Quantum radiation reaction force on a one-dimensional cavity with two relativistic moving mirrors

We consider a real massless scalar field inside a cavity with two moving mirrors in a two-dimensional spacetime, satisfying Dirichlet boundary condition at the instantaneous position of the boundaries, for arbitrary and relativistic laws of motion. Considering vacuum as the initial field state, we obtain formulas for the exact value of the energy density of the field and the quantum force acting on the boundaries, which extend results found in previous papers. For the particular cases of a cavity with just one moving boundary, non-relativistic velocities, or in the limit of infinity length of the cavity (a single mirror), our results coincide with those found in the literature.