Replica wormholes for an evaporating 2D black hole

Abstract

Quantum extremal islands reproduce the unitary Page curve of an evaporating black hole. This has been derived by including replica wormholes in the gravitational path integral, but for the transient, evaporating black holes most relevant to Hawking’s paradox, these wormholes have not been analyzed in any detail. In this paper we study replica wormholes for black holes formed by gravitational collapse in Jackiw-Teitelboim gravity, and confirm that they lead to the island rule for the entropy. The main technical challenge is that replica wormholes rely on a Euclidean path integral, while the quantum extremal islands of an evaporating black hole exist only in Lorentzian signature. Furthermore, the Euclidean equations for the Schwarzian mode are non-local, so it is unclear how to connect to the local, Lorentzian dynamics of an evaporating black hole. We address these issues with Schwinger-Keldysh techniques and show how the non-local equations reduce to the local ‘boundary particle’ description in special cases.

Real-time gravitational replicas: formalism and a variational principle

“…This may in particular be useful when the initial state can be prepared using a Euclidean path integral and when any sources are analytic functions of time. In that context, one may imagine computing entropies associated with general Euclidean choices of the region A and then analytically continuing parameters to obtain entropies for general Lorentzian regions as in [27,38,73]. Analytically continuing the Euclidean replica geometry in this way must give a solution to the variational problem described in section 2.…”

Real-time gravitational replicas: low dimensional examples

“…Real-time computation of correlation functions, both time-ordered and out-of-time-order, as well as density operator matrix elements and their moments, in any quantum system either with or without dynamical gravity, requires the use of a suitable timefolded contour, with segments of forward and backward evolution. One often however eschews the use of such contours, relying instead on computations in the Euclidean domain, and then analytically continuing the answers thus obtained into the real-time domain (see e.g., [2,3] for non-gravitational theories as well as the more recent analysis in gravitational context in [4]), a strategy that works well when the quantum evolution is not subject to nonanalytic sources. While this is strategy is efficient in extracting information about the non-perturbative aspects of the theory, it does not lend insight into the physical dynamical evolution directly.…”

Ergodic equilibration of Rényi entropies and replica wormholes

“…In JT gravity models [1][2][3] the suppression in powers of S R β agrees heuristically with the genus expansion in powers of the extremal entropy S 0 , so these terms might appear at the level of higher genus saddles of the gravitational path integral. However, there is a somewhat obscure minus sign for each handle arising from the global minus sign of the last term in (5.7).…”

“…such cases, but some of the [2, 1 n−2 ] partitions can also contribute. In particular, there are 3 2 such partitions that contribute. We therefore have (A.12) For a general partition π = {B 1 , .…”

“…Recent attempts to derive the Page curve from semiclassical Euclidean gravity in lowdimensional models have proven remarkably successful [1][2][3]. The lesson is that the sharp change in tendency of the Page curve for Rényi entropies can be reproduced á la Hawking-Page from an exchange in dominance at the Page time between the disconnected Euclidean black hole saddle and the so-called 'replica wormhole' saddle.…”