Differential Equations resulting from the interaction of Gravity with other Force Fields, and Shock-Waves in General Relativity

NSF Award Abstract - DMS-0103998 Mathematical Sciences：Differential Equations resulting from the interaction of Gravity with other Force Fields，and Shock Waves in General Relativity摘要DMS-0103998 Smoller这个项目关注的是爱因斯坦广义相对论所描述的，在两个不同尺度上的，涉及引力的数学问题：（A）基本粒子，重力与其他基本力耦合（例如，核力），以及通过狄拉克方程考虑量子力学效应;以及（B）天体物理学，特别是宇宙中的冲击波爆炸。 在第一部分，我们研究基本粒子（费米子）在旋转黑洞几何中的行为。 我们还研究了Teukolsky方程（适用于引力波，电磁波等）解的衰减和稳定性，在旋转的黑洞背景几何中。 在B部分，我们研究了一个不同于大爆炸模型的新宇宙学模型。 该模型与天文观测一致，基于可能发生在哈勃长度之外的冲击波。 我们还将研究霍金-彭罗斯奇点定理在涉及相互作用激波的情况下的有效性。 最后，我们将研究星星坍缩成黑洞的动力学，模型化为爱因斯坦-欧拉方程的初值问题，对初始数据有一定的约束。 这项研究还将帮助我们更多地了解爱因斯坦方程的所有初始数据必须满足的约束方程。这个项目涉及爱因斯坦广义相对论所描述的涉及引力的数学问题，在两个不同的尺度上：（A）基本粒子，引力与其他基本力耦合（例如，核力）和量子力学效应被考虑在内，（B）天体物理学，特别是宇宙中的冲击波爆炸。 在A部分，我们研究基本粒子（电子，质子等）的行为。靠近黑洞。 我们还研究了黑洞附近电磁波和引力波的稳定性。 在B部分，我们将探索一种新的宇宙学模型，它不同于通常的大爆炸模型，后者是基于冲击波爆炸。 我们还将研究大质量星星坍缩成黑洞的过程。

NSF Award Abstract - DMS-0103998 Mathematical Sciences: Differential Equations resulting from the interaction of Gravity with other Force Fields, and Shock Waves in General Relativity Abstract DMS-0103998 Smoller This project is concerned with mathematical problems involving gravity, as described by Einsteins theory of general relativity, on two different scales: (A) elementary particles, whereby gravity is coupled to other fundamental forces (e.g., nuclear forces), and quantum mechanical effects are taken into account, via the Dirac Equation; and (B) astrophysics, in particular, shock-wave explosions in the universe. In Part A, we study the behavior of elementary particles (fermions) in a rotating black-hole geometry. We also study the decay and stability of solutions of the Teukolsky equation (which applies to gravitational waves, electromagnetic waves, etc.), in a rotating black-hole background geometry. In Part B, we investigate a new cosmological model, different from the Big-Bang model. The model, which agrees with astronomical observations, is based on shock waves that can occur beyond the Hubble length. We will also investigate the validity of the Hawking-Penrose singularity theorem for situations involving interacting shock waves. Finally, we will study the dynamics of a star collapsing to a black hole, modeled as an initial-value problem for the Einstein-Euler equations, with certain constraints on the initial data. This study will also help us learn more about the constraint equations that all initial data for the Einstein equations must satisfy.This project is concerned with mathematical problems involving gravity, as described by Einsteins theory of general relativity, on two different scales: (A) elementary particles, whereby gravity is coupled to other fundamental forces (e.g., nuclear forces), and quantum mechanical effects are taken into account, and (B) astrophysics; in particular, shock-wave explosions in the universe. In Part A, we study the behavior of an elementary particle (electron, proton, etc.) near a black hole. We also study the stability of electromagnetic and gravitational waves near a black hole. In Part B, we explore a new model for cosmology, different from the usual Big Bang scenario, which is based on a shock-wave explosion. We will also study the collapse of a massive star to a black hole.