  We apply the background field method and the effective action formalism to
describe the four-dimensional dynamical Casimir effect. Our picture corresponds
to the consideration of quantum cosmology for an expanding FRW universe (the
boundary conditions act as a moving mirror) filled by a quantum massless GUT
which is conformally invariant. We consider cases in which the static Casimir
energy is repulsive and attractive. Inserting the simplest possible inertial
term, we find, in the adiabatic (and semiclassical) approximation, the
dynamical evolution of the scale factor and the dynamical Casimir stress
analytically and numerically (for SU(2) super Yang-Mills theory). Alternative
kinetic energy terms are explored in the Appendix.
