Strong cosmic censorship and causality violation

We investigate the instability of the Cauchy horizon caused by causality violation in the compact vacuum universe with the topology $B\times {\mathbf{S}}^1\times \mathbf{R}$, which Moncrief and Isenberg considered. We show that if the occurrence of curvature singularities are restricted to the boundary of the causality-violating region, the whole segments of the boundary become curvature singularities. This implies that strong cosmic censorship holds in the spatially compact vacuum space-time in the case of causality violation. This also suggests that causality violation cannot occur for a compact universe.