Hologram of a pure state black hole

TL;DR

This work advances holographic bulk reconstruction for pure-state AdS black holes formed by collapse by extending the HKLL smearing framework to AdS–Vaidya spacetimes. It first constructs local bulk operators in the large-$N$ limit and then incorporates finite-$N$ effects through early and late time cutoffs, linking decoherence and spectral discreteness to approximately local bulk observables with non-perturbative $O(e^{-N})$ corrections. By providing explicit boundary representations for bulk points in regions outside, behind, and inside the horizon, the paper clarifies how information about the collapsing state is encoded in a single CFT through complex-time continuation and shell matching, with a detailed analysis of how locality degrades at finite $N$ via $t_{min}$ and $t_{max}$ cutoffs. The discussion situates these results within the broader landscape of black hole information, comparing to Papadodimas–Raju mirror-operator constructions and outlining future work on state dependence, trans-Planckian issues, and scrambling dynamics. Overall, the work strengthens the link between boundary dynamics and near-horizon bulk locality for non-equilibrium black holes and highlights nonperturbative quantum gravity effects that become relevant for information retrieval and firewall considerations.

Abstract

In this paper we extend the HKLL holographic smearing function method to reconstruct (quasi)local AdS bulk scalar observables in the background of a large AdS black hole formed by null shell collapse (a "pure state" black hole), from the dual CFT which is undergoing a sudden quench. In particular, we probe the near horizon and sub-horizon bulk locality. First we construct local bulk operators from the CFT in the leading semiclassical limit, $N\rightarrow\infty$. Then we look at effects due to the finiteness of $N$, where we propose a suitable coarse-graining prescription involving early and late time cut-offs to define semiclassical bulk observables which are approximately local; their departure from locality being non-perturbatively small in $N$. Our results have important implications on the black hole information problem.

Figures (11)

• Figure 1: The Penrose diagram of an eternal AdS Schwarzschild black hole. The red curved lines at the top and the bottom are the future and past curvature singularities. $P$ and $Q$ are local bulk operator insertions outside the event horizon and the yellow segments on the boundary are their respective boundary smearing function support.
• Figure 2: The Penrose diagram of an eternal AdS Schwarzschild black hole with an operator inserted at $R$ inside the black hole horizon. a) Spacelike separated region to the right of the point $R$ exceeds the right boundary and spills over to the singularity. b) The smearing function support (yellow segments) of the inside horizon insertion, at $R$ - left boundary support is over timelike separated points and while right boundary support is over spacelike points. Similar smearing constructions are present for BTZ case as well as described below.
• Figure 3: The smearing function support for sub-horizon operators. Using complexified Schwarzschild time one can bring over the smearing integral over the left boundary, shown in yellow at the bottom left corner, to its symmetric region on the right boundary, shown in green at the bottom right corner.
• Figure 4: The smearing function support for sub-horizon operators. The contribution of the region shown in green at the bottom i.e $t<0$ of figure \ref{['fig:SAdS_left_right']} can be time-evolved and converted into a integral in the $t>0$ half, shown in this figure in green again.
• Figure 5: The alternative smearing function support for sub-horizon operators.