Structure and statistical properties of the semiclassical Einstein equations

TL;DR

This paper analyzes the semiclassical Einstein equations as a quantum-classical hybrid, clarifying their derivation and addressing criticisms, leading to the emergence of stochastic gravity as a natural extension.

Contribution

It demonstrates the formal equivalence of two derivation methods of the semiclassical Einstein equations and clarifies their interpretation as expectation values rather than actual realizations.

Findings

The semiclassical Einstein equations can be derived via two equivalent methods.

Criticisms of semiclassical gravity are addressed by interpreting the equations as expectation values.

Stochastic gravity naturally arises as an extension of the semiclassical framework.

Abstract

We treat the semiclassical Einstein equation as a quantum-classical hybrid and demonstrate the formal equivalence of its two derivation methods. This approach identifies the left-hand side of the equation as the expectation value of the Einstein tensor given the state of matter, and not its actual value in each realization of the set-up. As a result, standard criticisms of semiclassical gravity do not apply, and stochastic gravity emerges as a necessary extension