最大切片的拓扑障碍
Topological Obstructions To Maximal Slices
摘要：
整体双曲时空${\mathbb R}\times Σ$允许极大切片的必要条件是柯西切片$Σ$允许具有非负标量曲率的度量$R\ge 0$。本文考虑的两种情形是闭空间流形和渐近平坦空间流形。虽然这里的大多数结果将适用于四维或四维以上的时空，但这项工作将主要考虑四维时空。对于$Σ$封闭或渐近平坦的，所有拓扑都是场方程所允许的。由于所有的$Σ$都作为爱因斯坦方程解的柯西切片出现，而大多数$Σ$不允许度规具有$R\ge 0$，因此大多数全局双曲时空永远不允许最大切片，即平均外曲率为零的切片。特别是，允许最大切片的全局渐近平坦双曲时空是例外而不是规则。这样做的原因是由于构造这种切片的拓扑障碍。在渐近平坦的情况下，这将通过流形的光滑紧化来证明，以便将结果用于空间闭流形。
英文摘要：
A necessary condition for a globally hyperbolic spacetime ${\mathbb R}\times Σ$ to admit a maximal slice is that the Cauchy slice $Σ$ admit a metric with nonnegative scalar curvature, $R\ge 0$. In this paper, the two cases considered are the closed spatial manifold and the asymptotically flat spatial manifold. Although most results here will apply in four or more spacetime dimensions, this work will mainly consider 4-dimensional spacetimes. For $Σ$ closed or asymptotically flat, all topologies are allowed by the field equations. Since all $Σ$ occur as Cauchy slices of solutions to the Einstein equations and most $Σ$ do not admit metrics with $R\ge 0$, it follows that most globally hyperbolic spacetimes never admit a maximal slice, i.e. a slice with zero mean extrinsic curvature. In particular, asymptotically flat globally hyperbolic spacetimes which admit maximal slices are the exception rather than the rule. The reason for this is due to topological obstructions to constructing such slices. In the asymptotically flat case, this will be shown by smooth compactification of the manifold in order to use the results for spatially closed manifolds.
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