Extracting Spacetimes using the AdS/CFT Conjecture: Part II

TL;DR

This work advances bulk-mmetric reconstruction in AdS/CFT by showing how the static bulk metric function $f(z)$ of planar, asymptotically AdS spacetimes can be extracted from boundary entanglement data. It extends previous one-dimensional probes to higher-dimensional minimal surfaces anchored on belt and disk regions, deriving an Abel-inversion framework that maps entanglement entropy $S_A(l)$ to the bulk area $\mathcal{A}_\gamma(z_*)$ and hence to $f(z)$. The belt case yields an exact inversion, reproducing pure AdS and enabling a perturbative series for deformed geometries (e.g., planar black holes), while the disk case requires a perturbative treatment around AdS hemispheres to obtain analytic results. Together, these results illustrate how increasing boundary-dimensionality of probes depthens bulk access and outline extensions to other entangling shapes and covariant constructions for accessing $h(z)$.

Abstract

Motivated by the holographic principle, within the context of the AdS/CFT Correspondence in the large t'Hooft limit, we investigate how the geometry of certain highly symmetric bulk spacetimes can be recovered given information of physical quantities in the dual boundary CFT. In particular, we use holographic entanglement entropy proposal (relating the entanglement entropy of certain subsystems on the boundary to the area of static minimal surfaces) to recover the bulk metric using higher dimensional minimal surface probes within a class of static, planar symmetric, asymptotically AdS spacetimes. We find analytic and perturbative expressions for the metric function in terms of the entanglement entropy of straight belt and circular disk subsystems of the boundary theory respectively. Finally, we discuss how such extractions can be generalised.

Figures (4)

• Figure 1: These plots depict the shape of static minimal surfaces $\gamma_A$ anchored to the straight belt $A_S$ of width $l$ and the circular disk $A_D$ of radius $l$. The subsystems $A_S$ and $A_D$ sit on the boundary, where we have regulated the boundary directions to be of length $L$.
• Figure 2: In this figure we plot the profile of static minimal surfaces bounded to a straight belt $A_S$ of width $l=2$ for planar black holes of dimension $4+1$. The left and right figures show minimal surfaces of spacial dimension $2$ and $3$ respectively for different values of horizon depth $z_+=\{0.5(\text{green}),1(\text{blue}),2(\text{red})\}$. The dashed lines correspond to the static minimal surfaces in pure AdS space (i.e. $z_+\rightarrow\infty$).
• Figure 3: This is a plot of $z_\ast$ against the dimension of the minimal surface $m$ in a planar black hole spacetime of dimension $11+1$ for different depths of the horizon plane $z_+$ (shown here by the coloured dashed lines). The pure AdS plot (i.e. $z_+\rightarrow\infty$) given in \ref{['mvz']} is shown here by the black dashed line.
• Figure 4: The left diagram plots profile functions $z(r)$ anchored to the the circular disk $A_D$ of radius $l=1$ and dimension $m$, for the metric function $f(z)=(1-z^{10})^{-1/2}$. This spacetime describes a planar black hole in $10+1$ dimensions of horizon depth $z_+=1$. The dashed, red and blue curves show pure AdS, the numerical solution and the first order perturbed solution to \ref{['eomd']} respectively for $m=2$. The $r$ axis is the boundary of the spacetime. The right diagram shows how the maximum height $z_\ast$ changes for these curves as one increases the dimension of the circular disk up to the maximum allowed by the dimension of the spacetime.

Theorems & Definitions (3)

• Conjecture 1.1
• Conjecture 1.2
• Conjecture 1.3