From Quantum Relative Entropy to the Semiclassical Einstein Equations

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.

Figures (1)

• Figure 1: Sketch of a local region $\mathcal{U}\subset\mathcal{M}$, faintly shaded in gray, endowed with a local Rindler horizon, consisting of the null hypersurfaces $\mathcal{H}_A$ and $\mathcal{H}_B$ intersecting at the horizon cross section $\mathcal{S}$. Within $\mathcal{U}$, $\mathcal{H}_A$ is affinely parametrized by $V$, and likewise $\mathcal{H}_B$ by $U$, yielding a local double null coordinate system. Consequently, the region $\mathcal{U}$ is separated into two wedge-shaped regions, correspondingly decomposing $\mathcal{H}_A,\mathcal{H}_B$ into $\mathcal{H}_A^{\mathcal{R}},\mathcal{H}_A^{\mathcal{L}}$ and $\mathcal{H}_B^{\mathcal{L}},\mathcal{H}_B^{\mathcal{R}}$, respectively. The orange curve indicates the flow generated by the local approximate boost Killing vector field $\xi^a$, which becomes null on $\mathcal{H}_A$ and $\mathcal{H}_B$.