Probing the bubble interior with entanglement entropy and bulk-cone singularities

Abstract

In the thin wall approximation, we study a class of asymptotically AdS black holes which contain a spherically symmetric vacuum bubble with a different (positive or negative) cosmological constant. Collapsing, expanding, and static bubble solutions are considered. Among these, expanding bubbles with positive cosmological constant can provide a way to apply the AdS/CFT correspondence to describe the physics of an expanding universe. We systematically study the causal structure of the solutions as a function of the cosmological constant, the mass of the black hole, and the tension of the bubble. We then compute the holographic entanglement entropy for a class of boundary subregions using extremal codimension-two surfaces as a probe. For collapsing bubbles, we find examples in which the entanglement entropy also explores the geometry inside the black hole bifurcation surface. As a complementary way to probe the interior of the bubble, we investigate almost-null radial geodesics related to the bulk-cone singularities of boundary two-point correlators. While the bulk-cone singularities for collapsing and expanding bubbles are consistent with thermalization at late time, static bubbles violate thermalization and exhibit properties similar to those of scar states.

Figures (33)

• Figure 1: Phase diagram of the bubble solutions as a function of $(\kappa, \lambda)$.
• Figure 2: Penrose diagrams for expanding bubbles in the regions of parameter space where they exist.
• Figure 3: Penrose diagrams of an expanding bubble for an interior geometry with $\lambda=0$.
• Figure 4: Penrose diagrams for collapsing bubbles. We show in blue the trajectory of the domain wall, both in the interior geometry (left panel) and in the exterior AdS black hole (right panel). The interior background can be either a portion of empty AdS, dS or flat spacetime.
• Figure 5: Numerical scan in parameter space of the collapsing bubble for $d=2$, in the $(\kappa,\lambda)$ plane and for different values of $m=1.1$ (top-left), $m=1.5$ (top-right), $m=2$ (bottom-left) and $m=100$ (bottom-right). In the gray region, there is no time-reversal symmetric bubble. The classification I, II and III refers to the shape of the causal wedge, see fig. \ref{['Pen-Dia-Collapsing-Bubble']}.