Semiclassical relativity: The weak-field limit

Semiclassical relativity is a theory in which quantum matter fields interact with a classical gravitational field via the semiclassical Einstein equation $G_{\mathrm{ab}}=8\mathrm{}\pi 〈T_{\mathrm{ab}}〉$. We consider, for the case of massless quantum fields, the weak-field limit of this theory. It is shown that the linearized quantum stress energy can be determined in a manner which is essentially independent of the details of any regularization prescription. Using this expression for the quantum stress energy, we solve the linearized semiclassical Einstein equation. The solutions found include (1) perturbations that satisfy the linearized classical Einstein equation ${\dot{G}}_{\mathrm{ab}}=0\mathrm{}$, (2) perturbations that grow exponentially in time, and (3) perturbations that (in some sense) travel faster than the speed of light.