Analysis of Extremal Black Holes

自从爱因斯坦的广义相对论预言了黑洞以来，黑洞就吸引了科学家和公众的想象力。PI打算在著名的爱因斯坦方程的初值问题的背景下研究与黑洞动力学有关的基本数学问题。在这个方向上最突出的问题之一是黑洞稳定性猜想，即黑洞确实在外部扰动下是稳定的。这一猜想的解决将支持黑洞在物理理论中的相关性。PI的前人工作揭示了所谓的极值黑洞在一类特殊扰动下的一种新的不稳定性。世界各地的几个研究小组从严格数学和数值的角度对这种不稳定性进行了进一步的研究。PI将努力为极端黑洞的稳定性和不稳定性属性提供完整的描述，特别是通过数值模拟预测相关物理现象的严格数学结果。该项目还将研究爱因斯坦方程的重要全球方面，包括引力波的辐射性质。对这一提议的研究将提高我们社会对黑洞的理解，从而提高我们宇宙的结构。该项目的重点是研究与广义相对论中出现的双曲型偏微分方程的局部和整体性质有关的三个中心问题。首先，该项目将研究极端和亚极端黑洞的稳定性问题。在这个方向上，PI和他的合作者在球对称的背景下发展了一种新的方法，该方法解决了事件视界和零点无穷远的各种困难，并有望在双曲型偏微分方程(PDE)中有广泛的应用。为了得到黑洞稳定性问题的整体结果，PI打算对该方法进行推广，并得到线性波的更好的有界性和衰减性。这个项目将探索极端情况下的各种不稳定性，这些不稳定性源于非轴对称准正则模在实轴上收敛到有限值以及所谓的陷阱效应和超辐射的耦合。这些现象在黑洞动力学中具有基本的重要性，在物理文献中已经成为大量数值模拟的对象，但目前还没有得到严格的结果。其次，该项目将研究双曲型方程特征初始数据的粘连问题，这是几何分析中深入研究的黎曼粘连问题的推广。最后，该项目将开发爱因斯坦方程的散射理论，预计这将为爱因斯坦方程的全球行为，特别是引力波的研究提供新的见解。PI坚信，这项建议将导致发现真正新的分析技术，适用于更广泛的几何、分析、偏微分方程和数学物理问题。

Black holes have captured the imagination of scientists and the general public since they were predicted by Einstein's general theory of relativity. The PI intends to investigate fundamental mathematical problems concerning the dynamics of black holes in the context of the initial value problem for the celebrated Einstein equations. One of the most outstanding problems in this direction is the black hole stability conjecture, namely that black holes are indeed stable under perturbations from the outside. The resolution of this conjecture would underpin the relevance of black holes in the physical theory. Previous works of the PI uncovered a novel instability property of the so-called extremal black holes under a special class of perturbations. This instability has been further studied by several research groups worldwide both from the point of view of rigorous mathematics and numerics. The PI will strive to provide a complete description for the stability and instability properties of extremal black holes, obtaining in particular mathematically rigorous results for associated physical phenomena predicted by numerical simulations. The project will also investigate important global aspects of the Einstein equations including radiative properties of gravitational waves. Work on this proposal will improve our society's understanding of black holes, and consequently the structure of our universe. The focus of the project is the study of three central problems relating to local and global properties of hyperbolic partial differential equations arising in general relativity. First, the project will investigate the stability problem for extremal and sub-extremal black holes. In this direction, the PI and his collaborators have developed a new method in the context of spherical symmetry which addresses various difficulties at the event horizon and null infinity and is expected to have a wide range of applications in hyperbolic partial differential equations (PDEs). The PI intends to extend this method and also obtain finer boundedness and decay properties of linear waves in order to obtain global results for the black hole stability problem. This project will probe various instabilities in the extremal case which originate from the convergence of non-axisymmetric quasinormal modes to finite values on the real axis and the coupling of the so-called trapping effect and superradiance. These phenomena are of fundamental importance in black hole dynamics and have been the object of extensive numerical simulations in the physics literature, but no rigorous results are presently available. Second, the project will investigate the gluing problem for characteristic initial data of hyperbolic equations, which is an extension of the Riemannian gluing problem, the latter being intensively studied in geometric analysis. Finally the project will develop a scattering theory for the Einstein equations which is expected to provide new insights into the global behavior of the Einstein equations and in particular the study of gravitational waves. The PI strongly believes that the proposal will lead to the discovery of genuinely new analytical techniques applicable in a wider spectrum of problems in geometry, analysis, PDEs and mathematical physics.