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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 thedoi:10.1088/0264-9381/28/7/075005 fatcat:cxg6p4htuzasvnv73oqed2wtuu