Extremal Surfaces in Asymptotically AdS Charged Boson Stars Backgrounds

TL;DR

This work explores how extremal, codimension-2 spacelike surfaces behave in asymptotically AdS charged boson star backgrounds in $D=3$ and $D=4$. By solving the Einstein–Maxwell–scalar system and analyzing extremal surfaces via the area functional, the authors identify hollow phases where minimal surfaces do not cover the full bulk and map these against a stability threshold $\psi_c$; in $D=4$, they show $\psi_c<\psi_h$, indicating hollow configurations are dynamically unstable and thus unlikely physical. The results support the proposal that the bulk region encoded by the boundary entanglement structure is governed by the region $w(D_A)$, strengthening connections between holographic entanglement entropy and density-matrix duals. The study provides phase diagrams and stability insights that inform holographic interpretations and guide future investigations into zero-temperature transitions and boson-star holography.

Abstract

In this paper, inspired by the holographic dual of the entanglement entropy, we consider the behaviour of extremal, codimension two, spacelike surfaces in the background of three and four dimensional charged boson stars in asymptotically anti-de Sitter spacetime. We find conditions for which families of minimal area surfaces fail to contain the entire bulk spacetime and construct a phase diagram showcasing the transition between regimes. In addition, we use the relation between the star's mass and the central density of the scalar field to argue for a possible instability of such hollow solutions. Finally, we discuss the consequences of our findings for the study of holographic duals of reduced density matrices.

Figures (10)

• Figure 1: Plot of the star's mass (orange) and charge (green) versus the central value of the scalar field $\psi_0$ with $m^2=0$ and $q=0.2$ in $D=4$ dimensions.
• Figure 2: Plot of the star's mass (orange) and charge (green) versus the central value of the scalar field $\psi_0$ with $m^2=0$ and $q=0.2$ in $D=3$ dimensions.
• Figure 3: A plot of the Penrose diagram of a time slice of multiple extremal surfaces on an asymptotically AdS charged boson boson star background in global coordinates for $m^2=0,$$q=0.1$, and $\psi(0)=0.2$. From top to bottom we have $\theta=\pm0.251\pi, \pm0.355\pi, \pm0.446\pi, \pm0.5\pi$. In this particular case the central density of the scalar field is below the threshold $\psi_h$, therefore there are no degenerate extremal surfaces (see figure \ref{['geodesics']}), in other words, we observe a solid $w(D_A)$.
• Figure 4: Again, a plot of the Penrose diagram of a time slice of multiple extremal surfaces on an asymptotically AdS charged boson boson star background in global coordinates for $m^2=0,$$q=0.1$, and $\psi(0)=1.2$. However, in this example, the scalar central density is above the threshold $\psi_h$ and we observe the existence of degenerate extremal surfaces for a range of boundary anchor points $\theta$ (see figure \ref{['geodesics']}). In particular, for anchor points $\theta=\pm \pi/2$ there are three solutions two of which (blue line) lie on top of each other, have minimal area and do not penetrate the dashed small circle, while the third (red line) corresponds to a non minimal area extremal surface. As discussed in this paper, no minimal area surface penetrates the deepest bulk points within the small dashed circle.
• Figure 5: A similar plot as figures \ref{['sts']} and \ref{['hts']} highlighting the behaviour of all three extremal surfaces anchored at the same boundary points $\theta = \pm 0.49 \pi$. Although two of the solutions (red lines) penetrate the dashed circle, these are not the minimal area and therefore are not part of the $w(D_A)$ set.