Extensions of the Stability Theorem of the Minkowski Space in General Relativity
Extensions of the Stability Theorem of the Minkowski Space in General Relativity
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DOI：
10.1090/amsip/045
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发表时间：
2009-04
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影响因子：
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通讯作者：
L. Bieri
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作者：
L. Bieri
这本书包括两个独立的作品：第一部分是“爱因斯坦真空方程的解决方案”，由莉迪亚Bieri。第二部分是“爱因斯坦-麦克斯韦方程组的解”，由Nina Zipser。Christodoulou和Klainerman的一个著名结果是闵可夫斯基时空的全局非线性稳定性。在本书中，Bieri和Zipser对这个结果做了两个扩展。在第一部分中，Bieri用更一般的、渐近平坦的初值解了Einstein真空方程的Cauchy问题，并精确地描述了其渐近行为。特别地，她假设r的幂的衰减比Christodoulou-Klainerman的结果小，并且导数比Christodoulou-Klainerman的结果少一个。她证明，在这种情况下，同样，初始数据，是全球接近平凡的数据，产生一个解决方案，这是一个完整的时空，趋于闵可夫斯基时空在无穷沿着任何测地线。与原来的情况相反，这个证明中的某些估计在衰减方面是边界的，这表明关于初始数据在无穷远处衰减的主要定理中的条件是尖锐的。在第二部分中，Zipser证明了Einstein-Maxwell方程的光滑整体解的存在性。这些方程的一个非平凡解是一个带有电磁场的弯曲时空。为了证明爱因斯坦-麦克斯韦方程组解的存在性，Zipser遵循Christodoulou和Klainerman介绍的论点和方法。为了推广原来的结果，她需要考虑由于电磁场F的存在而产生的额外曲率项;在她的例子中，时空的里奇曲率不是恒为零，而是由F的分量的二次项表示。特别是Ricci曲率是应力-能量张量的常倍数。此外，黎曼曲率张量的无迹部分不再满足齐次比安奇方程，而是包含时空里奇曲率分量的非齐次方程。因此，本书的第二部分主要集中于推导由于电磁场的存在而产生的新项的估计。
This book consists of two independent works: Part I is 'Solutions of the Einstein Vacuum Equations', by Lydia Bieri. Part II is 'Solutions of the Einstein-Maxwell Equations', by Nina Zipser. A famous result of Christodoulou and Klainerman is the global nonlinear stability of Minkowski spacetime. In this book, Bieri and Zipser provide two extensions to this result. In the first part, Bieri solves the Cauchy problem for the Einstein vacuum equations with more general, asymptotically flat initial data, and describes precisely the asymptotic behavior. In particular, she assumes less decay in the power of $r$ and one less derivative than in the Christodoulou-Klainerman result. She proves that in this case, too, the initial data, being globally close to the trivial data, yields a solution which is a complete spacetime, tending to the Minkowski spacetime at infinity along any geodesic. In contrast to the original situation, certain estimates in this proof are borderline in view of decay, indicating that the conditions in the main theorem on the decay at infinity on the initial data are sharp. In the second part, Zipser proves the existence of smooth, global solutions to the Einstein-Maxwell equations. A nontrivial solution of these equations is a curved spacetime with an electromagnetic field. To prove the existence of solutions to the Einstein-Maxwell equations, Zipser follows the argument and methodology introduced by Christodoulou and Klainerman. To generalize the original results, she needs to contend with the additional curvature terms that arise due to the presence of the electromagnetic field $F$; in her case the Ricci curvature of the spacetime is not identically zero but rather represented by a quadratic in the components of $F$. In particular the Ricci curvature is a constant multiple of the stress-energy tensor for $F$. Furthermore, the traceless part of the Riemann curvature tensor no longer satisfies the homogeneous Bianchi equations but rather inhomogeneous equations including components of the spacetime Ricci curvature. Therefore, the second part of this book focuses primarily on the derivation of estimates for the new terms that arise due to the presence of the electromagnetic field.