On the Behavior of Solutions of Einstein's Equations and Other Geometric Partial Differential Equations

爱因斯坦方程及其他几何偏微分方程解的行为

基本信息

批准号：
0354659
负责人：
James Isenberg
金额：
$ 15万
依托单位：
University of Oregon Eugene
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2004
资助国家：
美国
起止时间：
2004-08-01 至 2008-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=0354659&HistoricalAwards=false
关键词：
Behavior Solutions Einstein Equations Other

项目摘要

In addition to providing us with an amazingly accurate and beautiful model for studying gravitational physics on both the astrophysical and the cosmological scale, Einstein?s gravitational field theory is a rich source of mathematically interesting questions. A number of the questions we are most interested in pertain to solutions of the Einstein constraint equations. These equations restrict the choice of initial data one can make for an evolving gravitational system. We are interested in parametrizing the set of solutions of the constraints (i.e., finding the degrees of freedom), developing algorithms for constructing solutions, and studying the behavior of solutions of the constraints. We hope to apply the technique of ?gluing? to implement the Einstein constraints. The idea of gluing is that, given a pair of solutions of the constraints, one attempts to connect the two solutions smoothly via a bridge joining a point in each of them. We have applied the gluing technique to the Einstein constraints, and have used it to construct multi-black hole initial data sets, to add wormholes to given solutions, to construct initial data for black holes in cosmological solutions, and to show that any given closed manifold (minus a point) admits an asymptotically flat solution of the constraints. Our work in gluing so far has assumed that the given solutions have a region of constant mean curvature, and are vacuum solutions. We are working to remove these assumptions, which should open up a much wider range of applications.

除了为我们提供一个非常准确，美丽的模型，用于研究天体物理和宇宙学量表的引力物理学外，爱因斯坦的引力场理论还是数学上有趣的问题的丰富来源。我们最感兴趣的许多问题与爱因斯坦约束方程的解决方案有关。这些方程式限制了人们可以选择不断发展的引力系统的初始数据。我们有兴趣参数约束（即找到自由度）的解决方案集，开发用于构建解决方案的算法以及研究约束解决方案的行为。我们希望应用“胶合”的技术？实施爱因斯坦的约束。粘合的想法是，鉴于一对约束的解决方案，一种尝试通过将两个解决方案通过连接到每个点的桥进行平稳连接。我们已经将胶合技术应用于爱因斯坦的约束，并使用它来构建多黑孔初始数据集，以在给定的溶液中添加虫洞，以在宇宙学解决方案中构造黑洞的初始数据，并表明任何给定的封闭歧管（MINUS A点）都承认，可以承认约束的无关固定溶液。到目前为止，我们在粘合方面的工作假设给定的溶液具有恒定平均曲率的区域，并且是真空溶液。我们正在努力删除这些假设，这些假设应该打开更广泛的应用程序。