Notes on semiclassical gravity

Abstract

In this paper we investigate the different possible ways of defining the semiclassical limit of quantum general relativity. We discuss the conditions under which the expectation value of the energy-momentum tensor can act as the source for a semiclassical, c-number, gravitational field. The basic issues can be understood from the study of the semiclassical limit of a toy model, consisting of two interacting particles, which mimics the essential properties of quantum general relativity. We define and study the WKB semiclassical approximation and the gaussian semiclassical approximation for this model. We develop rules for finding the back-reaction of the quantum mode q on the classical mode Q. We argue that the back-reaction can be found using the phase of the wave-function which describes the dynamics of q. We find that this back-reaction is obtainable from the expectation value of the hamiltonian if the dispersion in this phase can be neglected. These results on the back-reaction are generalised to the semiclassical limit of the Wheeler-DeWitt equation. We conclude that the back-reaction in semiclassical gravity is <T_ik> only when the dispersion in the phase of the matter wavefunctional can be neglected. This conclusion is highlighted with a minisuperspace example of a massless scalar field in a Robertson-Walker universe. We use the semiclassical theory to show that the minisuperspace approximation in quantum cosmology is valid only if the production of gravitons is negligible.