Abstract
It has been suggested in recent work that the Page curve of Hawking radiation can be recovered using computations in semi-classical gravity provided one allows for ``islands" in the gravity region of quantum systems coupled to gravity. The explicit computations so far have been restricted to black holes in two-dimensional Jackiw-Teitelboim gravity. In this note, we numerically construct a five-dimensional asymptotically AdS geometry whose boundary realizes a four-dimensional Hartle-Hawking state on an eternal AdS black hole in equilibrium with a bath. We also numerically find two types of extremal surfaces: ones that correspond to having or not having an island. The version of the information paradox involving the eternal black hole exists in this setup, and it is avoided by the presence of islands. Thus, recent computations exhibiting islands in two-dimensional gravity generalize to higher dimensions as well.
Highlights
The RT/HRT/EW formula [1,2,3] for computing entanglement entropies is a remarkable entry in the holographic dictionary
We will denote the area of the surface that penetrates the horizon by AH(x∂ ) and the area of the surface that ends on the Planck brane by A∂ M(x∂, yB), where yB is the value of y at which this surface intersects the brane
The resolution is that the area of the surface that ends on the Planck brane approaches approximately 2SBH at late times, as in [15]
Summary
The RT/HRT/EW formula [1,2,3] for computing entanglement entropies is a remarkable entry in the holographic dictionary. The entanglement entropy of the black hole was seen to undergo a first order phase transition following the appearance of a new nontrivial quantum extremal surface at late times. Following this idea, [14] considered a two-dimensional gravity+matter theory, where the matter sector has a three-dimensional holographic dual. The problem of setting up the above paradox in the 4d eternal black hole with a matter sector that has a 5d holographic dual reduces to finding a static 5d geometry with the correct boundary conditions We construct this geometry numerically using the DeTurck trick [22,23,24].
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