Extremal surfaces as bulk probes in AdS/CFT

TL;DR

This paper investigates how extremal surfaces and geodesics in asymptotically AdS spacetimes can serve as probes to understand the bulk geometry, revealing limitations like the inability to penetrate black hole horizons and how boundary region properties influence bulk reach.

Contribution

It provides a general analysis of extremal surfaces and geodesics as probes in AdS/CFT, including new insights into their depth limits and the influence of boundary region shape and size.

Findings

Spacelike geodesics at fixed time reach deepest into the bulk for a given boundary distance.

Geodesics cannot probe past the horizon in black hole spacetimes.

Higher-dimensional extremal surfaces reach deeper into the bulk for fixed boundary extent.

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.