  The Casimir energy is evaluated for massless scalar fields under Dirichlet or
Neumann boundary conditions, and for the electromagnetic field with perfect
conductor boundary conditions on one and two infinite parallel plates moving by
uniform proper acceleration through the Fulling--Rindler vacuum in an arbitrary
number of spacetime dimension. For the geometry of a single plate the both
regions of the right Rindler wedge, (i) on the right (RR region) and (ii) on
the left (RL region) of the plate are considered. The zeta function technique
is used, in combination with contour integral representations. The Casimir
energies for separate RR and RL regions contain pole and finite contributions.
For an infinitely thin plate taking RR and RL regions together, in odd spatial
dimensions the pole parts cancel and the Casimir energy for the whole Rindler
wedge is finite. In $d=3$ spatial dimensions the total Casimir energy for a
single plate is negative for Dirichlet scalar and positive for Neumann scalar
and the electromagnetic field. The total Casimir energy for two plates geometry
is presented in the form of a sum of the Casimir energies for separate plates
plus an additional interference term. The latter is negative for all values of
the plates separation for both Dirichlet and Neumann scalars, and for the
electromagnetic field.
