  We determine solutions to 5D Einstein gravity with a discrete fifth
dimension. The properties of the solutions depend on the discretization scheme
we use and some of them have no continuum counterpart. In particular, we find
that the neglect of the lapse field (along the discretized direction) gives
rise to Randall-Sundrum type metric with a negative tension brane. However, no
brane source is required. We show that this result is robust under changes in
the discretization scheme. The inclusion of the lapse field gives rise to
solutions whose continuum limit is gauge fixed by the discretization scheme. We
find however one particular scheme which leads to an undetermined lapse
reflecting the reparametrization invariance of the continuum theory. We also
find other solutions, with no continuum counterpart with changes in the metric
signature or avoidance of singularity. We show that the models allow a
continuous mass spectrum for the gravitons with an effective 4D interaction at
small scales. We also discuss some cosmological solutions.
