Analysis of extremal black holes

这个项目关注广义相对论中两个最杰出的猜想，即Kerr黑洞族的唯一性和稳定性，主要集中在极值黑洞这类特殊的黑洞。PI的最新贡献表明，波在事件视界上表现出不稳定的性质，这些性质源于一种新的守恒律系。此外，极端黑洞表现出非常强的陷阱，这与上述守恒定律和超辐射现象(即从黑洞中提取能量)相结合。已经开发出令人满意的技术来克服非极端情况下的陷阱问题；然而，鉴于前述的几何耦合，这些方法在极端情况下失效。国际和平研究所期望该项目将开发必要的工具和技术，以提供对此类时空演化方程的严格和明确的理解。特别是，PI打算继续对线性和非线性波进行严格的研究，并研究这些背景下的准正则模的分布。PI还将研究极值Kerr上拟线性方程的整体存在和破裂准则，希望对这一提议的最终目标，即极值黑洞的完全非线性稳定性和不稳定性和唯一性问题有更深入的了解。广义相对论是描述物理系统在引力作用下演化的经典理论。该理论最著名的预测之一是存在所谓的黑洞区域，即光线无法逃逸到无限远的区域。这些区域不仅激发了科学家的想象力，而且还在天文学、物理学和数学中找到了深刻的应用。一类特别重要的黑洞由所谓的极值黑洞组成，即零温度黑洞。后者是高能物理界的中心研究对象。PI启动了对这种时空上演化方程的严格数学研究，并出现了有趣和令人惊讶的结果。具体地说，在极端黑洞的视界上发现了一种新的不稳定性，在黑洞的数学和物理方面具有广泛的潜在应用。高能物理和数值相对论的专家们正在进一步研究这种不稳定性在其他情况下的应用。国际和平研究所预计，该项目将解开极端黑洞的分析、几何和物理之间的复杂相互作用。此外，国际和平研究所打算开始新的合作，这将导致知识交流，并增加与物理学、数值相对论和天体物理界成员的互动。PI计划在不同的地方举办研讨会，旨在向学生和研究人员介绍广义相对论非常丰富的数学结构。

This project concerns the investigation of two of the most outstanding conjectures in general relativity, namely the uniqueness and stability of the Kerr family of black holes, by concentrating on the special class of extremal black holes. Recent contributions of the PI showed that waves exhibit instability properties along the event horizon originating from a novel hierarchy of conservation laws. Moreover, extremal black holes exhibit very strong trapping which is coupled with the above conservation laws and also the phenomenon of superradiance (i.e. the extraction of energy from the black hole). Satisfactory techniques have been developed to overcome the problem of trapping in the non-extremal case; however, in view of the aforementioned geometric couplings, these methods break down in the extremal case. The PI expects that this project will develop the necessary tools and techniques to provide a rigorous and definitive understanding of evolution equations on such spacetimes. In particular, the PI intends to continue the rigorous study of linear and nonlinear waves and investigate the distribution of quasinormal modes on such backgrounds. The PI will also work on the global existence and breakdown criteria for quasilinear equations on extremal Kerr hoping to gain insights for the ultimate goal of this proposal, namely the fully non-linear stability and instability and uniqueness problem for extremal black holes. General relativity is the classical theory that describes the evolution of physical systems under the effect of gravity. One of the most celebrated predictions of the theory is the existence of so-called black hole regions, i.e. regions from where light cannot escape to infinity. Not only have these regions captured the imagination of scientists, but have also found profound applications to astronomy, physics, and mathematics. A particularly important class of black holes consists of the so-called extremal black holes, that is black holes with zero temperature. The latter are central objects of study in the high-energy physics community. The PI has initiated a rigorous mathematical study of evolution equations on such spacetimes and interesting and surprising results have emerged. Specifically, a novel instability has been discovered on the event horizon of extremal black holes bearing a wide range of potential applications regarding the mathematics and physics of black holes. Specialists in high-energy physics and numerical relativity are further researching the applications of this instability in other contexts. The PI expects that this project will unravel the complex interaction between the analysis, geometry and physics of extremal black holes. Moreover, the PI intends to start new collaborations which will lead to exchange of knowledge and also increase interactions with members of the physics, numerical relativity and astrophysics community. The PI plans to teach seminars at various places aiming at introducing students and researchers to the very rich mathematical structure of general relativity.