Volume-Distance-Ratio Asymptote and Spacetime Inextendibility

This paper develops geometric criteria for determining the inextendibility of spacetimes near singularities based on asymptotic analysis of volume-distance relationships. We introduce and analyze the asymptotic behavior of the volume-distance-ratio (VDR), defined as the ratio of volumes of small chronological diamonds to appropriate powers of distances between their vertices. In $\mathrm{C}^0$ and $\mathrm{C}^{0,1}$ spacetimes (which are weaker than the classical $\mathrm{C}^2$ regularity), we prove that VDR converges to the Minkowski value as chronological diamonds approach accumulation points. The central contribution is the establishment of inextendibility criteria showing that failure of VDR convergence to the Minkowski value implies inextendibility of the spacetime. These criteria apply to spacetime extensions satisfying $\mathrm{C}^0$ locally null-non-accumulating strongly-causal conditions and $\mathrm{C}^{0,1}$ strongly-causal conditions, where the local null-non-accumulation condition is introduced as a fundamental structural property ensuring the validity of VDR-based inextendibility criteria. Concrete applications demonstrate the power and scope of these methods. We prove that $2$-dimensional Misner spacetime is $\mathrm{C}^0$ strongly-causal inextendible and that spatially flat FLRW spacetimes with linear scale factor behavior are $\mathrm{C}^0$ locally null-non-accumulating strongly-causal inextendible. Furthermore, we establish $\mathrm{C}^{0,1}$ strongly-causal inextendibility for Christodoulou's class of spherically symmetric self-similar naked singularity spacetimes.