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

• 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

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在天体物理和宇宙尺度上研究引力物理的令人惊讶的准确和美丽的模型，爱因斯坦-S引力场理论还提供了丰富的数学有趣问题来源。我们最感兴趣的一些问题与爱因斯坦约束方程的解有关。这些方程限制了人们对演化的引力系统所能提供的初始数据的选择。我们感兴趣的是将约束的解集参数化(即寻找自由度)，开发构造解的算法，以及研究约束解的行为。我们希望应用胶合技术。来实现爱因斯坦约束。粘合的想法是，给定约束的一对解，人们试图通过连接每个解中的一个点的桥来平滑地连接两个解。我们将粘合技术应用到爱因斯坦约束中，并用它来构造多黑洞初始数据集，向给定的解添加虫洞，构造宇宙解中的黑洞的初始数据，并证明任何给定的闭流形(减去一点)都允许约束的渐近平坦解。到目前为止，我们在胶合方面的工作都假定给定解有一个常值平均曲率区域，并且是真空解。我们正在努力消除这些假设，这应该会打开更广泛的应用领域。