  We examine the continuum limit of the piecewise flat locally finite gravity
model introduced by 't Hooft. In the linear weak field limit we find the
energy--momentum tensor and metric perturbation of an arbitrary configuration
of defects. The energy--momentum turns out to be restricted to satisfy certain
conditions. The metric perturbation is mostly fixed by the energy--momentum
except for its lightlike modes which reproduce linear gravitational waves,
despite no such waves being present at the microscopic level.
