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Entanglement first law for timelike entanglement entropy and linearized Einstein's equation

Guo-Ying Li, Mei-Hui Xiao, Song He, Jia-Rui Sun

TL;DR

This work extends the entanglement first law to timelike subregions in holographic CFTs, showing that the timelike entanglement entropy satisfies a first-law-like relation with an entanglement temperature $T_{ m ent}$ scaling as $1/\Delta t$. By employing a double Wick rotation, the authors construct the modular Hamiltonian for a hyperbolic timelike subsystem and establish a precise timelike entanglement first law via relative entropy. The central result proves that, in asymptotically AdS spacetimes, this timelike first law is equivalent to the bulk linearized Einstein equations, with the equivalence shown first in AdS$_4$/CFT$_3$ and then generalized to arbitrary dimensions. The findings reveal deep dynamical ties between boundary entanglement and bulk gravity, and the analysis highlights the roles of analytic continuation and RG-flow-based formulations in rendering a robust entanglement-gravity correspondence for timelike regions.

Abstract

We extend the entanglement first law of conformal field theory (CFT) to timelike subregions. Focusing on intervals along the time direction of the boundary CFT, we show that the associated timelike entanglement entropy obeys a first-law-like relation, with an effective entanglement temperature inversely proportional to the temporal size of the interval. By implementing a double Wick rotation, we obtain the exact modular Hamiltonian for a suitable hyperbolic subsystem and use it to formulate the timelike entanglement first law precisely. Our central result is a detailed proof that, in asymptotically Anti-de Sitter spacetime, this timelike entanglement first law is equivalent to linearized Einstein's equations in the bulk: the first law follows from the linearized equations and, conversely, implies them. Our result further reveal the dynamical connections between entanglement and gravity.

Entanglement first law for timelike entanglement entropy and linearized Einstein's equation

TL;DR

This work extends the entanglement first law to timelike subregions in holographic CFTs, showing that the timelike entanglement entropy satisfies a first-law-like relation with an entanglement temperature scaling as . By employing a double Wick rotation, the authors construct the modular Hamiltonian for a hyperbolic timelike subsystem and establish a precise timelike entanglement first law via relative entropy. The central result proves that, in asymptotically AdS spacetimes, this timelike first law is equivalent to the bulk linearized Einstein equations, with the equivalence shown first in AdS/CFT and then generalized to arbitrary dimensions. The findings reveal deep dynamical ties between boundary entanglement and bulk gravity, and the analysis highlights the roles of analytic continuation and RG-flow-based formulations in rendering a robust entanglement-gravity correspondence for timelike regions.

Abstract

We extend the entanglement first law of conformal field theory (CFT) to timelike subregions. Focusing on intervals along the time direction of the boundary CFT, we show that the associated timelike entanglement entropy obeys a first-law-like relation, with an effective entanglement temperature inversely proportional to the temporal size of the interval. By implementing a double Wick rotation, we obtain the exact modular Hamiltonian for a suitable hyperbolic subsystem and use it to formulate the timelike entanglement first law precisely. Our central result is a detailed proof that, in asymptotically Anti-de Sitter spacetime, this timelike entanglement first law is equivalent to linearized Einstein's equations in the bulk: the first law follows from the linearized equations and, conversely, implies them. Our result further reveal the dynamical connections between entanglement and gravity.

Paper Structure

This paper contains 17 sections, 109 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Timelike entanglement first law and entanglement temperature
  3. Strip subsystem: $\left|t\right|<\frac{\Delta t}{2}$
  4. Hyperbolic subsystem: $-t^{2}+\sum_{i=1}^{d-2}x_i<-T_{0}^{2}$
  5. Comparison with CFT at finite $T$
  6. More on the timelike entanglement first law
  7. Linearized Einstein's equation and timelike entanglement first law
  8. The case of $d=3$
  9. Proof of linearized Einstein's equations will lead to the entanglement first law in AdS$_4$/CFT$_3$
  10. Deriving linearized Einstein's equations from the entanglement first law in AdS$_4/$CFT$_3$
  11. The cases of arbitrary $d>3$
  12. $\Delta S=\Delta \langle H\rangle$ from linearized Einstein's equations
  13. Linearized Einstein's equations from $\Delta S=\Delta \langle H\rangle$
  14. Conclusions and discussions
  15. Modular Hamiltonian for hyperbolic subregion
  16. ...and 2 more sections

Figures (5)

  • Figure 1: The entanglement temperature $T_{\rm ent}$=$\frac{d(\Delta E)}{d(\Delta S)}|_{\Delta t=\textrm{fixed}}$ of CFT$_2$ as a function of $\Delta t$ based on the holographic calculation: the bulk dual is the BTZ black hole at temperature $T$, where we have set $T=1,2,3$.
  • Figure 2: The domain of dependence of subregion $\mathcal{V}$ is the red region, and the modular flow is the blue line.
  • Figure 3: The geometric meaning of multiplying $i$.
  • Figure 4: The diagram on the left illustrates the procedure of the double Wick rotation, while the one on the right shows the resulting coordinate system.
  • Figure 5: The blue region is the domain of dependence of $\mathcal{V'}_E$, and the modular flow is the blue curve. Here $r^2=t_E^2+\sum_{i=1}^{d-2}x_i^2$.