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与重力有关的数学问题。具体地说，我的研究将涉及(1)证明在克尔旋转黑洞几何中的麦克斯韦电磁方程和引力波的初值问题解的衰减性；(2)研究广义相对论中产生的冲击波和膨胀波；(3)磁性旋转和非旋转的牛顿恒星。第一部分继续以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背景几何中的衰变结果的工作。第二部分讨论广义相对论中的冲击波和膨胀波，并将这些波解应用于天体物理学中的问题；例如，在不需要暗能量或宇宙常数的情况下，所谓的宇宙反常加速。它是基于我严格获得的爱因斯坦方程的新解。第三部分源于我最近关于旋转牛顿恒星的存在和稳定性结果的结果，目的是将这些结果推广到磁恒星(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。这项研究的重点是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送电磁波或引力波，黑洞是否会保持基本完好，或者它会辐射出很大一部分能量？第二是继续研究爱因斯坦方程式的新解，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殊的方式修改了爱因斯坦的方程式，并涉及到“暗能量”的概念，这是一种未被观测到的反引力，以与天文观测相一致。这种方法没有任何物质基础。我感兴趣的最后一个领域是从磁场开始研究，目的是从理论上解释太阳黑子和太阳耀斑。这些现象仍然非常神秘，而且很重要，因为当它们发生时，它们会扰乱卫星和无线电通信。目前还不知道如何预测太阳黑子和太阳耀斑，对相关方程的坚定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.