Fooling the Censor: Going beyond inner horizons with the OPE

TL;DR

This work extends holographic OPE methods to charged AdS black branes, showing that the T+J sector of thermal two-point functions develops a complex-time singularity whose location τ_c(μ,q) matches the time shift of bouncing null geodesics bouncing off the timelike singularity. By deriving and summing the OPE coefficients for stress-tensor and current exchanges, the authors link boundary singularities to bulk geodesic structure, revealing smooth behavior across extremality and signaling that inner-horizon and naked-singularity features can imprint on boundary observables in the large-N limit. The results demonstrate nontrivial non-analytic in q^2 contributions tied to inner horizons, provide explicit large-n asymptotics for OPE coefficients, and propose a general ansatz for general μ and q that captures phase and power-law structure of the boundary data. These findings offer a boundary-accessible window into black hole interiors and naked-singularity regimes, with potential implications for understanding quantum resolutions of singularities via holographic probes.

Abstract

The analytic structure of holographic correlation functions at finite temperature contains information about curvature singularities of black holes in AdS. We compute the Operator Product Expansion (OPE) coefficients of the holographic two-point function of scalar operators at finite temperature and finite chemical potential. We show that the stress-tensor and current (T+J) sector of the OPE contains a singularity in the complex time plane at a location that can be identified with the time-shift of a bouncing geodesic in the charged black hole geometry: The geodesic starts at a boundary of a charged black hole in AdS, bounces off the timelike singularity, before returning to a different asymptotic boundary on the same side of the Penrose diagram. We show that the singularity in the T+J sector is smooth across the point where black hole becomes extremal, indicating that the analytic properties of holographic correlators could potentially probe naked singularities.

Figures (18)

• Figure 1: The Penrose diagram of an eternal Schwarzschild-AdS black hole in four dimensions or higher. The curvature singularity is bent inwards compared to the asymptotic boundary. Spacelike geodesics (blue) with endpoints on different boundaries probe the interior region. As their energy, $E$ is increased, the geodesics tend to a null geodesic that reflects off the black hole singularity (red-dashed).
• Figure 2: The Penrose diagram for a charged black hole with real charge $q$. The outer horizons are depicted in blue dashed lines, while the inner horizons are in dashed gray. The timelike singularity is bent out compared to the asymptotic boundary Brecher:2004gn. We will argue that the boundary correlation functions contain a singularity which coincides with the time shift of a bouncing geodesics that starts at the asymptotic boundary of region I, reflects off the singularity in region VII and reaches the asymptotic boundary on the same side of the diagram, but at a different asymptotic boundary -- that of region VIII.
• Figure 3: Determining the spacelike geodesics in black hole can be mapped to the scattering of a classical particle in a potential $V(r)$. On the left (a), we depict the potential for a neutral black brane in AdS, where as the energy in increased, the geodesics probe closer and closer to the singularity at the origin, where they approach the bouncing geodesic. On the right (b), we show the potential for the charged black brane, where instead there exists a maximum between the inner ($r_-$) and the outer horizon $r_+$. Spacelike geodesics can only probe to a minimal distance, set by the maximum of the potential and do not tend to a null geodesic.
• Figure 4: The Penrose diagram for the extremal black hole in AdS. On the left of the diagram there are infinitely many copies of the boundary of AdS, while on the right we find the singularities, which are bent out relative to the boundary. $\tau_c(\mu,q_{\rm ext})$ can be interpreted as the time-shift of a null geodesic (red) starting at one copy of the asymptotic boundary, crossing the degenerate horizon and bouncing off the singularity towards a different copy of the asymptotic boundary.
• Figure 5: Penrose diagram for a black hole with imaginary charge. With $t_L$ we denote the Lorentzian time and mark the direction in which it increases on both boundaries.