Holographic Horizons

TL;DR

This paper investigates how different boundary conditions in (2+1)-dimensional AdS space affect the coverage of the space by Fefferman-Graham coordinates, revealing a connection between excluded regions and boundary horizon entropy.

Contribution

It demonstrates that in certain boundary geometries, the Fefferman-Graham coordinates do not cover the entire AdS space and links the boundary horizon entropy to the length of the excluded region.

Findings

Excluded regions correspond to boundary horizons.

The length of the excluded region matches the boundary CFT entropy.

Fefferman-Graham coordinates do not fully cover AdS in some cases.

Abstract

We examine how the (2+1)-dimensional AdS space is covered by the Fefferman-Graham system of coordinates for Minkowski, Rindler and static de Sitter boundary metrics. We find that, in the last two cases, the coordinates do not cover the full AdS space. On a constant-time slice, the line delimiting the excluded region has endpoints at the locations of the horizons of the boundary metric. Its length, after an appropriate regularization, reproduces the entropy of the dual CFT on the boundary background. The horizon can be interpreted as the holographic image of the line segment delimiting the excluded region in the vicinity of the boundary.