Spacetime inextensibility criteria by volume-distance-ratio asymptote and applications to naked singularities and FLRW spacetimes

We study the inextensibility problem of the spacetime at a future boundary point. We detect the inextensibility of the spacetime by the volume-distance-ratio asymptote of the timelike diamond approaching the future boundary point. The fundamental idea is to compare the asymptote with the one in Minkowski spacetime. By this idea, we establish the inextensibility criteria for both $C^{0,1}$ and $C^0$ regularities. As applications, we prove that i) $C^{0,1}$-inextensibility of the interior solution of the spherically self-similar naked singularity in the gravitational collapse of a massless scalar field constructed by Christodoulou. The key estimate is on the volume form of the interior solution of the naked singularity in a self-similar coordinate system. ii) $C^0$-inextensibility of the spatially flat FLRW spacetime with asymptotically linear scale factor $a(t) \sim |t|$. The key estimate is the volume comparison with the spatially hyperbolic FLRW spacetime with the scale factor $|t|$ which is the causal past of a point in the Minkowski spacetime.