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

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. Moreover, for geodesics joining time-like separated points, if there is a photon ring then the timelike entanglement entropy in the $AdS_3/CFT_2$ 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.

Figures (10)

• Figure 1: In the left panel we have plotted the effective potential $V_{\rm eff\,1}$ as a function of the radial coordinate and for different values of the impact parameter $\sigma$. The light blue curves represent the $V_{\rm eff\,1}$ for $\sigma=1.2$ and $\sigma=2$. The dashed line corresponds to the value $\sigma=1$ and the dark blue curves correspond to the values $\sigma=0.5$ and $0.8$. In the right panel we have plotted $\left\vert\frac{\Delta t}{\Delta \phi}\right\vert$ as a function of $\sigma$. The peak of the function is at $\sigma=1$. In any case there is never a point where it reaches $1$. The inset plot in the right panel correspond to the return point $r_t$ as a function of $\sigma$.
• Figure 2: We have plotted the potential $V_{\rm eff\,5}$ for different values of $\sigma$. We have set $\mathcal{K}=1$ and fixed the value of the momenta to $L=10$, therefore each different value of $\sigma$ corresponds to different value of the energy $E$. We have also set $kx^2=0$. In the left panel we have plotted the potential $V_{\rm eff\,5}$ for $\sigma$ values lower than one. In the right panel the potential is plotted for $\sigma$ values higher than one.
• Figure 3: The plots show results found by numerical integration. We have set $\kappa x_1^2=0$. In the left panel we have plotted $\Delta t$ defined in \ref{['dt']}. The lower integration limit is calculated also numerically, which corresponds to the highest root of $V_{\rm eff\,5}$. Every point corresponds to $\sigma=1,\dots,20$. We have joined all points with a curve. Each curve corresponds to the values of $L=10,L=20,L=30$. In the center panel we plotted $\Delta \phi$ defined in \ref{['dp']}. The parameters $\sigma$ and $L$ are the same in the three panels. In the right panel we plot the quotient $\Delta t/\Delta \phi$
• Figure 4: The plots show results found by numerical integration. We have set $\kappa x_1^2=1$.In the left panel we have plotted $\Delta t$ defined in \ref{['dt']}. The lower integration limit is calculated also numerically, which corresponds to the highest root of $V_{\rm eff\,5}$. Every point corresponds to $\sigma=1,\dots,20$. We have joined all points with a curve. Each curve corresponds to the values of $L=10,L=20,L=30$. In the center panel we plotted $\Delta \phi$ defined in \ref{['dp']}. The parameters $\sigma$ and $L$ are the same three panels. In the right panel we plot the quotient $\Delta t/\Delta \phi$
• Figure 5: Potentials $V_{\rm eff\,6}$ and $V_{\rm eff\,7}$ for $L=1,4,7,10$ and $\sigma=2$. The left panel (red curves) corresponds to the effective potential $V_{\rm eff\,6}$ with $\kappa x_1^2=-1$. In the right panel the effective potential $V_{\rm eff\,7}$ is plotted (blue curves) with the same parameters as $V_{\rm eff\,6}$ but with $\kappa x_1^2=1$.