### Gottesman & Preskill

Dan Gottesman and John Preskill have a nice little paper which seems to be a quite devastating critique of a recent proposal by Horowitz and Maldacena on blackhole information.

The blackhole information paradox, in a nutshell, is the apparent difficulty in finding a unitary evolution from the initial state of the infalling matter, which forms the blackhole, to the final state of the outgoing Hawking radiation. Horowitz and Maldacena’s proposal involves a particularly clever decomposition of the Hilbert space of the stuff “inside” the horizon, and the imposition of a final state condition at the blackhole singularity.

Let $\mathcal{H}_M$ be the Hilbert space of the infalling matter, $\mathcal{H}_{\mathrm{in}}$ and $\mathcal{H}_{\mathrm{out}}$ be the Hilbert spaces of the infalling and outgoing Hawking radiation, respectively. Each of these Hilbert spaces are purported to be of dimension $N=e^S$, where $S$ is the entropy of the blackhole. We can take them to have orthonormal bases $| e_i \rangle$, $| f_i \rangle$ and $| g_i \rangle$, respectively. The Unruh state, $| \mathcal{U} \rangle \in \mathcal{H}_{\mathrm{in}}\otimes \mathcal{H}_{\mathrm{out}}$ of the Hawking radiation is

Horowitz and Maldacena propose that the final state, $\langle F | \in \mathcal{H}_M \otimes \mathcal{H}_{\mathrm{in}}$, at the singularity be given by

In that case, the evolution operator, $T: \mathcal{H}_M\to \mathcal{H}_{\mathrm{in}}$ is determined by

with the important proviso that we should consider only conditional probabilities: *given* that the final state at the singularity is $\langle F |$, what is the amplitude to go from some initial state in $\mathcal{H}_M$ to some final state in $\mathcal{H}_{\mathrm{out}}$? That has the effect of multiplying $\hat{T}$ by $N$, with the result that

is unitary.

*But*, say Gottesman and Preskill, the Hamiltonian governing the time-evolution inside the horizon is unlikely to be diagonal $H \neq H_M\otimes 1 + 1\otimes H_{\mathrm{in}}$. Instead, interactions between the infalling Hawking fluctuations and the infalling matter will produce some more general state of the stuff inside the horizon. Instead of $T$, the transition amplitude is given by

for some unitary operator $U: \mathcal{H}_M \otimes \mathcal{H}_{\mathrm{in}} \to \mathcal{H}_M \otimes \mathcal{H}_{\mathrm{in}}$.

Depending on the nature of $U$, we could have no information loss ($\tilde{T}$ unitary) or total information loss ( the final state of the Hawking radiation being totally independent of the initial state in $\mathcal{H}_M$) or anything in between.

Perhaps one can compensate for the effect of interactions by adjusting the choice of final state $\langle F |$, but it’s not exactly clear how one is supposed to do that.

The one thing that String Theory definitely *doesn’t* hand us is a description of quantum gravitational processes (like the formation and subsequent evaporation of a blackhole) in terms of Hilbert spaces, much less finite-dimensional ones which can be decomposed in the fashion described above. Maybe that’s a sign we need to get beyond our silly attachment to such ideas.

But I don’t know how to do that *either*.

## Re: Gottesman & Preskill

In retrospect, the business of imposing a final state boundary condition (‘postselection’) on an arbitrary collection of collapsing matter seems pretty damn suspicious.

I seem to remember Tipler saying something vaguely similar at the Cambridge M-theory Cosmology conference (for quite different reasons, though). The upshot was that the collapsing matter at the end of the (closed) Universe would have to reach the Omega Point in order to rearrange itself (with infinite intelligence and potency) into the unique state for which the final collapse was allowed. Bunkum, of course.

Although, in the face of the intractable difficulties of black hole singularities and information loss, we should try everything once.

If there is to be a final state of gravitational collapse, surely a physically reasonable one would be a state of maximal entropy. But that’s as far from solving the information problem as it can be, and may conflict with the string-theoretic picture of entropy living ‘around the horizon’.

What’s to stop us bringing a collection of collapsing matter in from asymptotically flat past infinity and watching the radiation come out at asymptotically flat future infinity? Could string theory, in principle, give us this S-matrix? (I use ‘in principle’ in its loosest sense…) Or is there some fundamental obstruction there?