  Recent work in the literature had evaluated the energy-momentum tensor of a
Casimir apparatus in a weak gravitational field, for an electromagnetic field
subject to perfect conductor boundary conditions on parallel plates. The
Casimir apparatus was then predicted to experience a tiny push in the upwards
direction, and the regularized energy-momentum tensor was found to have a trace
anomaly. The latter, unexpected property made it compelling to assess what
happens in a simpler case. For this purpose, the present paper studies a free,
real massless scalar field subject to homogeneous Dirichlet conditions on the
parallel plates. Working to first order in the constant gravity acceleration,
the resulting regularized and renormalized energy-momentum tensor is found to
be covariantly conserved, while the trace anomaly vanishes if the massless
scalar field is conformally coupled to gravity. Conformal coupling also ensures
a finite Casimir energy and finite values of the pressure upon parallel plates.
