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

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.

💡 Research Summary

This paper presents a novel methodology for probing the interior of a black hole using thermal correlation functions in the context of the AdS/CFT correspondence. The central challenge addressed is that the interior of a single-sided black hole, dual to a thermal state in the boundary conformal field theory (CFT), is causally inaccessible from the boundary. The authors demonstrate that certain fine-tuned four-point correlators in the boundary CFT can encode local scattering events occurring inside the black hole.

The core technique involves two key steps. First, the authors employ “smeared” boundary operators created by convolving local CFT operators with directed wavepackets (Eq. 2.1). In the high-frequency (WKB) limit, these operators correspond to creating particles with specific momenta that propagate along geodesics into the bulk. By arranging these smeared operators on specific spacelike curves on the boundary called “boundary hyperboloids” (which are intersections of the boundary with lightcones emanating from a bulk point), one can design boundary correlation functions that factorize into a flat-space-like S-matrix element describing scattering around an exterior bulk point X.

The second and most innovative step is an analytic continuation that maps an exterior scattering experiment to an interior one. The authors propose a specific integral transform (Eq. 1.1) that acts on a late-time operator on the right (R) boundary. This transform effectively re-interprets it as an operator on the left (L) boundary, corresponding to the other side of a two-sided black hole geometry. Applying this transform to a well-defined exterior 4-point correlator (e.g., a “radar experiment” correlator shown in Fig. 1) yields a new correlator (Fig. 3) whose bulk description involves a scattering event around a point X’ located inside the black hole interior. The transformed boundary operator is shown to act as a local creation/annihilation operator for “right-moving” modes near the interior point (Table 1).

A major result is the factorization formula showing that this transformed thermal correlator decomposes into a product of a local flat-space scattering amplitude in the interior and the appropriate WKB phases. This provides a concrete “dictionary” translating specific boundary CFT measurements into local interior physics. However, accessing the interior information in this way comes at an exponential cost, incurring a suppression factor of O(e^{-βω/2}), which can be interpreted as an energy barrier related to the Hawking temperature.

The paper further explores the geometric implications of interior probing by studying the shape of the boundary hyperboloids for points deep inside the black hole, including near the singularity. The required placement of operators on these curves provides a distinct signature of the interior geometry in the boundary data.

In conclusion, this work establishes a precise, albeit experimentally challenging (due to the exponential suppression), link between thermal boundary correlation functions and local events inside a black hole. It advances the holographic toolkit for studying black hole interiors by moving beyond two-point functions and leveraging the richer structure of higher-point correlators to extract information about locality and scattering behind the horizon. The findings connect to broader topics in quantum gravity, such as the nature of the black hole singularity and the encoding of interior geometry in boundary observables.