LOCALIZED EXOTIC SMOOTHNESS
LOCALIZED EXOTIC SMOOTHNESS
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DOI：
10.1088/0264-9381/11/7/015
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发表时间：
1994-07-01
影响因子：
3.5
通讯作者：
BRANS, CH
中科院分区：
文献类型：
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作者：
BRANS, CH
Gompf的端和技术被用来建立一个无限的非同构流形的存在，所有具有相同的平凡R4拓扑，但其中的奇异微分结构被限制在一个区域，这是空间有限的。因此，光滑度在拓扑上（但不光滑）为B3 x R1的区域之外是标准的，其中B3是紧凑的三球。这个区域的外部与标准R1 X S2 X R1同构。在时空图中，受限的奇异性扫过一个世界管，它被证明可能是爱因斯坦方程某些非标准解的来源。结果表明，光滑的洛伦兹签名度量可以从适当定义的区域，包括外部（标准）区域上给出的度量在全球范围内继续。类似的构造提供了拓扑S2 x R2的克鲁斯卡尔形式的史瓦西解决方案。这导致了爱因斯坦度量的存在，这些度量在外部与标准黑洞的度量相同，但没有一个可以与这样的标准对象全局同构。柯西问题的某些方面也讨论了R（θ）4模型是“半标准”，说所有的t < 0，但其中t不能是全局光滑的。
Gompf's end-sum techniques are used to establish the existence of an infinity of non-diffeomorphic manifolds, all having the same trivial R4 topology, but for which the exotic differentiable structure is confined to a region which is spatially limited. Thus, the smoothness is standard outside of a region which is topologically (but not smoothly) B3 x R1, where B3 is the compact three ball. The exterior of this region is diffeomorphic to standard R1 X S2 x R1. In a spacetime diagram, the confined exoticness sweeps out a worldtube which, it is conjectured, might act as a source for certain non-standard solutions to the Einstein equations. It is shown that smooth Lorentz signature metrics can be globally continued from ones given on appropriately defined regions, including the exterior (standard) region. Similar constructs are provided for the topology S2 x R2 of the Kruskal form of the Schwarzschild solution. This leads to conjectures about the existence of Einstein metrics which are externally identical to standard black hole ones, but none of which can be globally diffeomorphic to such standard objects. Certain aspects of the Cauchy problem are also discussed in terms of R(THETA)4 models which are 'half-standard', say for all t < 0, but for which t cannot be globally smooth.