  We propose a quantitative test for the validity of the semi-classical
approximation in gravity, namely that the solutions to the semi-classical
equations should be stable to linearized perturbations, in the sense that no
gauge invariant perturbation should become unbounded in time. We show that a
self-consistent linear response analysis of these perturbations based upon an
invariant effective action principle 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. The properties of this
correlation function are discussed and it is shown on general grounds that it
contains no state-dependent divergences and requires no new renormalization
counterterms beyond those required in the leading order semi-classical
approximation.
