Mathematical Questions in Gravitation, Black Holes, Cosmology, and Rotating Stars

我的研究涉及天体物理学中出现的涉及重力的数学问题。具体来说，我的研究将涉及(1)证明麦克斯韦电磁学方程和引力波方程初值问题解的衰减，两者都是在克尔旋转黑洞几何中；(2)研究广义相对论中产生的激波和膨胀波；(3)磁旋转和非旋转牛顿恒星。第1部分继续之前关于狄拉克和标量波动方程、麦克斯韦方程和线性化爱因斯坦方程在KBH背景几何中的衰减结果的工作。第二部分讨论广义相对论中的激波和膨胀波，以及将这些波解应用于天体物理问题；例如，所谓的宇宙反常加速，而不需要暗能量或宇宙常数。它是基于我严格获得的爱因斯坦方程的新解。第3部分源于我最近关于旋转牛顿恒星的存在性和稳定性的结果，旨在将这些结果扩展到磁性恒星（如我们的太阳）。这项研究的动机是试图从严格的数学角度更好地理解太阳黑子和太阳耀斑。该提案主要关注天体物理学的三个领域。第一个涉及研究黑洞在电磁和引力扰动下是否稳定。也就是说，如果一个人向一个旋转的黑洞发射电磁波或引力波，这个黑洞会保持完整，还是会辐射掉很大一部分能量？第二件事是继续研究爱因斯坦方程的新解，解释所谓的宇宙反常加速。也就是说，标准物理模型以一种特别的方式修改了爱因斯坦的方程，并包含了“暗能量”的概念，一种未被观测到的反引力，以便与天文观测相一致。这种方法没有任何物理基础。我感兴趣的最后一个领域是从磁场开始研究，目的是得到太阳黑子和太阳耀斑的理论解释。这些现象仍然是高度神秘的，而且很重要，因为当它们发生时，它们会干扰卫星和无线电通信。目前还不知道如何预测太阳黑子和太阳耀斑，对相关方程的牢固的数学理解将是朝着这个方向迈出的重要一步。

My research deals with mathematical problems involving gravity which arise in astrophysics. Specifically my research will involve (1) proving decay of solutions to the initial-value problem for Maxwell's equations of electromagnetism and for gravitational waves, both in a Kerr rotating black hole geometry; (2) studying shock waves and expansion waves arising in general relativity; (3) magnetic rotating and non-rotating Newtonian stars. Part 1 continues previous work on decay results for the Dirac and scalar wave equations, to Maxwell's equations and to the linearized Einstein equations in a KBH background geometry. Part 2 deals with shock waves and expansion waves in general relativity, and applying these wave solutions to problems in astrophysics; e.g. the so-called anomalous acceleration of the universe, without needing dark energy or the cosmological constant. It is based on my rigorously obtained new solutions of the Einstein equations. Part 3 arises from my recent results on existence and stability results for rotating Newtonian stars and is aimed at extending these results to magnetic stars (like our sun). This study is motivated by an attempt to better understand sunspots and solar flares, from a rigorous mathematical point of view.The proposal is focused on three areas of astrophysics. The first involves studying whether black holes are stable under electromagnetic and gravitational perturbation. That is, if one sends an electromagnetic or gravitational wave into a rotating black hole, will the black hole remain pretty much intact, or will it radiate away a good portion of its energy? The second is to continue my work on new solutions of Einstein's equations explaining the so-called anomalous acceleration of the universe. That is, the standard physical model modifies Einstein's equations in an ad hoc manner and involves the notion of "dark energy", an unobserved anti-gravitational force, in order to agree with astronomical observations. This approach has no physical basis whatsoever. The final area of interest to me is to study start with magnetic fields with an aim to getting a theoretical explanation of sunspots and solar flares. These phenomena remain highly mysterious, and are important since when they occur, they disrupt satellite as well as radio communication. It is not known how to predict sunspots and solar flares, and a firm mathematical understanding of the associated equations would be an important step in this direction.