Replica wormholes for an evaporating 2D black hole

TL;DR

The paper extends the replica-wormhole/island framework to evaporating two-dimensional black holes in JT gravity coupled to a large-$N$ CFT, addressing how Euclidean replicas and nonlocal conformal welding relate to Lorentzian boundary dynamics via the Schwarzian action. By constructing shockwave-induced evaporating geometries and employing Schwinger-Keldysh techniques, the authors derive the quantum extremal surface (QES) conditions from replica equations in the $n\to1$ limit and verify that the island entropy reproduces the island formula through a gravitational Ward identity. They demonstrate that the Page curve emerges for the evaporating black hole and show a consistent, Lorentzian-real-time interpretation of replica wormholes, including a no-mixing welding solution and a factorization of two-interval wormholes in the late-time regime. The results also establish that two-interval geometries in an eternal black hole factorize into two copies of the single-interval solution, reinforcing the physical picture that wormholes factorize in the OPE limit of twist operators and supporting the universality of the island mechanism in dynamical settings.

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 are non-local, so it is unclear how to bridge the gap between the Euclidean path integral and 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.

Figures (19)

• Figure 1: A black hole in AdS$_2$ that is created by a shockwave, then evaporates. The AdS$_2$ region is initially in vacuum. We will calculate the von Neumann entropy of region $R$.
• Figure 2: A shockwave thrown into an AdS$_2$ black hole. In this case, the AdS$_2$ region starts at finite temperature. Here we show a time-symmetric shockwave, with both ingoing and outgoing shocks, which has a straightforward Euclidean continuation.
• Figure 3: Eternal black hole glued to flat space in Euclidean (left) and Lorentzian (right). The region within the red triangle is covered by the Poincare coordinates $x^\pm$. The coordinates $y^\pm = t \pm \sigma$ cover the blue wedge and the right Minkowski half-space.
• Figure 4: The island $I$ for a single interval placed in the right flat region: $R=[B,B']$ in an eternal black hole. The island has two ends $A$ and $A'$ determined by the extremal conditions. As we will see, the island is placed outside the original horizon of the right black hole, which is depicted as the dashed null lines.
• Figure 5: The Page curve is reproduced by the island formula for the entropy of the Hawking radiation.