Black Hole Information Paradox 1. Susskind vs. Hawking

Stephen Hawking showed that there is a big problem with the information, if black holes are present. If singularities swallow matter without returning it, information vanishes too. If the initial state of a system is pure, after a part of the system falls in the black hole, it is entangled with the part remaining outside. After the part falling inside vanishes in the singularity, the outside part remains in a mixed state (its state depends on the state of the inside of the black hole). This was not considered to be a problem before Hawking's analysis, because it was assumed that the infalling part is there, somewhere inside the black hole (I represented this in figure 1, A., in a Penrose-Carter diagram). It became an issue after Hawking (and his precursors, Zel'dovich and Starobinski) found that the black holes evaporate. The problem was that the black hole may completely evaporate, vanishing with the information it swallowed, and the outside part remains in a mixed state. This means that the information is lost forever, and the unitary evolution is broken (fig. 1, B.).

Figure 1. A. In the case of a non-evaporating black hole, by choosing an appropriate foliation of spacetime in space+time, unitarity is not broken. B. Unitarity is broken for an evaporating black hole, because when the initial state is pure, the final state is mixed.

Hawking's result was bad news for many physicists of good-faith. Hawking obtained it by combining the most trusted and proven theories we know, General Relativity and Quantum Theory. So, his result became acknowledged as a paradox, because he seemed to force us to choose between things we considered to be true. On the one hand, Quantum Theory is based on unitary evolution, and we don't want to be broken. On the other hand, Hawking impeccably combined GR and QT to prove that the radiation obtained during evaporation doesn't carry information out of the black hole.

Two sides were formed. While many didn't accept the idea that information is lost, another side, made mostly of General Relativists (who also happens to support Quantum Field Theory), were more willing to accept that information is lost. Among those who don't think that violation of unitarity is such a catastrophe that would destroy the world, there were Hawking, Kip Thorne, Roger Penrose, who explains his position in Cycles of Time, also Robert Wald in his recent talk at the conference Fuzzorfire workshop, seems not to be bothered by this, etc. The side which rejected the possibility that information may be lost was made of John Preskill, Leonard Susskind, Gerard 't Hooft, and others. In 1997, Thorne and Hawking bet against Preskill that information is really lost in the black hole.

In 2008, at 15 years after he, with his collaborators, came with a solution, Susskind wrote a book, The Black Hole War: My Battle with Stephen Hawking to Make the World Safe for Quantum Mechanics. In the book, he recalls that, while discussing with L. Thorlacius and J. Uglum what they will do at a conference in 1993, he told them

Damn it, I don't want to talk about CGHS or RST. It's a dead end. I want to do something that really will shake things up. Let's go way out on a limb and say something very bold that will really get their attention.

They did. They proposed a very intriguing solution to the black hole information paradox, described in the paper "The Stretched Horizon and Black Hole Complementarity". It is also explained very well in the book L. Susskind, J. Lindesay, An introduction to black holes, information and the string theory revolution, World Scientific, 2005. I will discuss in more detail this solution in the next post. Their ideas were very clever, and convinced a large part of the community of physicists that it was the correct solution.

In 2004, Hawking announced at a conference that he conceded the bet. Kip Thorne haven't change his position.In his book The Black Hole War: My Battle with Stephen Hawking to Make the World Safe for Quantum Mechanics, Susskind presents Hawking, in an amusing style, as a tyrant, an enemy of Quantum Mechanics. He seems to believe that he defeated Hawking, and that this is why Hawking conceded the bet with Preskill. Others followed Susskind's declarations, and spread the idea that he defeated Hawking.

In fact, Hawking was probably not convinced by the black hole complementarity, but rather by Maldacena's AdS/CFT correspondence, and wrote a paper in which he explained his own solution saving the information. In the paper, Hawking didn't need to use black hole complementarity, but he mentioned Maldacena's results, which rely on Susskind's and 't Hooft holographic principle.

In his proposal, Hawking uses the method of sum over histories, originated by Feynman, but developed as an approach to quantum gravity by himself and J. B. Hartle. To solve the information loss problem, he proposes that one should sum over topologies of spacetime. While it is clear that the trivial topologies, those without black holes, don't break unitarity, the non-trivial ones, including those containing black holes, break it. So, Hawking tries to argue that the non-trivial topologies don't contribute to the sum over histories. In other words, if a possible history has black holes, information is lost, but if we consider all of them, it is not lost, because the histories losing information don't contribute to the overall sum.

His solution was criticized, for instance for not being well supported mathematically, by John Preskill. Indeed, Hawking's paper from 2005 looks rather like a sketch of a research program, where the key points are merely conjectured. Not much progress was made in that direction since then.

My main objection to his proposal is the following. In general, when we sum over histories, we have to impose some boundary conditions. For example, in the case of the two-slit experiment, we take into consideration only paths that go through the slits. Similarly, if we want to see what happens with information in the presence of black holes, using the sum over histories, one should impose conditions compatible with the presence of the black hole. Or, Hawking claims that the only contributions are given precisely by the histories which are not compatible with the presence of black holes, and ignores exactly the histories which actually should be considered.

When making his proposal, it seems that Hawking was not aware neither that he was the "evil side" in a black hole war, nor that Susskind defeat him. Recently, at the Fuzzorfire workshop, Hawking asked for the definition of the stretch horizon, to which Susskind replied that he already told the definition 20 years ago, then left (minute 45).

I think that, when Hawking raised the problem of information loss, he did a great job. This is a very good problem indeed, and fueled plenty of research. In following posts, I will argue that in this process, Susskind did an excellent job too, by finding something very important, in my opinion, but then he lost it, by inventing the black hole complementarity principle. Next, in Stretched Complementarity, I will explain why I don't buy Susskind's solution.