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arXiv 0904.0620gr-qc
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广义相对论中Minkowski空间稳定性定理的推广

An Extension of the Stability Theorem of the Minkowski Space in General Relativity

Lydia Bieri

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50 pages
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摘要：

本文简述了作者在以前的工作中对广义相对论中Minkowski空间稳定性定理的证明。我们讨论了爱因斯坦真空（EV）方程的解。我们解决柯西问题更一般，渐近平坦的初始数据比在开创性的工作“的闵可夫斯基空间的全局非线性稳定性”的D。Christodoulou和S.克莱纳曼或比任何其他工作。此外，我们精确地描述了渐近行为。我们对初始数据的放松假设产生了在$L^{\infty}(M)$中没有界的时空曲率。作为一个主要的结果，我们在我们的工作中遇到的边界情况下，我们在本文中讨论。事实上，我们的某些估计是边界上的衰减，表明我们的主要定理的条件是尖锐的，只要假设在无穷大的衰减在初始数据的关注。因此，边界情况是我们对数据放松假设的结果。他们没有出现在其他作品中，因为他们都对他们的数据进行了更强的假设。我们工作的EV方程的不变公式。我们的主要证明是基于一个自举参数。为了结束这个论证，我们必须证明时空曲率和相应的几何量具有所要求的衰减。为了做到这一点，爱因斯坦方程相对于时空的特定叶理被分解。这个结果推广了D. Christodoulou和S.克莱纳曼

英文摘要：

In this paper, we sketch the proof of the extension of the stability theorem of the Minkowski space in General Relativity done explicitly in previous work by the present author. We discuss solutions of the Einstein vacuum (EV) equations. We solve the Cauchy problem for more general, asymptotically flat initial data than in the pioneering work `The Global Nonlinear Stability of the Minkowski Space' of D. Christodoulou and S. Klainerman or than in any other work. Moreover, we describe precisely the asymptotic behaviour. Our relaxed assumptions on the initial data yield a spacetime curvature which is not bounded in $L^{\infty}(M)$. As a major result, we encounter in our work borderline cases, which we discuss in this paper as well. The fact that certain of our estimates are borderline in view of decay indicates that the conditions in our main theorem are sharp in so far as the assumptions on the decay at infinity on the initial data are concerned. Thus, the borderline cases are a consequence of our relaxed assumptions on the data. They are not present in the other works, as all of them place stronger assumptions on their data. We work with an invariant formulation of the EV equations. Our main proof is based on a bootstrap argument. To close the argument, we have to show that the spacetime curvature and the corresponding geometrical quantities have the required decay. In order to do so, the Einstein equations are decomposed with respect to specific foliations of the spacetime. This result generalizes the work `The Global Nonlinear Stability of the Minkowski Space' of D. Christodoulou and S. Klainerman.

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