Extracting spacetimes using the AdS/CFT conjecture

TL;DR

The paper tackles the problem of reconstructing bulk AdS geometries from boundary data within the AdS/CFT framework, focusing on static spherically symmetric spacetimes. It develops inversion techniques that map boundary-probe information from bulk-cone singularities and, in the AdS$_3$ case, entanglement-entropy data via the minimal-surface length $\\mathcal{L}$ into integral equations for the metric functions $f(r)$ and $h(r)$, with Abel/Volterra-type solutions. Analytic checks are provided for pure AdS, and extensive numerical tests demonstrate recovery of the metric down to the local maximum of the effective potential, with full recovery in the two-function case possible in AdS$_3$ using entanglement data. The approach offers a principled route to read off bulk geometry from boundary observables, highlighting dimensional limitations and outlining paths toward generalization to less symmetric or higher-dimensional settings.

Abstract

We present analytic methods for extracting a class of bulk geometries given information of certain physical quantities in the boundary CFT. More specifically we look at singular correlators and entanglement entropy in the CFT to provide information of null and spacelike geodesics repectively in the bulk. We show that static spherically symmetric, asymptotically AdS spacetimes which do not admit null circular orbits can be fully recovered, and that any spacetime can be recovered up to the local maximum of the potential. We provide analytical and numerical examples to verify the methods used.

Figures (5)

• Figure 1: $(a)$ is a $2+1$-dimensional plot of boundary to boundary null geodesics in an AdS-black hole geometry with metric $d s^2=-f(r)\,d t^2+\frac{d r^2}{f(r)}+r^2d\Omega^2_n$ and $f(r)=1+r^2-\frac{1}{10r^2}$, where $n-1$ angular directions have been projected out. To plot the null trajectories, the spacetime has been compactified using the coordinate transformation $r=\mathrm{tan}(\tilde{r})$, thus the boundary is a timelike cylinder with radius $\tilde{r}=\frac{\pi}{2}$. $(b)$ illustrates the quantity $\Delta t(\alpha)$ via projection of the null geodesics on the $r$-$t$ plane, where $\alpha=\frac{E}{J}$ parameterises the null curves. $(c)$ illustrates the quantity $\Delta\phi(\alpha)$ via projection of the null geodesics on the $r$-$\phi$ plane. Notice that the boundary null curve ($\alpha=1$), we have $\Delta t(\alpha)=\Delta\phi(\alpha)=\pi$, which is as expected from a pure AdS calculation. (Colour coding for values of $\alpha$: $\{1,1.3,1.7,1.85,1.87\}=\{$Black, Red, Green, Blue, Purple$\}$)
• Figure 2: Extraction of $V(r)$ where $f(r)=1+r^2-\dfrac{1}{r^2+1}$. In this case we can extract the entire spacetime.
• Figure 3: Extraction of $V(r)$ where $f(r)=1+r^2-\dfrac{1}{5r^2}+r^2\sum_{i=1}^3\dfrac{\mathrm{A_i}}{\sqrt{2\pi}\sigma_i}\,e^{-\frac{(r-\mu_i)^2}{2{\sigma_i}^2}}$ and $\mathrm{A_i}=(0.05,0.03,0.1),\,\mu_i=(1,1.3,2)\,\,\text{with}\,\,\sigma_i=(0.1,0.1,0.3)$.
• Figure 4: Testing equation \ref{['exhr']} for $h(r) = 1 + r^2 - \left[\dfrac{\mathrm{cos}(2.8r)}{r}\right]^2$
• Figure 5: Testing equation \ref{['star']} for $V(r) = 1 + \dfrac{1}{r^2} - \left[\dfrac{\mathrm{cos}(2.8r)}{r^2}\right]^2$ and $h(r) = 1+r^2-\dfrac{1}{r^2}$