From Quantum Relative Entropy to the Semiclassical Einstein Equations

TL;DR

This paper derives the semiclassical Einstein equations from quantum relative entropy principles, linking quantum information theory with gravitational dynamics through modular theory and horizon energy flux analysis.

Contribution

It introduces a quantum field theoretic derivation of Einstein equations using relative entropy, extending Jacobson's thermodynamic approach with quantum information concepts.

Findings

Relative entropy between vacuum and excitations equals horizon energy flux.

Energy flux is proportional to horizon area variation under Bekenstein-Hawking law.

Semiclassical Einstein equations emerge naturally from quantum relative entropy considerations.

Abstract

We provide arguments indicating that the semiclassical Einstein equations follow from quantum relative entropy and its proportionality to an area variation. Using modular theory, we establish that the relative entropy between the vacuum state and coherent excitations of a scalar quantum field on a bifurcate Killing horizon is given by the energy flux across the horizon. Under the assumption of the Bekenstein-Hawking entropy-area formula, this energy flux is proportional to a variation in the surface area of the horizon cross section. The semiclassical Einstein equations follow automatically from this identification. Our approach provides a quantum field theoretic generalization of Jacobson's thermodynamic derivation of the Einstein equations, replacing classical thermodynamic entropy with the well-defined quantum relative (Araki-Uhlmann) entropy. This suggests that quantum information plays a central role in what is often seen as a zeroth order approximation of a theory of quantum gravity, namely quantum field theory in curved spacetimes.