Extremal Surface Barriers

TL;DR

The paper derives a purely geometric criterion for barrier surfaces that limit how far boundary-anchored extremal surfaces can reach in Lorentzian spacetimes, showing that a splitting surface with $^{\Sigma}K_{\mu\nu}v^{\mu}v^{\nu}\le 0$ blocks $\,\Sigma\-\,$deformable extremal surfaces from crossing. It further proves that totally geodesic or negative-extrinsic-curvature barriers constrain extremal probes, and that trapped-surface structures can generate barriers for codimension-2 surfaces, with outermost barriers necessarily possessing a nonnegative curvature direction. The work connects these barriers to the presence of singularities or marginally trapped surfaces and discusses their implications for AdS/CFT bulk reconstruction and firewall scenarios, suggesting that some bulk regions are not accessible via extremal surfaces and may require non-extremal probes or larger Hilbert-space factors. Overall, the results indicate a fundamental limit on reconstructing the full bulk geometry from boundary extremal observables, motivating further study of alternative probes and the role of barriers in holography.

Abstract

We present a generic condition for Lorentzian manifolds to have a barrier that limits the reach of boundary-anchored extremal surfaces of arbitrary dimension. We show that any surface with nonpositive extrinsic curvature is a barrier, in the sense that extremal surfaces cannot be continuously deformed past it. Furthermore, the outermost barrier surface has nonnegative extrinsic curvature. Under certain conditions, we show that the existence of trapped surfaces implies a barrier, and conversely. In the context of AdS/CFT, these barriers imply that it is impossible to reconstruct the entire bulk using extremal surfaces. We comment on the implications for the firewall controversy.

Figures (7)

• Figure 1: The family of deformations $\{N_{r}\}$, all anchored at Ext$(\Sigma)\cap I$. For $^{\Sigma}K_{\mu\nu}v^{\mu}v^{\nu}\leq 0$, $N_{I}$ and $N$ do not exist.
• Figure 2: A zoom-in near a neighborhood where $\Sigma$ and $N$ coincide. The horizontal lines at the point $p$ represent the covector $^{\Sigma}\!k_{\mu}(p)=\ ^{\Sigma}\!k^{\nu}(p)g_{\mu\nu}$, where $^{\Sigma}k^{\nu}(p)$ is the null generator for null $\Sigma$, or the normal for timelike or spacelike $\Sigma$. The horizontal lines at $s$ represent the covector $^{\Sigma}\!k_{\mu}(s)$, which is obtained by parallel transporting $^{\Sigma}\!k_{\mu}(p)$ along $\Sigma$.
• Figure 3: A illustration of a possible situation in which the extremal surface (in red) are anchored at $\mathcal{R}$, but they give rise to an outermost barrier $\Sigma$ (in blue above) whose exterior contains $\mathcal{R}$ as a proper subset.
• Figure 4: The isotropic AdS$_{d+1}$ cosmology. $\Sigma$ is the surface where $a(t)$ is maximized.
• Figure 5: The Schwarzschild-AdS$_{d+1}$ black hole. $\Sigma_{1}$ and $\Sigma_{2}$ are totally geodesic splitting surfaces, and are therefore by Theorem 2.2 extremal surface barriers.

Theorems & Definitions (8)

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