Semiclassical gravity in static spacetimes as a constrained initial value problem

We study the semiclassical Einstein field equations with a Klein-Gordon field in ultrastatic and static spacetimes. In both cases, the equations for the spacetime metric become constraint equations. In the ultrastatic case, the Hadamard singular structure can be characterised explicitly, which allows one to in principle give initial data for the Wightman function that has correct distributional singularities, such that the expectation value of the renormalised stress-energy tensor of the solution can be defined, and that hence the semiclassical Einstein equations make sense. Assuming a "positive energy" condition for the Klein-Gordon operator, we characterise the states for which, if the constraints hold for initial data, they hold everywhere in spacetime. These turn out to be time-translation invariant states. The static case is analysed by conformal techniques, effectively reducing the problem to an ultrastatic one.