Replica wormholes and the black hole interior

TL;DR

The paper demonstrates that replica wormholes provide a gravitational derivation of the Page curve and island formula, using simple JT gravity with end-of-the-world branes, and extends these results to more complex UV-complete theories like JT gravity with matter and the SYK model. It develops a planar resummation framework to track Renyi entropies across Page transitions and shows how entanglement-wedge reconstruction can be realized via the Petz map, with wormholes enabling interior access from radiation. It also explores extensions to de Sitter space and discusses the interpretation of wormholes as part of an ensemble average, while addressing factorization issues and the limitations in non-averaged systems. Together, these results illuminate how bulk wormholes underpin information recovery from black holes and shape reconstruction of interior operators, with implications for holography and quantum gravity at large.

Abstract

Recent work has shown how to obtain the Page curve of an evaporating black hole from holographic computations of entanglement entropy. We show how these computations can be justified using the replica trick, from geometries with a spacetime wormhole connecting the different replicas. In a simple model, we study the Page transition in detail by summing replica geometries with different topologies. We compute related quantities in less detail in more complicated models, including JT gravity coupled to conformal matter and the SYK model. Separately, we give a direct gravitational argument for entanglement wedge reconstruction using an explicit formula known as the Petz map; again, a spacetime wormhole plays an important role. We discuss an interpretation of the wormhole geometries as part of some ensemble average implicit in the gravity description.

Figures (19)

• Figure 1: Euclidean and Lorentzian geometries for a black hole with an EOW brane behind the horizon.
• Figure 2: We show the expected entanglement wedges for our simple model based on the "island" conjecture. In the $k \ll e^{S_{BH}}$ phase, the entanglement wedge of the $\sf{B}$ system (boundary dual of the gravity theory) is the whole spacetime, shown hatched. In the $k \gg e^{S_{BH}}$ phase, an island develops. The entanglement wedge of the $\sf{B}$ system retreats to the exterior of the horizon, and the entanglement wedge of the auxiliary $\sf{R}$ system is the shaded blue island behind the horizon.
• Figure 3: The boundary conditions for $|\langle \psi_i|\psi_j\rangle|^2$ are shown at left, and two ways of filling in the geometry are shown at right. To compute the purity, we want to sum over $i,j$ by connecting the dashed lines together. For the disconnected geometry, this will lead to a single $k$ index loop, and for the connected geometry it will lead to two loops.
• Figure 4: The boundary conditions for $\text{Tr}(\rho_{\sf{R}}^n)$ with $n = 6$ are shown at left, along with two extreme ways of filling in the geometry, corresponding to the completely connected and completely disconnected options. Note that the geometry at right contains a fixed point of the $\mathbb{Z}_n$ symmetry that rotates the replicas.
• Figure 5: The left figure is an example of planar geometry that we need to include in our analysis. The middle figure has an extra handle and is down by $e^{-2S_0}$. The right figure involves a crossing, and is down by $k^{-2}$ (it has two dashed index loops instead of four).