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。其中一个项目研究时空的演化，以观察是否只有在黑洞内部才能发生标志着GR崩溃的奇点的发展，并且在黑洞内部，只有进入黑洞的观察者才能看到。这就是所谓的宇宙审查猜想。这个问题将通过继续发展爱因斯坦方程的光锥近似来解决，爱因斯坦方程与已经发展的杨-爱因斯坦米尔斯-希格斯方程相平行。我们的目标是要表明，时空曲率不能在爱因斯坦方程的（真空）解中（逐点）爆炸，直到曲率的积分（贝尔-罗宾逊）能量爆炸。另一个项目将是继续发展研究黑洞碰撞的改进数值方法。最近的工作建立了（保形变换）ADM发展方程在零无穷远的正则性。这个结果及其可能的推广将被应用到实际的数值计算中。宇宙监督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。对于爱因斯坦方程来说，宇宙审查是闵可夫斯基空间场论的全局存在性质的自然类比，因此，在分析闵可夫斯基空间场论中已经证明成功的技术应该对前者有重要贡献，这似乎是非常合理的。此外，在开发用于分析黑洞碰撞的改进数值方法方面的任何重大进展，都可以预期将强烈影响全球正在进行的对这些重要天体物理过程进行数值模拟的努力，并最终影响LIGO和丽莎观测。这些研究的大部分应该引起数学家的兴趣。改进计算碰撞黑洞产生的引力波的数值方法可能会对这种辐射的实验研究产生影响。博士后研究员和研究生有望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.