Black Hole Singularity and Timelike Entanglement

TL;DR

The work probes black hole singularities through timelike entanglement entropy within AdS/CFT, using analytically tractable BTZ and AdS–Schwarzschild–like spacetimes. It shows that timelike entanglement captures singularity structure in single-sided setups and reveals intricate saddle-point structure, including multiple complex saddles in two-sided configurations that can challenge Lewkowycz–Maldacena assumptions. The analysis connects to holographic prescriptions by matching geodesic constructions (with careful handling of imaginary parts) and discusses conceptual remedies such as end-of-the-world branes to resolve ambiguities. Overall, the results illuminate how boundary entanglement quantities encode interior singularity data and suggest new avenues for understanding information aspects of black holes beyond standard holographic entanglement entropy.

Abstract

We study timelike and conventional entanglement entropy as potential probes of black hole singularities via the AdS/CFT correspondence. Using an analytically tractable example, we find characteristic behavior of holographic timelike entanglement entropy when the geometry involves a curvature singularity. We also observe interesting phenomena that, in some particular setups, holographic timelike and conventional entanglement entropy are determined from multiple complex saddle points, which fall outside the assumptions of the Lewkowycz-Maldacena type argument.

Figures (7)

• Figure 1: Penrose diagrams for (a) BTZ black hole and (b) AdS-Schwarzschild black hole. The crucial difference between them is the presence of a curvature singularity at the origin $r=0$ (wavy lines). Another important characteristic of the AdS-Schwarzschild black hole is that the $r=0$ surface concaves the Penrose diagram Fidkowski:2003nf. This difference in the global structure also affects the structure of geodesics discussed later.
• Figure 2: 3d plots of $\operatorname{Re}[\Delta \tau(\tilde{E}+iE)]$ for (a) BTZ black hole (first-sheet) and (b) AdS-Schwarzschild-like black hole (first and second sheet). In BTZ case, the real part are just divided into $|E|<1$ and others, where $\operatorname{Re} [\Delta \tau(iE)] = \pi (=\beta/2)$ if $|E|<1$ and $\operatorname{Re}[\Delta \tau(iE)] = 0$ if $|E|>1$. In Schwarzschild-like case, a simple analytic continuation (starting from a particular value $\tilde{E}$, and rotation to imaginary axis) can end up in the range of orange sheet (first sheet) or move beyond the branch to blue sheet (second sheet). This gives $\operatorname{Re}[\Delta \tau(iE)] = 0$ if $E$ is above a certain value $E_{\rm cr}$, and $\operatorname{Re}[\Delta \tau(iE)] = \pi/2 (=\beta/2)$ if $E$ is below a certain value $E_{\rm cr}$.
• Figure 3: Contour plots of $\operatorname{Im}[A_{\text{min.}}(\tilde{E}+iE)]$ for (a) BTZ black hole and (b) AdS-Schwarzschild-like black hole. Each dashed line corresponds to $\operatorname{Im}[A_{\text{min.}}(\tilde{E}+iE)]=0$, hence expected to be no contribution from the timelike geodesics.
• Figure 4: Schematic picture of holographic prescription proposed in Doi:2023zaf. The candidates of geodesics $\Gamma_A$ consist of a union of spacelike (orange curves) and timelike (blue curve in the center) geodesics such that $\partial\Gamma_A=\partial A$ as like the standard extremal surface. Then, we consider variation with respect to the joining points (black dots around the center). We require they are all stationary.
• Figure 5: Holographic entanglement entropy (or timelike entanglement entropy) for a two-sided setup.