On the nature of Hawking’s incompleteness for the Einstein-vacuum equations: The regime of moderately spatially anisotropic initial data
On the nature of Hawking’s incompleteness for the Einstein-vacuum equations: The regime of moderately spatially anisotropic initial data
复制标题
关于爱因斯坦真空方程的霍金不完备性的本质：中等空间各向异性初始数据的状态
DOI：
10.4171/jems/1092
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发表时间：
2018
影响因子：
2.6
通讯作者：
Jared Speck
中科院分区：
文献类型：
--
作者：
I. Rodnianski;Jared Speck
在数学物理学文献中，有启发性的论点，可以追溯到三十年前，表明对于高维爱因斯坦真空方程的初始光滑解的开放集合，稳定的，近似单调的曲率奇点可以动态地形成沿着类空超曲面。在这篇文章中，我们研究了柯西问题，并在足够高的维度上给出了这一现象的严格证明，从而提供了作为纯引力效应的稳定曲率爆破（没有对称性假设）沿着类空超曲面的第一个构造性证明。我们的证明适用于满足霍金著名的“奇点”定理的假设的正则初始数据的开子集，这表明该解决方案是测地线不完整的，但没有揭示的性质的不完整性。具体而言，我们的主要结果是一个动态稳定性的证明的Kasner曲率奇异性的一个子集的Kasner解决方案的度量表现出只有适度（而不是严重）的空间各向异性行为。独立的兴趣是我们的证明方法，这是更强大的比以前的方法，因为i）它不依赖于近似单调性恒等式和ii）它容纳的可能性，解决方案开发非常奇异的高阶空间导数，其爆破率是允许的，在我们的引导参数的范围内，比那些驱动基本爆破的基本水平的量差得多。由于这些原因，我们的方法可以用来获得类似的爆破结果为各种爱因斯坦物质系统在任何数量的空间维度的解决方案对应于一个开放的一组适度的空间各向异性的初始数据，从而超越了近空间各向同性制度在早期的工作。
In the mathematical physics literature, there are heuristic arguments, going back three decades, suggesting that for an open set of initially smooth solutions to the Einstein-vacuum equations in high dimensions, stable, approximately monotonic curvature singularities can dynamically form along a spacelike hypersurface. In this article, we study the Cauchy problem and give a rigorous proof of this phenomenon in sufficiently high dimensions, thereby providing the first constructive proof of stable curvature blowup (without symmetry assumptions) along a spacelike hypersurface as an effect of pure gravity. Our proof applies to an open subset of regular initial data satisfying the assumptions of Hawking's celebrated "singularity" theorem, which shows that the solution is geodesically incomplete but does not reveal the nature of the incompleteness. Specifically, our main result is a proof of the dynamic stability of the Kasner curvature singularity for a subset of Kasner solutions whose metrics exhibit only moderately (as opposed to severely) spatially anisotropic behavior. Of independent interest is our method of proof, which is more robust than earlier approaches in that i) it does not rely on approximate monotonicity identities and ii) it accommodates the possibility that the solution develops very singular high-order spatial derivatives, whose blowup rates are allowed to be, within the scope of our bootstrap argument, much worse than those of the base-level quantities driving the fundamental blowup. For these reasons, our approach could be used to obtain similar blowup results for various Einstein-matter systems in any number of spatial dimensions for solutions corresponding to an open set of moderately spatially anisotropic initial data, thus going beyond the nearly spatially isotropic regime treated in earlier works.