Aspects of holographic timelike entanglement entropy in black hole backgrounds

TL;DR

This paper explores the holographic construction of timelike entanglement entropy in black hole backgrounds, revealing how extremal surfaces encode real and imaginary components, and analyzing their behavior across different dimensions and near-horizon dynamics.

Contribution

It introduces a holographic framework for timelike entanglement entropy in black holes, including extremal surfaces with spacelike and timelike branches, and studies their properties in various dimensions.

Findings

Extremal surfaces extend into black hole interiors, reproducing field-theoretic results.

Boundary subsystem length diverges at a critical point near the horizon, shifting with dimension.

Finite tEE exhibits volume-plus-area structure with dimension-dependent coefficients.

Abstract

We study the holographic construction of timelike entanglement entropy (tEE) in black hole backgrounds in Lorentzian geometries. The holographic tEE is realized through extremal surfaces consisting of spacelike and timelike branches that encode its real and imaginary components, respectively. In the BTZ black hole, these surfaces extend into the interior of the black hole and reproduce the field-theoretic results. The analysis is further generalized to higher-dimensional AdS-Schwarzschild black holes, where the characteristics of tEE are obtained with increasing size of the boundary subsystem. Besides, we also show that the boundary subsystem length diverges at a dimension-dependent critical turning point. Notably, this critical point moves closer to the black hole horizon as the dimensionality of the bulk increases. For large subsystem lengths, the finite part of the tEE displays a characteristic volume-plus-area structure, with a real volume term and a complex coefficient of the area term approaching constant values at large dimensions. Besides, we also study the monotonicity of a new quantity, timelike entanglement density, which offers insights into a timelike area theorem in specific limits. Subsequently, we investigate the near-horizon dynamics in various black hole backgrounds, where the spacelike and timelike surfaces exhibit exponential growth of the form $e^{\frac{2\pi}{\beta} \Delta t}$ with inverse black hole temperature $\beta$.