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Holographic Timelike Entanglement Entropy from Rindler Method

Peng-Zhang He, Hai-Qing Zhang

TL;DR

This work extends entanglement entropy to timelike regions within AdS/CFT by introducing a timelike cut-off and proposing $S^{(t)} = S(\mathcal{D},\varepsilon^{t})$ which equals the thermal entropy after a Rindler transformation plus a constant $\frac{i c \pi}{6}$. Using the Rindler method in AdS$_3$/CFT$_2$, the authors derive a holographic dual where the real part arises from spacelike geodesics and the imaginary part from a timelike geodesic, culminating in $S_{TEE} = S_{thermal} + i \frac{\pi c}{6}$. To address the imaginary contribution, they modify the Rindler map (e.g., $l_v \to -l_v$) so the horizon corresponds to two spacelike geodesics, reproducing the same final relation for timelike entanglement entropy. The results provide a framework for timelike entanglement and suggest connections to the emergence of time from entanglement, with applicability to finite-size and finite-temperature CFTs and a consistent holographic dual.

Abstract

For a Lorentzian invariant theory, the entanglement entropy should be a function of the domain of dependence of the subregion under consideration. More precisely, it should be a function of the domain of dependence and the appropriate cut-off. In this paper, we refine the concept of cut-off to make it applicable to timelike regions and assume that the usual entanglement entropy formula also applies to timelike intervals. Using the Rindler method, the timelike entanglement entropy can be regarded as the thermal entropy of the CFT after the Rindler transformation plus a constant $icπ/6$ with $c$ the central charge. The gravitational dual of the `covariant' timelike entanglement entropy is finally presented following this method.

Holographic Timelike Entanglement Entropy from Rindler Method

TL;DR

This work extends entanglement entropy to timelike regions within AdS/CFT by introducing a timelike cut-off and proposing which equals the thermal entropy after a Rindler transformation plus a constant . Using the Rindler method in AdS/CFT, the authors derive a holographic dual where the real part arises from spacelike geodesics and the imaginary part from a timelike geodesic, culminating in . To address the imaginary contribution, they modify the Rindler map (e.g., ) so the horizon corresponds to two spacelike geodesics, reproducing the same final relation for timelike entanglement entropy. The results provide a framework for timelike entanglement and suggest connections to the emergence of time from entanglement, with applicability to finite-size and finite-temperature CFTs and a consistent holographic dual.

Abstract

For a Lorentzian invariant theory, the entanglement entropy should be a function of the domain of dependence of the subregion under consideration. More precisely, it should be a function of the domain of dependence and the appropriate cut-off. In this paper, we refine the concept of cut-off to make it applicable to timelike regions and assume that the usual entanglement entropy formula also applies to timelike intervals. Using the Rindler method, the timelike entanglement entropy can be regarded as the thermal entropy of the CFT after the Rindler transformation plus a constant with the central charge. The gravitational dual of the `covariant' timelike entanglement entropy is finally presented following this method.
Paper Structure (9 sections, 59 equations, 3 figures)

This paper contains 9 sections, 59 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Brief review of timelike entanglement entropy
  3. Alternative perspective on timelike entanglement entropy: Rindler method
  4. Deriving holographic entanglement entropy using Rindler method
  5. Timelike entanglement entropy from a timelike curve
  6. Holographic timelike entanglement entropy from the Rindler method
  7. Conclusion and discussion
  8. Define timelike entanglement entropy via Wick rotation
  9. The origin of $ic\pi/6$

Figures (3)

  • Figure 1: Penrose diagrams of the geodesics connecting the boundary subregions, where the boundary subregion $A$ is denoted as the red line segment. The blue curves represent the spacelike geodesics, while the purple curve represents the timelike geodesic.
  • Figure 2: A schematic diagram of a spatial interval (horizontal line) and its domain of dependence (blue shape), with cut-offs $\varepsilon_{u},\varepsilon_{v}$ marked in the diagram.
  • Figure 3: Illustration of the spacelike curve $\mathcal{I}_s$ and its dual timelike curve $\mathcal{I}_t$. The cut-offs of $\mathcal{I}_{t}$ are marked by the blue line segment.