Global Studies of Einstein Spacetimes

该奖项支持探索爱因斯坦广义相对论(GR)基本问题的项目。其中一个项目考察了时空的演化，以了解发出GR破裂信号的奇点的发展是否只在自然界中发生在黑洞内部，并且在黑洞中，只有遇到黑洞的观察者才能看到。这被称为宇宙审查猜想。这个问题将通过继续开发爱因斯坦方程的光锥估计来解决，这些估计与已经为杨-米尔斯-希格斯方程开发的光锥估计类似。我们的目标将是证明，在爱因斯坦方程的(真空)解中，时空曲率不会(按点)爆炸，直到曲率的积分能量(Bel-Robinson)爆炸。另一个项目将是继续开发用于研究黑洞碰撞的改进的数值方法。最近的工作建立了(共形变换的)ADM发展方程在零无穷远的正则性。这一结果及其潜在的推广将应用于实际的数值计算。宇宙审查猜想通常被认为是经典广义相对论中主要的开放数学问题。宇宙审查是爱因斯坦方程的自然类比，也是明可夫斯基空间场理论的整体存在性质的自然类比，因此，似乎很有可能，在对后者的分析中被证明成功的技术应该对前者有重要的贡献。此外，在开发用于分析黑洞碰撞的改进的数值方法方面取得的任何重大进展，预计都将极大地影响正在进行的对这些重要天体物理过程进行数值模拟的全球努力，并最终影响LIGO和LISA观测。这项研究的大部分应该会引起数学家的兴趣。改进的计算黑洞碰撞引力波的数值方法的发展可能会对这种辐射的实验搜索产生影响。预计博士后研究员和研究生将在这些项目中发挥关键作用。

This award supports projects to explore fundamental issues in Einstein's general theory of relativity (GR). One project examines the evolution of spacetimes to see if the development of singularities that signal the breakdown of GR can occur in nature only inside black holes and, within black holes, can be visible only to observers who run into them. This is called the Cosmic Censorship Conjecture. This question will be addressed through continued development of light­cone­ estimates for the Einstein equations that parallel those that have already been developed for the Yang-­Mills-­Higgs equations. The objective will be to show that space­time curvature cannot blow up (point­wise) in a (vacuum) solution to Einstein's equations until the integrated (Bel­-Robinson) energy for curvature blows up. Another project will be to continue the development of improved numerical methods for the study of black hole collisions. Recent work has established the regularity of the (conformally transformed) ADM evolution equations at null infinity. This result and its potential generalizations will be applied to actual numerical calculations.The cosmic censorship conjecture is often considered to be the main open mathematical problem in classical general relativity. Cosmic censorship is the natural analogue, for Einstein's equations, of the global existence property for Minkowski space field theories and so it seems very plausible that techniques that have proven successful in the analysis of the latter should have something important to contribute to the former. In addition, any significant progress on the development of improved numerical methods for the analysis of black hole collisions can be anticipated to strongly influence the ongoing worldwide efforts to numerically model these important astrophysical processes and ultimately to impact LIGO and LISA observations. Much of this research should be of interest to mathematicians. Development of improved numerical methods for the computation of gravitational waves from colliding black holes could have an impact on the experimental search for such radiation. Postdoctoral fellows and graduate students are expected to play key roles in these projects.