable, and you can imagine doing this by coating the books with some sort of phosphorescent substance before you throw them into the fire: red for "Hamlet", green for "Origins". Now, you just sit back and watch the color of the flame, and then you know which book was just burned.
The thing you have to understand here is that coating the books in this manner is not cheating, because information was never lost in principle, only in practice. We can make things more practical using coding, and this way we will be able to recover information with arbitrary accuracy.
Now let's throw the books into a black hole. You may think: "Oh, the Hawking radiation is just like the fire, we can encode the information in some way and just watch the 'color' of the Hawking radiation". Only this does not work at all. The Hawking radiation is not burning the books. The stuff that is emitted has absolutely nothing to do with what falls in, because for all I know the radiation was just created while the books were thrown in eons ago. There is no causal connection whatsoever between the books and the vacuum fluctuations. In fact Hawking himself acknowledged this right away: the radiation is completely and utterly thermal, which means that it depends on absolutely nothing, except the temperature. And the temperature of the black hole is set precisely by the mass, and the mass of each book is the same. I don't know about you, but I find such a situation absolutely untenable, because if this were all true, we would now have broken the law of "you can reverse anything". When I first read about this, I decided that it could not possibly be right, and embarked on figuring out why.
First, I replace the two books by just two particles, identifiable in some way. You can think of a particle or its anti-particle (of equal mass of course), or of a photon with one or the other polarization. Then I mentally throw them into the black hole. And nothing coming out and the black hole just sitting there almost makes me physically sick, so I realize that just before the particle disappears before the horizon, it must emit something, it simply must. Then I start reading. It's 2003, so I can't Google around. And I quickly happen upon the literature of the quantum theory of radiation, which describes how a black body responds to radiation. And I read a superb article by Einstein from 1917, where he describes how he derived Planck's radiation law using only what now looks like common sense assumptions, but which at the time must have looked like pure magic. In this ground-breaking paper, Einstein shows that when radiation is incident on a black body, three things happen: absorption, reflection, the spontaneous emission of radiation, and the stimulated emission of radiation. Stimulated emission is what give you a laser: a particle comes in, two (identical ones) come out. Put a mirror on one side, and two particles re-enter, and four come out. Put a mirror on the other side... and you get my drift. (To make a real laser, you have to make one of the mirrors a little permeable, so that the beam can finally get out).
Now, Hawking radiation has precisely Planck's form, but in Hawking's paper you only read about spontaneous emission. What happened to the stimulated part? In fact, I then realize, that emitting stimulated particles is precisely what I need to get rid of that queasy feeling in my stomach! So I read Hawking's paper again and again, and there is no stimulated emission. Zilch, nada.
So then I sit down and redo Hawking's calculation, but I take care not to throw out the bath water when, umm, there's still something in it. The calculation usually goes like this: You write down the vacuum in flat space time (far away from the black hole), and then you transform it into another basis, namely the one in the far future, in the presence of a black hole. This transformation is called a "Bogoliubov transformation" and it creates the future vacuum in which there are particles, from a past vacuum where there are none. Except that if any particles are actually forming the black hole, there should be some particles in the past too! So I just take the past vacuum with a single particle present and evolve it into the future, then I take the vacuum with a single anti-particle, and evolve it into the future. And lo and behold, everything changes! Suddenly, the radiation outside of the black hole at future infinity depends on what I threw in! Of course it has to, because the particle stimulated the emission of another particle before it went down the rabbit hole. Stimulated emission is just like making xerox copies. It's as if physics strips off the information from the particle (which is still falling into the hole) to make sure that the laws of physics are upheld. And I don't feel so terrible.
Then I read some more, and I find that I'm not the only who has noticed this. In fact, Jacob Bekenstein (working with his student Meisels) wrote a beautiful paper just a year after Hawking wrote his, where he essentially writes "Hold your horses Mr. Hawking, you... kinda... forgot something". Using just statistical arguments of the form Einstein used in his <looking for adjectives> really swell 1917 paper <that's a fail>, Bekenstein shows that if you have absorption, reflection, and spontaneous emission of radiation, then you must have stimulated emission. If not, you might get some, umm, paradoxes. Then my student Greg ver Steeg (who is helping me derive all the known results and deriving in parallel with me our new ones) and I discover that Panangaden and Wald have derived Bekenstein's result less than a year later in quantum field theory. But both expressions look very different from the result that we derived. First, we are worried that we have nothing new, then we worry that our calculation, which uses methods completely different from what Bekenstein and Meisels, as well as Pangaden and Wald have used, may be wrong. They look utterly different. The first thing we notice is that Bekenstein's result can be simplified enormously using some of the things we discovered. Then Greg codes both expressions into Mathematica and evaluates them numerically. And they agree excactly!
To make a long story slightly shorter, it took us another year to actually prove that the two expressions (ours and that of Bekenstein & Meisels, which was the same as that of Panangaden & Wald) can be turned into each other analytically, but there it was. Now, all we had to do is prove that including stimulated emission leads to a non-vanishing capacity of the information transmission channel. Because if you can do that, then black holes are exonerated, proven innocent, free to go! Well, perhaps we still have to show that the whole initial state of the black hole can be reconstructed from the final state in principle, but one step at a time! Let's first convince ourselves that the most basic laws aren't fractured, first. So that we can sleep again, and not tiptoe downstairs in the middle of the night to check a calculation that is too hard to do in your head. Really!
Part 3 is here.