Curvature, causality and completeness in space-times with causally complete spacelike slices
Curvature, causality and completeness in space-times with causally complete spacelike slices
复制标题
具有因果完整类空间切片的时空曲率、因果关系和完整性
DOI：
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发表时间：
1986
影响因子：
0.8
通讯作者：
G. Galloway
中科院分区：
文献类型：
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作者：
G. Galloway
设S是时空M中的类空切片（在第2节中正式定义）。我们将说S在M中是未来因果完备的，如果对于每个p ε J+（S），集合J-（p）<$S在S中的闭包是紧的。定义时间对偶的过去因果完备性。则S是因果完备的，如果它既是未来因果完备的，又是过去因果完备的。一个紧致类空切片必然是因果完备的，就像任何柯西曲面一样，但是因果完备的概念比这两个条件都要广泛得多。例如从闵可夫斯基空间移去原点得到的时空切片t = const.t0是因果完备的，尽管它们既不是柯西的也不是紧的。前面例子中的切片t = 0和闵可夫斯基空间中的双曲面（其中（t，x1，...，xn）是标准惯性坐标）是因果不完备切片的例子。从物理上讲，一个无边片S是未来因果完备的，如果来自S的信息到达S的未来的一个点，来自S中的一个有限的非奇异区域。最大Reissner-Nordstrom时空是一个著名的例子，其中任何渐近平坦的部分柯西曲面都不满足这个有限性条件。事实上，对于任何这样的部分柯西曲面S，J-（p）<$S对于任何p ε H+（S）都是非紧的。然而，正如文献中所讨论的（例如[17]，第625页f），人们相信，在这种情况下，柯西视界对于S上初始数据的扰动是不稳定的。
Let S be a spacelike slice (defined formally in Section 2) in a space-time M. We will say that S is future causally complete in M if for each p ε J+(S) the closure in S of the set J-(p) ∩ S is compact. Define past causal completeness time-dually. Then S is causally complete if it is both future and past causally complete. A compact spacelike slice is necessarily causally complete, as is any Cauchy surface, but the concept of causal completeness is much broader than either of these two conditions. For example the slices t = const. ≠ 0 in the space-time obtained by removing the origin from Minkowski space are causally complete, although they are neither Cauchy nor compact. The slice t = 0 in the previous example and the hyperboloid in Minkowski space (where (t, x1, …, xn) are standard inertial coordinates) are examples of slices which are not causally complete. Physically speaking, an edgeless slice S is future causally complete if the information from S which reaches a point in the future of S comes from a finite nonsingular region in S. The Maximal Reissner-Nordstrom space-time is a well-known example in which this finiteness condition is not fulfilled by any of its asymptotically flat partial Cauchy surfaces. Indeed for any such partial Cauchy surface S, J-(p) ∩ S is non-compact for any p ε H+(S). However, as has been discussed in the literature (e.g. [17], p. 625 f), it is believed that the Cauchy horizon in this situation is unstable with respect to perturbations of the initial data on S.