Extremal surfaces as bulk probes in AdS/CFT

TL;DR

The paper investigates how bulk AdS geometry can be inferred from boundary CFT data by analyzing probes that are geometric in the bulk—geodesics and extremal surfaces. It shows that spacelike geodesics anchored at the boundary can reach deeper into the bulk than null geodesics, but in static spacetimes they cannot cross black hole horizons; near-horizon probing is possible only under specific parameter limits and often constrained by boundary data. Extending to higher-dimensional extremal surfaces, the depth reached depends on dimensionality, region shape, and background, with round boundary regions (balls) and higher-dimensional surfaces enabling deeper bulk reach; however, in static backgrounds with horizons, extremal surfaces cannot penetrate the horizon. A key result is a conjecture that, for fixed boundary area, round balls maximize the reach z_* of extremal surfaces, with perturbative and exact checks suggesting this optimality holds beyond pure AdS. The work thus delineates which boundary observables and region shapes best encode bulk geometry, highlighting both the power and limits of geometric probes in static AdS/CFT contexts and pointing toward future time-dependent extensions.

Abstract

Motivated by the need for further insight into the emergence of AdS bulk spacetime from CFT degrees of freedom, we explore the behaviour of probes represented by specific geometric quantities in the bulk. We focus on geodesics and n-dimensional extremal surfaces in a general static asymptotically AdS spacetime with spherical and planar symmetry, respectively. While our arguments do not rely on the details of the metric, we illustrate some of our findings explicitly in spacetimes of particular interest (specifically AdS, Schwarzschild-AdS and extreme Reissner-Nordstrom-AdS). In case of geodesics, we find that for a fixed spatial distance between the geodesic endpoints, spacelike geodesics at constant time can reach deepest into the bulk. We also present a simple argument for why, in the presence of a black hole, geodesics cannot probe past the horizon whilst anchored on the AdS boundary at both ends. The reach of an extremal n-dimensional surface anchored on a given region depends on its dimensionality, the shape and size of the bounding region, as well as the bulk metric. We argue that for a fixed extent or volume of the boundary region, spherical regions give rise to the deepest reach of the corresponding extremal surface. Moreover, for physically sensible spacetimes, at fixed extent of the boundary region, higher-dimensional surfaces reach deeper into the bulk. Finally, we show that in a static black hole spacetime, no extremal surface (of any dimensionality, anchored on any region in the boundary) can ever penetrate the horizon.

Figures (9)

• Figure 1: Sketch of possible behaviour of spacelike geodesics (red curves) on a Penrose diagram of an AdS black hole.$^{12}$ Suppose a spacelike geodesic has crossed the future horizon (a). There are several qualitatively different a-priori possibilities: it can end at the singularity (b), it can continue to the boundary of the other asymptotic region (c), it can return to the same boundary through the future horizon (d), or it can return to the same boundary via the past horizon (e). We argue in the text that only (b) and (c) are viable possibilities.
• Figure 2: Effective potentials for global Schwarzschild-AdS$_5$, as $L$ is varied at fixed $E$ (left) and as $E$ is varied at fixed $L$ (right), both for fixed $r_+=1$. (left): $L=0,\ldots,3$ in increments of $1/4$ from bottom right (red) to top right (purple) at $E=2$. (right): $E=0,\ldots,4$ in increments of $1/2$ from top (red) to bottom (purple) at $L=2$.
• Figure 3: Spacelike $E=0$ geodesics in various backgrounds, for various values of $L$. We plot the $(r,\varphi)$ plane with the radial coordinate given by $\tan^{-1}r$. The outer circle is the AdS boundary while the inner disk represents a black hole of radius $r_+ = 1/2$ in AdS units. Specific spacetimes used are BTZ (left), Schwarzschild-AdS$_5$ (middle), and extremal Reissner-Nordstrom-AdS$_5$ (right). The values of angular momenta $L$ are chosen so as to vary $\Delta \varphi$ in increments of $\frac{2 \pi}{10}$. The values of $L$ which gives $\Delta \varphi = 2\pi$ (purple curve) are $L_{\rm BTZ}= 1.09 \, r_+$, $L_{\rm SAdS} = 1.002 \, r_+$, and $L_{\rm RNAdS} = 1.07 \, r_+$, respectively.
• Figure 4: Sketch of the general set-up used in Section \ref{['s:ExtSurf']}. To a given $n$-dimensional region ${\cal R}$ on the boundary we associate an extremal surface ${\cal S}$ in the bulk. A useful quantity characterizing the surface ${\cal S}$ is its maximal reach $z_{\ast}$ into the bulk, and we will characterize the boundary regions ${\cal R}$ by their shape, area, or extent $X({\cal R})$.
• Figure 5: Left: cross-section of $n$-dimensional extremal surfaces in Poincare AdS$_{d+1}$, with varying dimensionality $n=1,2,\ldots,5$: the outermost (red) curve corresponds to $n=1$ while the innermost (purple) curve to $n=5$. Note that $d$ (as long as it is large enough to accommodate the surface) does not enter. Right: corresponding ratio of maximal bulk radial extent $z_{\ast}$ to its boundary size $\Delta x$ grows approximately linearly with $n$.