Continuing past the inner horizon using WKB

TL;DR

The paper develops a WKB-based framework to extract interior information of rotating and charged AdS black holes from boundary two-point functions by implementing analytic continuations in both position and momentum space. Through detailed analyses of BTZ, AdS RN, and AdS Kerr geometries, it shows how timelike interior geodesics thread inner horizons and contribute to boundary correlators via shifted geodesic lengths and boundary-time phases. It also extends the method to black holes without inner horizons, revealing complex or bouncing geodesics and the need for image-sum corrections to capture interior-like effects in the boundary theory. The work suggests that boundary observables could encode signatures of inner-horizon instability and quantum corrections, offering a potential holographic probe of strong-gravity phenomena beyond the exterior spacetime. Overall, the approach provides a concrete route to connect bulk interior structure with boundary correlator analytic structure and motivates further study of quantum backreaction and higher-dimensional generalizations.

Abstract

Features of the black hole interior can be extracted from the analytic structure of boundary correlation functions. Working in the geodesic approximation, we find analytic continuations that probe the interior of rotating and charged black holes. These generate contributions from timelike geodesics that thread the interior and emerge in a future universe. We implement these continuations on the momentum space two-point function and exemplify this in several black hole backgrounds. We also identify position space analytic continuations achieving the same task that incorporate different continued momentum space correlators. These correspond to non-perturbative corrections to the WKB approximation. We demonstrate this explicitly in the rotating BTZ black hole by showing that the interior geodesics contribute to the continued position space correlator and motivate a picture for how these contributions arise in higher dimensions. For AdS Schwarzschild, we identify the analytically continued solution that captures the bouncing geodesic. We discuss the possibility of using these continuations to probe the instability of inner horizons from the boundary.

Figures (9)

• Figure 1: Shown here are some geodesics that probe behind the inner horizon of a charged or rotating black hole in the maximally extended spacetime. These geodesics cannot reemerge in the same universe.
• Figure 2: (a) The analytic structure of the position space correlator $G^+(\tau,\varphi)$. The correlator has branch cuts separated by the chiral temperatures $\{\beta,\bar{\beta}\}$. (b) The analytic continuation that generates the Lorentzian excursion. In the $\tau$ plane, the continuations shifts the branch points of the holomorphic part of $G^+$ (labeled in green) in addition to moving $\tau$. Note that the anti-holomorphic part of $G^+$ is left unchanged.
• Figure 3: (a) Path of the steepest descent contour in the $\bar{p}$ plane. Under the continuation, the left side of the steepest descent contour peels off of the real axis and bends clockwise through the branch cut of $A_-$. The contour eventually breaks due to a Stokes phenomenon and it becomes unclear in the $\bar{p}$ coordinate if it is deformable to the defining contour. (b) The steepest descent contour in the unwrapped coordinate $\bar{q}$. The principal sheet (a) is mapped to the blue shaded region. This coordinate system makes clear that the steepest descent path (red) continues to be anchored to the asymptotic regions (stars) of the defining contour and can be deformed to it.
• Figure 4: This shows the trajectory of geodesics that leave the Euclidean section after continuation. They generically emerge from times different from zero.
• Figure 5: (a) The path of analytic continuation on the $E$ plane to generate the null geodesic that probes the singularity. We start with $E=0$ and $L=i\epsilon$ and wind around the branch point $E=E^*$ before sending $E\to \infty$. (b) The corresponding change in the $\tau$ coordinate. Different stages of analytic continuation are colored the same in the two figures. The path in the right pane winds around the branch point at $\tau-i\varphi=0$. As one sends $E\to\infty$ we have $\varphi\to0$ and $\tau\to\frac{1}{2}(\beta_+-\beta_-+i\beta_c)$. This plot is generated with $L=0.1i$ and $(r_+,r_-)=(\sqrt{10},1)$.