Physical-Space Estimates on Black Hole Perturbations

数学广义相对论领域涉及对空间和时间的数学描述，重点是分析爱因斯坦方程的解，如黑洞。该领域的一个重要问题是黑洞稳定性的数学证明，这对于将它们理解为现实的物理对象是必不可少的。如果是稳定的，受到引力(或其他类型)辐射扰动的黑洞可能会出现暂时的变化，但最终会恢复到它们最初的状态。这项工作的成果将通过同行评议的出版物和研讨会分享给数学和物理界，并将通过媒体文章、公开讲座和学校外展活动向广大社区传播。双曲型偏微分方程的物理空间估计已被成功地应用于最近证明慢旋转的Kerr族的非线性稳定性，以及分析带电Kerr-Newman黑洞中引力辐射和电磁辐射的相互作用。研究人员计划开发一种稳健的方法，基于对耦合波方程系统的能量-动量张量的定义，这种方法有可能应用于其他物质领域。他们还计划将这些方法的适用性扩展到快速旋转的黑洞解决方案，方法是利用对能量方法和综合局部能量衰减估计如何在存在旋转的情况下相互作用的精细分析。扰动黑洞的辐射可以用偏微分方程组来描述，其性质可以通过各种方法来研究。虽然一些涉及更简单形式的分解的技术，如模式，已被证明是非常成功的，但它们在应用于非线性问题和不同种类的辐射的相互作用方面存在局限性。这项研究旨在扩展已知的涉及物理空间估计的方法来研究黑洞解的稳定性。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The field of Mathematical General Relativity concerns the mathematical description of space and time, focusing on the analysis of solutions to the Einstein equation, such as black holes. One important problem in the field is the mathematical proof of the stability of black holes, which is essential towards understanding them as realistic physical objects. If stable, black holes which are perturbed with gravitational (or other kinds of) radiation could present a temporary change but would eventually return to their initial status. The result of this work will be shared to the mathematical and physical community through peer-reviewed publications and seminars and will be disseminated to the general community through media articles, public lectures and outreach events in schools. Graduate students and postdocs will be involved in this research.Physical-space estimates for hyperbolic partial differential equations have been successfully applied to the recent proof of non-linear stability of the slowly rotating Kerr family and to the analysis of the interaction of gravitational and electromagnetic radiations in the charged Kerr-Newman black hole. The investigators plan to develop a robust approach, based on the definition of a combined energy-momentum tensor for a system of coupled wave equations, that has the potential to be applied to other matter fields. They also plan to extend the applicability of these methods to rapidly spinning black hole solutions by making use of a refined analysis of how energy methods and integrated local energy decay estimates interact in the presence of rotation. Radiations perturbing black holes are described by partial differential equations, whose properties can be studied through various techniques. Although some techniques involving decomposition in simpler forms, such as modes, have proved to be very successful, they have limitations in applications to non-linear problems and to the interaction of different kinds of radiation. This research aims to extend the known methods involving physical-space estimates for the study of stability of black hole solutions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.