Stability of the spacetime positive mass theorem in spherical symmetry
Stability of the spacetime positive mass theorem in spherical symmetry
复制标题
球对称时空正质量定理的稳定性
DOI：
10.1007/s12220-020-00431-0
复制
发表时间：
2021
影响因子：
1.1
通讯作者：
Sormani, Christina
中科院分区：
文献类型：
--
作者：
Bryden, Edward;Khuri, Marcus;Sormani, Christina
正质量定理的刚性声明断言，具有零adm质量且满足主导能量条件的爱因斯坦方程的一个渐近平坦的初始数据集一定来自于嵌入到Minkowski空间中。在这篇文章中，我们讨论了当质量仅仅很小时会发生什么的问题。特别地，我们给出了与正质量定理的时空版本相关的稳定性命题的一个猜想，并举例说明了它在真的情况下是如何基本尖锐的。然后，这个猜想建立在所有维度的球对称假设下。更确切地说，证明了一个满足主导能量条件的渐近平坦的初始数据序列，除了可能在内边界外没有视界，且adm质量趋于零的情况下，一定是从等距嵌入到在保持固有平坦意义的指向体积收敛到Minkowski空间的静态时空序列中产生的。由嵌入和初始数据得到的第二基本形式的差必收敛到零。此外，还给出了一些较小的切线结果，包括球对称初值在所有维度上具有刚性声明的时空版本的Penrose不等式，以及最外层视界的对称性继承性质。
The rigidity statement of the positive mass theorem asserts that an asymptotically flat initial data set for the Einstein equations with zero ADM mass, and satisfying the dominant energy condition, must arise from an embedding into Minkowski space. In this paper, we address the question of what happens when the mass is merely small. In particular, we formulate a conjecture for the stability statement associated with the spacetime version of the positive mass theorem, and give examples to show how it is basically sharp if true. This conjecture is then established under the assumption of spherical symmetry in all dimensions. More precisely, it is shown that a sequence of asymptotically flat initial data satisfying the dominant energy condition, without horizons except possibly at an inner boundary, and with ADM masses tending to zero must arise from isometric embeddings into a sequence of static spacetimes converging to Minkowski space in the pointed volume preserving intrinsic flat sense. The difference of second fundamental forms coming from the embeddings and initial data must converge to zero in,. In addition some minor tangential results are also given, including the spacetime version of the Penrose inequality with rigidity statement in all dimensions for spherically symmetric initial data, as well as symmetry inheritance properties for outermost apparent horizons.