Abstract
We discuss holographic models of extremal and non-extremal black holes in contact with a bath in d dimensions, based on a brane world model introduced in [1]. The main benefit of our setup is that it allows for a high degree of analytic control as compared to previous work in higher dimensions. We show that the appearance of quantum extremal islands in those models is a consequence of the well-understood phase transition of RT surfaces, and does not make any direct reference to ensemble averaging. For non-extremal black holes the appearance of quantum extremal islands has the right behaviour to avoid the information paradox in any dimension. We further show that for these models the calculation of the full Page curve is possible in any dimension. The calculation reduces to numerically solving two ODEs. In the case of extremal black holes in higher dimensions, we find no quantum extremal islands for a wide range of parameters. In two dimensions, our results agree with [2] at leading order; however a finite UV cutoff introduced by the brane results in subleading corrections. For example, these corrections result in the quantum extremal surfaces moving further outward from the horizon, and shifting the Page transition to a slightly earlier time.
Highlights
This is in tension with the assumption that to an outside observer, the black hole looks like an ordinary, unitary quantum mechanical system, e.g., as suggested by the AdS/CFT correspondence [6, 7]
Unitary evolution would require that the thermodynamic entropy of the black hole, which is proportional to its horizon area [9,10,11], set an upper bound on the entanglement entropy of the radiation
Since the former decreases as the black hole radiates, at some time — known as the Page time — the thermodynamic entropy of the black hole will equal the entropy of the radiation, and the latter entropy must decrease in the subsequent evolution, reaching zero when the black hole has disappeared
Summary
Let us review the holographic model discussed in [1]. Beginning with the bulk gravity perspective our setup is described by (d + 1)-dimensional Einstein gravity with a negative cosmological constant,. The solution for the backreacting brane is constructed by first cutting off the spacetime along an AdSd slice near the asymptotic boundary θ = 0, i.e. at θ = θB 1 where θB is determined by the brane tension To — see below Two such spaces are joined together along this surface, and the brane is realized as the interface between the two geometries. We arrive at the brane perspective by replacing the conformal defect in the boundary perspective by its gravitational dual This description includes the boundary CFT on the asymptotic AdSd+1 boundary, and two copies of the boundary CFT on the brane, as dictated by the usual Randall-Sundrum (RS) scenario. It is possible that in the Karch-Randall model, the graviton mass depends on the effective gravitational coupling on the brane, and is correlated with the island size, but not responsible for the island
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