Entanglement Entropy in Quantum Gravity and the Plateau Problem

TL;DR

The paper argues that entanglement entropy in quantum gravity can be defined as a macroscopic, geometry-driven quantity $S={\cal A}({\cal B})/(4G)$, where ${\cal B}$ is a minimal (or extremal) co-dimension-2 surface separating regions in a Euclidean section of the spacetime. Using a Gibbons–Hawking Euclidean path integral and replica trick, the author shows that, in the semiclassical limit, the dominant geometries are those with ${\cal B}$ of least area, yielding a direct geometric link to the Plateau problem. The static-spacetime analysis places ${\cal B}$ on a constant-time slice, ensuring a von Neumann entropy-like structure, while black hole spacetimes reveal that ${\cal B}$ can include the horizon, reproducing the Bekenstein–Hawking entropy and clarifying information-loss interpretations. The work also presents variational formulae, explores entropy inequalities, and discusses quantum corrections and renormalization of divergences, highlighting a coherent framework in which gravity, entanglement, and minimal-surface geometry are intertwined. Overall, the paper connects entanglement entropy in quantum gravity to minimal-surface geometry, with implications for holography and the emergence of gravitational dynamics from microscopic degrees of freedom.

Abstract

In a quantum gravity theory the entropy of entanglement $S$ between the fundamental degrees of freedom spatially divided by a surface is discussed. The classical gravity is considered as an emergent phenomenon and arguments are presented that: 1) $S$ is a macroscopical quantity which can be determined without knowing a real microscopical content of the fundamental theory; 2) $S$ is given by the Bekenstein-Hawking formula in terms of the area of a co-dimension 2 hypesurface $\cal B$; 3) in static space-times $\cal B$ can be defined as a minimal hypersurface of a least volume separating the system in a constant time slice. It is shown that properties of $S$ are in agreement with basic properties of the von Neumann entropy. Explicit variational formulae for $S$ in different physical examples are considered.

Figures (9)

• Figure 1: A constant time slice of a system spatially divided by a hypersurface $\cal B$.
• Figure 2: Construction of the space ${\cal M}_n$ for a 2D finite-temperature system set on an interval $\Sigma$. For $n=3$ the space ${\cal M}_n$ is obtained from 3 copies of the cylinders $\Sigma \times S^1$. The cylinders are cut and glued along the part $\Sigma_1$ of a "constant time" slice.
• Figure 3: In a quantum gravity one can specify the entangled regions of a system by dividing its spatial boundary ${\cal S}$ onto domains $D_1$ and $D_2$. The separating surface $\cal B$ is determined by a dynamical problem with the condition that $\cal B$ ends at the boundary $\cal P$ between $D_1$ and $D_2$.
• Figure 4: Some examples of minimal surfaces corresponding to a single contour in a flat space.
• Figure 5: The Carter-Penrose diagram for a Schwarzschild black hole. The black hole in cavity corresponds to a part of the diagram between the lines $r=r_0$. Shown at the diagram is one of the Cauchy hypersurfaces $\Sigma_t^L\bigcup \Sigma_t^R$.