Back in 2001, in a truly beautiful paper, Juan Maldacena formulated a version of Hawking’s information paradox, which has the added advantage that it could be discussed and analyzed in the context of a complete background independent theory of quantum gravity, namely that of the AdS/CFT correspondence.
This variant is similar to the original paradox, formulated for black holes surrounded by flat space, in that it displays a sharp conflict between properties of black holes in classical General Relativity, and basic postulates of quantum mechanics. Alas, it is also different in many crucial ways from the original paradox. Despite that, Juan’s proposed resolution to his paradox seems to have led to Hawking’s arguments, who managed to convince himself (though I think it is fair to say not too many others, unless they were already convinced) that information is not lost after all in the process of black hole formation and evaporation.
To present the paradox, first I’ll have to tell you just one fact about AdS spaces. AdS space has a center and a boundary, and a gravitational potential which makes it difficult to approach the boundary. Anything you throw out from the center towards the boundary of AdS space will eventually come back and hit you. It is as if you are surrounded by reflecting mirrors. This fact is a useful in many ways, in particular in stabilizing black holes.
For certain black holes (the so-called large ones, which are the ones we’ll discuss) this means that the Hawking radiation emitted by the black hole is constantly reflected back towards the black hole. Soon enough the system will achieve an equilibrium state, where the black hole exists is in detailed balance with its Hawking radiation, it absorbs and emits the same amount of radiation.
Which means that the black hole never disappears, unlike its cousin in asymptotically flat space, it is eternal. Therefore, the original information loss paradox (roughly, what happens to all the information stored in the BH after it completely evaporates?) cannot be formulated in this context (at least using the black holes I am describing). Tough luck.
So, instead of looking at the complicated process of black hole formation and evaporation, let us start with one of those stable black hole and perturb it, add to it one bit of information, poke it just a little bit, you get the picture. Then we can try and follow the history of the perturbation and ask: after some time has passed, can we trace back the nature of the perturbation? can we retrieve the information we sent towards the black hole from the set of measurements we can do later in time?
Turns out, this process is puzzling in the same way the original paradox was, but the advantage is that the new puzzle can be formulated in a complete microscopic theory of quantum gravity, namely in the gauge theory variables of the gauge-gravity duality. Let me describe the process and show where the problem lies.
The original black hole, formulated in gauge theory variables, is simply a thermal state, it is a thermal equilibrium of quarks and gluons at a temperature which equals the black hole Hawking temperature. Now, imagine shooting a very energetic particle into that thermal gas. What will happen? it will start colliding with all the gas particles, thereby slowing down while speeding up the average speed of the other particles. Wait long enough and you will find a gas of particles in a new thermal equilibrium state, in a slightly higher temperature.
This is exactly what happens in the gauge theory, and it has a simple interpretation in the gravity language: you send some bundle of energy towards the black hole, and it falls behind the horizon. The black hole “rings” for a little bit and then settles back to an approximate equilibrium state, which is a slightly larger black hole. The perturbation decays exponentially as function of time (the exponents are known as quasi-normal modes), after a little while you forget the black hole was perturbed, the information has disappeared behind the horizon.
But now comes the paradoxical part: in quantum mechanics this description can only be an approximation to the complete situation, because information is never lost. In principle you can always perform precise enough measurements on that seemingly thermal state to retrieve the information about the original perturbation. This is the principle of quantum mechanical unitarity, evolution in quantum mechanics is reversible.
The clearest way to see what is going on is looking at the behavior of the perturbation after a very long time. In the story we told, the effect of the perturbation dies off exponentially with time. In the complete quantum mechanical system that is what happens initially: for a very long time the effect dies off, but it cannot go all the way to zero, lest the information really be lost. Instead, after very long time the system will undergo Poincare recurrences: it will come back arbitrarily close to its initial configuration. So, looking at the behavior of the perturbation after extremely long times, of the order of the so-called Poincare recurrence time, is one clear way of deciding whether information is lost or not.
This is very much “in theory” type of situation, the common (unfortunate) analogy seems to be trying to retrieve the contents of a burnt book from its ashes. Nevertheless, this is a question of principle: is there a set of measurements we can do on that perturbed black hole which can be used to retrieve the complete information in the original perturbation?
Looking at the black hole solution of General relativity, if we take this seriously as the complete description of the situation, the precise answer is that the effect of perturbation dies of exponentially, for all times. Wait long enough and the system is arbitrarily close to an equilibrium state. In that case, you can never retrieve the information, it is lost. You can say words like “it continue to live behind the horizon”, but for you, the observer staying in the safety of the outer region, that information is lost.
On the other hand, the system has a dual which is a more complete and microscopic description of the theory. When looking at the process in the complete theory you can see that information is never really lost. In fact, you can do more – you can also truncate the complete theory, taking into account classical gravity only, and observe that in fact with this truncation information really is lost. In other words, you can see the precise details of how the illusion of information loss comes about.
So, what is the current state of affairs? in the dual language, it is clear that the system is unitary, but it also became clear that in the limit corresponding to classical gravity, there is an apparent information loss. In fact, this information loss looks like a generic phenomena of that (so-called large N) limit in a whole slew of models, even those not dual to any known gravity theories. So, it looks like this paradox is pretty much solved, at least no mystery as to whether information is really lost or not.
What is missing at the moment is a phrasing of the resolution in the original, gravitational variables. Since we have a complete description of the story, it is known in detail that information is not lost, and why it seems to be lost as an artifact of the specific approximation we are making. What we are missing is the always elusive question of the “mecahnism” of retoring unitarity. What are the essential ingredients of the complete story that are needed in order to restore unitarity. Can we phrase the solution in terms of some coherent narrative phrased in the gravitational language? would be nice if we could, maybe we are not that far off, stay tuned.