Stability of Minkowski space and polyhomogeneity of the metric
Stability of Minkowski space and polyhomogeneity of the metric
复制标题
DOI：
10.1007/s40818-020-0077-0
复制
发表时间：
2020-06-01
期刊：
影响因子：
2.8
通讯作者：
Vasy, Andras
中科院分区：
文献类型：
--
作者：
Hintz, Peter;Vasy, Andras
研究了（3+1）维闵可夫斯基时空作为爱因斯坦真空方程解的非线性稳定性。类似于我们以前的宇宙学黑洞的稳定性的工作，我们构造的非线性初值问题的解决方案，使用迭代方案，我们在每一步解决一个线性化方程的全局;我们使用广义调和规范和实施约束阻尼固定的几何零无穷大。线性分析主要是基于能量和向量场方法起源于工作克莱纳曼。Lindblad和Rodnianski的弱零条件自然地产生为零无穷远处线性模型算子中某些度量分量的幂零耦合。紧化R4的一个流形的角落，与边界超曲面对应的类空，零，类时无穷大，我们表明，使用框架的梅尔罗斯的b-分析，多齐次初始数据产生一个多齐次时空度量。最后，我们将Bondi质量与度量在零无穷远处展开的对数项联系起来，证明了Bondi质量损失公式。
We study the nonlinear stability of the (3+1)-dimensional Minkowski spacetime as a solution of the Einstein vacuum equation. Similarly to our previous work on the stability of cosmological black holes, we construct the solution of the nonlinear initial value problem using an iteration scheme in which we solve a linearized equation globally at each step; we use a generalized harmonic gauge and implement constraint damping to fix the geometry of null infinity. The linear analysis is largely based on energy and vector field methods originating in work by Klainerman. The weak null condition of Lindblad and Rodnianski arises naturally as a nilpotent coupling of certain metric components in a linear model operator at null infinity. Upon compactifying R4 to a manifold with corners, with boundary hypersurfaces corresponding to spacelike, null, and timelike infinity, we show, using the framework of Melrose's b-analysis, that polyhomogeneous initial data produce a polyhomogeneous spacetime metric. Finally, we relate the Bondi mass to a logarithmic term in the expansion of the metric at null infinity and prove the Bondi mass loss formula.