Space-time extensions II

The global extendibility of smooth causal geodesically incomplete spacetimes is investigated. Denote by $\gamma$ one of the incomplete non-extendible causal geodesics of a causal geodesically incomplete spacetime $(M,g_{ab})$. First, it is shown that it is always possible to select a synchronised family of causal geodesics $\mathrm{\Gamma }$ and an open neighbourhood $\mathcal{U}$ of a final segment of $\gamma$ in $M$ such that $\mathcal{U}$ is comprised by members of $\mathrm{\Gamma }$, and suitable local coordinates can be defined everywhere on $\mathcal{U}$ provided that $\gamma$ does not terminate either on a tidal force tensor singularity or on a topological singularity. It is also shown that if, in addition, the spacetime, $(M,g_{ab})$, is globally hyperbolic, and the components of the curvature tensor, and its covariant derivatives up to order $k-1$ are bounded on $\mathcal{U}$, and also the line integrals of the components of the $k^{th}$-order covariant derivatives are finite along the members of $\mathrm{\Gamma }$—where all the components are meant to be registered with respect to a synchronised frame field on $\mathcal{U}$—then there exists a $C^{k-}$ extension $\Phi: (M,g_{ab}) \rightarrow (\widehat{M},\widehat{g}_{ab})$sothatforeach$\bar\gamma\in\Gamma$, which is inextendible in $(M,g_{ab})$, the image, $\mathrm{\Phi }\circ \bar{\gamma }$, is extendible in $(\widehat{M},{\widehat{g}}_{ab})$. Finally, it is also proved that whenever $\gamma$ does terminate on a topological singularity $(M,g_{ab})$ cannot be generic.