Abstract
Recent work has established a route towards the semiclassical validity of the Page curve, and so provided evidence that information escapes an evaporating black hole. However, a protocol to explicitly recover and make practical use of that information in the classical limit has not yet been given. In this paper, we describe such a protocol, showing that an observer may reconstruct the phase space of the black hole interior by measuring the Uhlmann phase of the Hawking radiation. The process of black hole formation and evaporation provides an invertible map between this phase space and the space of initial matter configurations. Thus, all classical information is explicitly recovered. We assume in this paper that replica wormholes contribute to the gravitational path integral.
Highlights
R is a region of space containing all of the Hawking radiation, while I is a separate region called the ‘island’
A protocol to explicitly recover and make practical use of that information in the classical limit has not yet been given. We describe such a protocol, showing that an observer may reconstruct the phase space of the black hole interior by measuring the Uhlmann phase of the Hawking radiation
We have shown that the curvature of the Uhlmann phase of Hawking radiation is equivalent to the symplectic form of the island and the radiation
Summary
The set-up we consider consists of a spacetime divided into a gravitational region and an asymptotic region, which are coupled together at a ‘cutoff surface’. We form a black hole in the gravitational region, and allow the asymptotic region to absorb the resulting Hawking radiation, until the black hole evaporates. This is illustrated in figure 1.4 The spacetime depicted lies between two. The map representing the evolution from Σ1 to Σ2 is not invertible, so from a naive perspective classical information appears to have been lost down the black hole. This is the problem we intend to solve in this paper. One can think of HR as being the Hilbert space of the Hawking radiation itself
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