Imprint of the black hole interior on thermal four-point correlators

TL;DR

The paper develops a boundary-based protocol to image the black hole interior by mapping high-frequency boundary correlators to local flat-space amplitudes near an interior bulk point. Central to the approach is a boundary integral transform that analytic-continues exterior operators into interior left-moving modes, yielding a factorization of the thermal four-point function into flat-space scattering data with an overall $\mathcal{O}(e^{-\beta\omega/2})$ suppression. The construction hinges on directed wavepackets smeared on boundary hyperboloids and a WKB/geodesic-based dictionary linking boundary operators to interior bulk oscillators, enabling a 3→1 or 2→2 interior amplitude description on an in-in contour. The work also analyzes boundary hyperboloids, AdS-Rindler geometry, and Planar BTZ examples to illustrate how interior physics can be diagnosed from boundary data, with implications for bulk locality, holographic cameras, and the flat-space limit in finite-temperature holography. Overall, the framework offers a concrete, though intricate, route to probe interior bulk dynamics via thermal correlators and two-sided boundary constructions, while noting limitations near the singularity and the need for further generalization to dynamical collapse and more general wormhole geometries.

Abstract

We consider correlators smeared against directed wavepackets over a thermal state dual to a single-sided planar AdS black hole. In the large frequency limit, our measurement is simplified using a bulk WKB description. We propose a dictionary that maps the action of smeared boundary operators to flat-space oscillators near an interior bulk point on the thermal state, by analytically continuing late-time operators from the right to the left boundary via an integral transform. Using the dictionary the smeared correlator factorizes to a flat-space like scattering amplitude about the interior event. Our transformed correlators describe local physics in the two-sided black hole interior, while incurring a suppression of $\mathcal{O}(e^{-βω/ 2})$. These measurements necessitate a non-trivial time ordering of operators living on boundary hyperboloids which are causally connected to the past light cone of the bulk point, as well as on a corresponding future branch.

Figures (11)

• Figure 1: (a) Radar scattering process $\langle \Psi | \, O_{x_4, p_4, \sigma_4} \, O_{y_3, q_3, \sigma_3}\, O^\dagger_{x_2, p_2, \sigma_2} \, O^\dagger_{x_1, p_1, \sigma_1} \, | \Psi \rangle$ with a bulk point in black hole exterior as in Caron-Huot:2025hmkCaron-Huot:2025she, see equation \ref{['eq:bdryops']} for the definition of smeared operators. The dashed line is trace over all final states that may fall behind the horizon. The process has three early-time (two emission and one absorption) operators and a single late-time (absorption) operator. (b) Out of time-ordered CFT correlator on an in-in timefold. (c) The bulk scattering amplitude with same time-ordering.
• Figure 2: (a) and (b) On a Schwinger Keldysh (SK) fold with a thermal identification, the transform \ref{['transformaton']} takes the high frequency operator insertions $y$ on the right future boundary as operators introduced at the left boundary. There is an energy cost for both operations of $\mathcal{O}(e^{-\beta \omega / 2})$, which we absorb in our operational definition \ref{['finform']}. While the transform is a robust operation, geometrically these operations on saddles violate path integral rules (for instance KSW criteria in the bulk Kontsevich:2021dmbWitten:2021nzp). (c) The transformation in \ref{['finform']} takes us from an exterior radar experiment defined on the SK fold to a deformed SK fold, thereby implementing $t_3 \mapsto -t_3 + {\mathrm{i} \beta / 2}$ such that $O_{y_3, q_3, \sigma_3}$ is shifted to $O^\dagger_{Ly_3, q_3, \sigma_3}$.
• Figure 3: The analytic continuation can be thought of as pushing the bulk point $X$ from a well-defined exterior scattering experiment on single timefold in Fig \ref{['fig:extbpcontour']} to an interior scattering experiment with bulk point $X'$. The transformation introduces an extra folding (dashed orange line), and local right-moving modes near $X'$. Note that the boundary hyperboloids for exterior bulk point experiment and the interior bulk point experiment are different, and we need to change the operator positions and the shooting momenta appropriately to go from $X$ to $X'$.
• Figure 4: (a) Going from the Euclidean to the radar configuration. Crossing these light cones takes us to the second sheet and eventually to the bulk point discontinuity for scattering configuration as in Fig \ref{['fig:extbpcontour']}. (b) Going from the Euclidean to the otoc configuration on two timefolds. Crossing lightcones takes us to the second sheet and from there to the bulk point singularity.
• Figure 5: Analytic continuation along the complexified paths $P^\pm(\theta)$ (shown in red) takes us from the bulk point within the Rindler wedge to the future wedge by analytically continuing to the complex coordinates. This takes observers from support on future time in right Rindler wedge to the left Rindler wedge.