The global stability of the Minkowski spacetime solution to the Einstein-nonlinear system in wave coordinates

In this article, we study the coupling of the Einstein field equations of general relativity to a family of models of nonlinear electromagnetic fields. The family comprises all covariant electromagnetic models that satisfy the following criteria: they are derivable from a sufficiently regular Lagrangian, they reduce to the linear Maxwell model in the weak-field limit, and their corresponding energy-momentum tensors satisfy the dominant energy condition. Our main result is a proof of the global nonlinear stability of the 1 + 3-dimensional Minkowski spacetime solution to the coupled system for any member of the family, which includes the linear Maxwell model. This stability result is a consequence of a small-data global existence result for a reduced system of equations that is equivalent to the original system in our wave coordinate gauge. Our analysis of the spacetime metric components is based on a framework recently developed by Lindblad and Rodnianski, which allows us to derive suitable estimates for tensorial systems of quasilinear wave equations with nonlinearities that satisfy the weak null condition. Our analysis of the electromagnetic fields, which satisfy quasilinear first-order equations, is based on an extension of a geometric energy-method framework developed by Christodoulou, together with a collection of pointwise decay estimates for the Faraday tensor developed in the article. We work directly with the electromagnetic fields, and thus avoid the use of electromagnetic potentials.

在本文中，我们研究广义相对论的爱因斯坦场方程与一族非线性电磁场模型的耦合。该族包含所有满足以下标准的协变电磁模型：它们可从一个足够正则的拉格朗日量导出，在弱场极限下退化为线性麦克斯韦模型，并且它们相应的能量 - 动量张量满足主能量条件。我们的主要结果是证明了对于该族中任何成员（包括线性麦克斯韦模型），耦合系统的1 + 3维闵可夫斯基时空解的全局非线性稳定性。这个稳定性结果是一个简化方程组的小数据全局存在性结果的推论，在我们的波动坐标规范下，该简化方程组与原始系统等价。我们对时空度规分量的分析基于林德布拉德（Lindblad）和罗德尼亚斯基（Rodnianski）最近开发的一个框架，这使我们能够对具有满足弱零条件的非线性项的拟线性波动方程的张量系统推导出合适的估计。我们对满足拟线性一阶方程的电磁场的分析基于克里斯托多罗（Christodoulou）开发的一个几何能量方法框架的扩展，以及本文中为法拉第张量开发的一系列逐点衰减估计。我们直接处理电磁场，从而避免使用电磁势。