  The vacuum expectation values of the energy--momentum tensor are investigated
for massless scalar fields satisfying Dicichlet or Neumann boundary conditions,
and for the electromagnetic field with perfect conductor boundary conditions on
two infinite parallel plates moving by uniform proper acceleration through the
Fulling--Rindler vacuum. The scalar case is considered for general values of
the curvature coupling parameter and in an arbitrary number of spacetime
dimension. The mode--summation method is used with combination of a variant of
the generalized Abel--Plana formula. This allows to extract manifestly the
contributions to the expectation values due to a single boundary. The vacuum
forces acting on the boundaries are presented as a sum of the self--action and
interaction terms. The first one contains well known surface divergences and
needs a further regularization. The interaction forces between the plates are
always attractive for both scalar and electromagnetic cases. An application to
the 'Rindler wall' is discussed.
