Photon spheres and bulk probes in $\text{AdS}_3$/$\text{CFT}_2$: the quantum BTZ black hole

TL;DR

This paper investigates the existence and properties of boundary-anchored geodesics in the quantum BTZ black hole within AdS3/CFT2, analyzing implications for entanglement entropy and photon rings.

Contribution

It provides a comprehensive analysis of geodesic existence in the quantum BTZ black hole and explores conditions related to photon rings and entanglement entropy.

Findings

Conditions for geodesic existence in all branches of quBTZ.

Identification of geodesic types based on boundary point separation.

Insights into the role of photon rings in entanglement entropy.

Abstract

The entanglement entropy in $d+1$ dimensional conformal field theories can be calculated using the area of $d$ dimensional minimal surfaces in $AdS_{d+2}$. Therefore, the existence of surfaces anchored in the boundary of an asymptotically anti-de Sitter (AdS) spacetime is crucial for the calculation of entanglement entropy. In particular, in $d=3$ the extremal surfaces are geodesics with two ends in the boundary. In the Schwarzschild-AdS black hole the space-like geodesics can connect timelike-separated points by winding around the horizon multiple times. This result can be extended to other asymptotically AdS spacetimes. For geodesics joining time-like separated points, if there is a photon ring then the timelike entanglement entropy in the $\text{AdS}_3/\text{CFT}_2$ will not have an imaginary part. We present an exhaustive analysis about the existence of geodesics anchored in the boundary of the three dimensional quantum BTZ (quBTZ) black hole and its charged counterpart. We found conditions for the existence of geodesics with two ends in the boundary in all branches of the quBTZ and determine the type of distance between the points in the boundary. We use a criteria for the existence of light rings to shed some light over the conjecture for spacetimes that are spherically symmetric and have a photon sphere: there are always points with time-like separation that can be connected by space-like or null geodesics.