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

美国国家科学基金会奖摘要-DMS-0103998数学科学奖：引力场与其他力场相互作用产生的微分方程，以及广义相对论中的冲击波DMS-0103998斯莫勒这个项目涉及爱因斯坦广义相对论所描述的与引力有关的数学问题，在两个不同的尺度上：(A)基本粒子，即重力与其他基本力(如核力)耦合，并通过狄拉克方程考虑量子力学效应；(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(费米子)在旋转黑洞几何中的行为。我们还研究了旋转黑洞背景几何中Teukolsky方程(适用于引力波、电磁波等)解的衰减性和稳定性。在B部分，我们研究了一个不同于大爆炸模型的新的宇宙学模型。该模型基于哈勃长度以外的冲击波，与天文观测结果一致。我们还将研究霍金-彭罗斯奇点定理在包含相互作用的冲击波的情况下的有效性。最后，我们将研究恒星塌缩为黑洞的动力学，它被建模为爱因斯坦-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，其中重力与其他基本力(如核力)耦合，并考虑量子力学效应，以及(B)天体物理学，特别是宇宙中的冲击波爆炸。在A部分，我们研究基本粒子(电子、质子等)的行为。在一个黑洞附近。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，我们探索一种新的宇宙学模型，不同于通常的基于冲击波爆炸的大爆炸场景。我们还将研究大质量恒星向黑洞的崩塌。

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.