Musings

No Information Lost Here!

The blogosphere — or, at least, that little corner of it that I pay attention to — is all a-twitter about Hawking’s announcement that he has solved the blackhole information problem (30 years after posing it). Normally, I try not to devote attention to such things, but the flurry of discussion prompted me to take a look at the transcript of Hawking’s talk.

He starts off with the amusing comment,

I adopt the Euclidean approach, the only sane way to do quantum gravity non-perturbative [sic].

Of course, among its other defects, the Euclidean path-integral he wishes to do is horribly infrared divergent. So his first step is to introduce an infrared regulator, in the form of a small negative cosmological constant. This is not merely a technicality. None of the subsequent arguments make any sense without it.

Anyone who hasn’t been asleep for the past 6 years knows that quantum gravity in asymptotically anti-de Sitter space has unitary time evolution. Blackholes may form and evaporate in interior, but the overall evolution is unitary and is holographically dual to the evolution in a gauge theory on the boundary.

With the large accumulation of evidence for AdS/CFT, I doubt there are many hold-outs left who doubt that the above statement holds, not just in the semiclassical limit that Hawking considers, but in the full nonperturbative theory.

Nonetheless, a “bulk” explanation of what is going on is desirable, and Hawking claims to provide one. Hawking devotes a long discussion to the point that trivial topology dominates the Euclidean path-integral (at zero temperature). Since the trivial topology can be foliated by spacelike surfaces, one can straightforwardly Wick-rotate and it follows that Minkowski-signature time evolution is unitary. Presumably, Hawking is aware of, but neglected to mention Witten’s old paper, which not only show the dominance of the trivial topology at low temperature, but shows that, at high temperature, the path integral is dominated by the Hawking-Page instanton (the analytic continuation of the AdS blackhole) and that, moreover, the phase transition which separates these two regimes (which Hawking and Page argued for, in the context of semiclassical gravity in AdS) is related to the confinement/deconfinement transition in the large-N gauge theory.

All of these are true facts, well-known to anyone familiar with AdS/CFT. But the latter goes well beyond the semiclassical approximation that Hawking uses. No one (at least, no one I talk to) has the slightest doubt that quantum gravity has unitary time evolution in asymptotically AdS space. The blackhole information paradox is solved in AdS, and it was “solved” long ago.

However, most people agree that the extrapolation to zero cosmological constant is not straightforward. There still room to doubt that time evolution in asymptotically flat space is unitary. On the thorny issue of extrapolating to zero cosmological constant, Hawking is silent.