时空扩张II
Space-time extensions II
摘要：
研究了光滑因果测地线不完备时空的整体可拓性。用$γ$表示因果测地线不完全时空的不完全不可扩展因果测地线之一$(M,g_{ab})$。首先，它表明，它总是可以选择一个同步的家庭的因果测地线$Γ$和开放的邻域$\mathcal{U}$的$γ$在$M$的最后一段，使$\mathcal{U}$是由$Γ$的成员，并提供$γ$不终止于任何地方的$\mathcal{U}$的潮汐力张量奇点或拓扑奇点。它还表明，如果，此外，时空，$(M,g_{ab})$，是全球双曲，和曲率张量的分量，其协变导数的顺序$k-1$上有界的$\mathcal{U}$，并且$k^{th}$阶协变导数的分量的线积分沿着$Γ$的成员是有限的。其中所有分量都打算相对于$\mathcal{U}$上的同步帧字段被注册-则存在$C^{k-}$扩展$Φ: (M,g_{ab}) \rightarrow (\widehat{M},\widehat{g}_{ab})$，使得对于在$(M,g_{ab})$中不可扩展的每个$\barγ\inΓ$，图像$Φ\circ\barγ$在$(\widehat{M},\widehat{g}_{ab})$中可扩展。最后，证明了当$γ$终止于拓扑奇点时，$(M,g_{ab})$不可能是通用的.
英文摘要：
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.
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