Space-time extensions II

Abstract

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 $\Gamma$ and an open neighbourhood $\mathcal{U}$ of a final segment of $\gamma$ in $M$ such that $\mathcal{U}$ is comprised by members of $\Gamma$, 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 $\Gamma$---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 $\Phi :(M,g_{ab})\to (\hat{M},{\hat{g}}_{ab})$ so that for each $\bar{\gamma }\in \Gamma$, which is inextendible in $(M,g_{ab})$, the image, $\Phi \circ \bar{\gamma }$, is extendible in $(\hat{M},{\hat{g}}_{ab})$. Finally, it is also proved that whenever $\gamma$ does terminate on a topological singularity $(M,g_{ab})$ cannot be generic.

42 pages, no figures, small changes to match the published version

References in corpus (9)

• Loop Quantum Cosmology
• The use of Generalised Functions and Distributions in General Relativity
• Analytic description of singularities in Gowdy spacetimes
• Globally hyperbolic spacetimes can be defined as "causal" instead of "strongly causal"
• Limit curve theorems in Lorentzian geometry
• Asymptotic Behavior in Polarized {\bf T}${}^2$-symmetric Vacuum Spacetimes
• Stationary Black Holes as Holographs
• An extension theorem for conformal gauge singularities
• A global conformal extension theorem for perfect fluid Bianchi space-times

Cited by in corpus (3)

• Stationary Black Holes as Holographs II
• Spacetime singularities and curvature blow-ups
• Generalised hyperbolicity in spacetimes with Lipschitz regularity