  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.
