Further Results on the Smoothability of Cauchy Hypersurfaces and Cauchy Time Functions
Further Results on the Smoothability of Cauchy Hypersurfaces and Cauchy Time Functions
复制标题
柯西超曲面和柯西时间函数光滑性的进一步结果
DOI：
10.1007/s11005-006-0091-5
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发表时间：
2005
影响因子：
1.2
通讯作者：
Miguel Sánchez Caja
中科院分区：
文献类型：
--
作者：
A. Bernal;Miguel Sánchez Caja
近年来，关于整体双曲时空M_i的柯西超曲面和时间函数的光滑性的民间问题得到了解决。本文给出了进一步的结果，这些结果适用于以下几个问题：（1）任何带边界的紧致类空因果子流形H都可以扩张到类空柯西超曲面S。如果H只是无时的，则存在光滑扩张的反例，但连续扩张（事实上，对任何紧致的无时子集K都有效）仍然是可能的。（2）给定任何类空柯西超曲面S，一个柯西时间函数
$$\mathcal{T}$$（即，光滑函数，处处具有过去定向类时梯度，并且Cauchy超曲面作为水平），
$$S= \mathcal{T}^{-1}（0）$$被构造-因此，时空正交分裂为
$$\mathbb{R} \times S$$以规范的方式。更进一步，如果柯西超曲面S是非类空的（包括非光滑的，或非共时的但非因果的），则得到了最后一个结果的精确版本。
AbstractRecently, folk questions on the smoothability of Cauchy hypersurfaces and time functions of a globally hyperbolic spacetime M, have been solved. Here we give further results, applicable to several problems: (1)Any compact spacelike acausal submanifold H with boundary can be extended to a spacelike Cauchy hypersurface S. If H were only achronal, counterexamples to the smooth extension exist, but a continuous extension (in fact, valid for any compact achronal subset K) is still possible.(2)Given any spacelike Cauchy hypersurface S, a Cauchy temporal function
$$\mathcal{T}$$ (i.e., a smooth function with past-directed timelike gradient everywhere, and Cauchy hypersurfaces as levels) with
$$S= \mathcal{T}^{-1}(0)$$ is constructed – thus, the spacetime splits orthogonally as
$$\mathbb{R} \times S$$ in a canonical way. Even more, accurate versions of this last result are obtained if the Cauchy hypersurface S were non-spacelike (including non-smooth, or achronal but non-acausal).