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

TL;DR

This paper extends the entanglement first law to timelike regions in conformal field theory, demonstrating its equivalence to linearized Einstein's equations in the bulk AdS spacetime, thus linking entanglement dynamics with gravity.

Contribution

It introduces a timelike entanglement first law and proves its equivalence to linearized Einstein's equations in asymptotically AdS spacetime.

Findings

Timelike entanglement entropy obeys a first-law-like relation.

The timelike entanglement first law is equivalent to linearized Einstein's equations.

The results reveal a dynamical link between entanglement and gravity.

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.