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E~3-Complete Timelike Surfaces in E_1~3 Are Globally Hyperbolic

机译:E_1〜3中的E〜3完全时似曲面是全局双曲的

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摘要

A C~∞ immersion Z of a surface S into Minkowski 3-space E_1~3 can also be viewed as a C~∞ immersion of S into Euclidean 3-space E~3. If the metric I induced on S by Z: S → E_1~3 is Lorentzian, then Z:S → E_1~3 is called timelike. If the metric I_ε induced on S by Z: S → E~3 is complete, then Z: S → E_1~3 is said to be E~3-complete. In all results below, S is a surface provided with the Lorentzian metric I and the time orientation induced by a timelike C~∞ immersion Z: S → E_1~3. Among the conformally invariant properties definable on a time-oriented Lorentzian surface are the causality conditions of interest in general relativity. Two of these conditions, stable causality and global hyperbolicity (defined in [1, pp. 63-65]) are dealt with in this paper.
机译:曲面S在Minkowski 3空间E_1〜3中的C〜∞浸入Z也可以看作S在欧氏3空间E〜3中的C〜∞浸入。如果我通过Z:S→E_1〜3在S上诱导的度量是洛伦兹式的,则Z:S→E_1〜3称为时态。如果由Z:S→E〜3在S上诱导的度量I_ε已完成,则称Z:S→E_1〜3为E〜3完全。在下面的所有结果中,S是具有洛伦兹度量I的表面,并且是由类似时间的C〜∞浸入Z诱导的时间方向:S→E_1〜3。在时间取向的洛伦兹表面上可定义的共形不变性质是广义相对论中所关注的因果条件。本文讨论了其中两个条件,即稳定因果关系和全局双曲线(在[1,pp。63-65]中定义)。

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