A Geodesic Route toward Holography beyond AdS: Tessellating the Schwarzschild Black Hole
Abstract
In vacuum anti-de Sitter (AdS) space, the geometry admits a perfect tessellation by a geodesic network. This tessellation characterizes the background exactly, and a partial entanglement entropy (PEE) tensor-network toy model of AdS/CFT can be defined on it. It has so far been unclear whether this construction can be extended beyond vacuum AdS. In this paper we take the first step in this direction. We show that the exterior region of the Schwarzschild black hole and its Einstein--Rosen bridge are perfectly tessellated by a specific geodesic gas emitted from the boundary, so that Crofton reconstruction holds in these regions. We then prove that such a geodesic tessellation and the Crofton reconstruction it defines extend to more generic Riemannian manifolds. This allows us to construct a PEE tensor-network model of holographic duality on more generic geometric backgrounds. In the case of the PEE tensor-network model for the Schwarzschild background, we can concretely realize the Bekenstein--Hawking entropy, the Ryu--Takayanagi formula and the ER=EPR proposal. Our approach opens a new route towards holographic toy models beyond AdS/CFT.
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BibTeX
@misc{gu2026geodesic,
title = {{A Geodesic Route toward Holography beyond AdS: Tessellating the Schwarzschild Black Hole}},
author = {Shi Gu and Qiang Wen and Haocheng Zhong and Yiwei Zhong},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21471},
eprint = {2609.21471},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21471},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Geodesic Route toward Holography beyond AdS: Tessellating the Schwarzschild Black Hole
AU - Shi Gu
AU - Qiang Wen
AU - Haocheng Zhong
AU - Yiwei Zhong
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.21471
UR - https://arxiv.org/abs/2609.21471
AB - In vacuum anti-de Sitter (AdS) space, the geometry admits a perfect tessellation by a geodesic network. This tessellation characterizes the background exactly, and a partial entanglement entropy (PEE) tensor-network toy model of AdS/CFT can be defined on it. It has so far been unclear whether this construction can be extended beyond vacuum AdS. In this paper we take the first step in this direction. We show that the exterior region of the Schwarzschild black hole and its Einstein--Rosen bridge are perfectly tessellated by a specific geodesic gas emitted from the boundary, so that Crofton reconstruction holds in these regions. We then prove that such a geodesic tessellation and the Crofton reconstruction it defines extend to more generic Riemannian manifolds. This allows us to construct a PEE tensor-network model of holographic duality on more generic geometric backgrounds. In the case of the PEE tensor-network model for the Schwarzschild background, we can concretely realize the Bekenstein--Hawking entropy, the Ryu--Takayanagi formula and the ER=EPR proposal. Our approach opens a new route towards holographic toy models beyond AdS/CFT.
ER -