Linear response, validity of semiclassical gravity, and the stability of flat space

A quantitative test for the validity of the semiclassical approximation in gravity is given. The criterion proposed is that solutions to the semiclassical 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 semiclassical 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 semiclassical 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 semiclassical Einstein equations for $n=4\mathrm{}$ is stable to all perturbations on distance scales much larger than the Planck length.