Semiclassical Einstein Equations Derived from Quantum Relative Entropy

Derivation from Quantum Relative Entropy

A new theoretical argument indicates that the semiclassical Einstein equations can be derived from quantum relative entropy. This derivation relies on the proportionality of quantum relative entropy to a variation in area.

Energy Flux and Horizon Area

Using modular theory, the relative entropy between the vacuum state and coherent excitations of a scalar quantum field on a bifurcate Killing horizon is shown to be equivalent to the energy flux across that horizon. Under the assumption of the Bekenstein-Hawking entropy-area formula, this energy flux is directly proportional to a variation in the surface area of the horizon's cross section.

Generalization of Previous Work

This approach provides a quantum field theoretic generalization of Jacobson's thermodynamic derivation of the Einstein equations. It replaces the classical thermodynamic entropy used in Jacobson's original work with the quantum relative (Araki-Uhlmann) entropy.

Implications for Quantum Gravity

The findings suggest that quantum information plays a central role in quantum field theory in curved spacetimes, which is often considered a zeroth-order approximation of a theory of quantum gravity. This indicates a deeper connection between quantum information and the fundamental equations governing spacetime.