Semiclassical Gravity Theory and Quantum Fluctuations

TL;DR

This paper examines the validity of semiclassical gravity, showing it works well only when stress tensor fluctuations are small, and highlights large fluctuations in cases with negative energy density, like the Casimir effect.

Contribution

It introduces a measure of stress tensor fluctuations, analyzes their behavior in various quantum states, and proposes a stochastic approach to describe fluctuating gravitational fields.

Findings

Stress tensor fluctuations are small near coherent states.

Large fluctuations occur when local energy density is negative.

Gravitational fields with negative energy density exhibit significant metric fluctuations.

Abstract

We discuss the limits of validity of the semiclassical theory of gravity in which a classical metric is coupled to the expectation value of the stress tensor. It is argued that this theory is a good approximation only when the fluctuations in the stress tensor are small. We calculate a dimensionless measure of these fluctuations for a scalar field on a flat background in particular cases, including squeezed states and the Casimir vacuum state. It is found that the fluctuations are small for states which are close to a coherent state, which describes classical behavior, but tend to be large otherwise. We find in all cases studied that the energy density fluctuations are large whenever the local energy density is negative. This is taken to mean that the gravitational field of a system with negative energy density, such as the Casimir vacuum, is not described by a fixed classical metric but is undergoing large metric fluctuations. We propose an operational scheme by which one can describe a fluctuating gravitational field in terms of the statistical behavior of test particles. For this purpose we obtain an equation of the form of the Langevin equation used to describe Brownian motion.