Linear Response, Validity of Semi-Classical Gravity, and the Stability of Flat Space

TL;DR

This paper proposes a stability-based criterion for the validity of semi-classical gravity, demonstrating that flat spacetime remains stable under linear perturbations in Minkowski space, supporting the approximation's consistency.

Contribution

It introduces a gauge-invariant linear response framework using an invariant effective action to test semi-classical gravity's validity, applied specifically to Minkowski spacetime with a scalar field.

Findings

Flat space solutions are stable to all perturbations at large scales.

The linear response criterion involves no additional divergences or counterterms.

The spectral representation of vacuum polarization is explicitly computed.

Abstract

A quantitative test for the validity of the semi-classical approximation in gravity is given. The criterion proposed is that solutions to the semi-classical Einstein equations should be stable to linearized perturbations, in the sense that no gauge invariant perturbation should become unbounded in time. A self-consistent linear response analysis of these perturbations, based upon an invariant effective action principle, necessarily involves metric fluctuations about the mean semi-classical geometry, and brings in the two-point correlation function of the quantum energy-momentum tensor in a natural way. This linear response equation contains no state dependent divergences and requires no new renormalization counterterms beyond those required in the leading order semi-classical approximation. The general linear response criterion is applied to the specific example of a scalar field with arbitrary mass and curvature coupling in the vacuum state of Minkowski spacetime. The spectral representation of the vacuum polarization function is computed in n dimensional Minkowski spacetime, and used to show that the flat space solution to the semi-classical Einstein equations for n=4 is stable to all perturbations on distance scales much larger than the Planck length.