Space-time extensions II

Abstract

The global extendibility of smooth causal geodesically incomplete spacetimes is investigated. Denote by $γ$ 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 $Γ$ and an open neighbourhood $\mathcal{U}$ of a final segment of $γ$ in $M$ such that $\mathcal{U}$ is comprised by members of $Γ$, and suitable local coordinates can be defined everywhere on $\mathcal{U}$ provided that $γ$ 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 $Γ$---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 $Φ: (M,g_{ab}) \rightarrow (\widehat{M},\widehat{g}_{ab})$ so that for each $\barγ\inΓ$, which is inextendible in $(M,g_{ab})$, the image, $Φ\circ\barγ$, is extendible in $(\widehat{M},\widehat{g}_{ab})$. Finally, it is also proved that whenever $γ$ does terminate on a topological singularity $(M,g_{ab})$ cannot be generic.