U.S.-France Cooperative Research: Global Studies of Einstein Spacetimes

该奖项将支持耶鲁大学的文森特·蒙克里夫博士和法国巴黎六世大学的伊冯·乔奎特-布鲁哈特博士在数学物理方面的合作。该项目的主要目标是在爱因斯坦方程的大类宇宙学解的整体行为方面获得新的结果。研究人员的目标是通过提供广义相对论整体存在定理的类比来刻画这种时空的最大柯西发展。他们还将研究奇异解的边界行为，并希望推导出展开方法，使人们能够对时空奇点本身“放上显微镜”，并以几何上自然的方式对它们进行分类。这些目标一直是广义相对主义者长期面临的挑战，但现在数学方法正在变得可行，这些方法可能会在这些基本问题上取得真正的进展。具有高度对称性的时空的早期整体存在性结果现在可能扩展到更大的解族，而新的技术，如“多尺度”方法的应用，可能为时空奇点的分析研究提供所需的工具。这项研究的基本方法将是努力扩大现有数学方法的适用范围，例如使用光锥和更高阶能量估计，以适用于更大和更不对称的时空系列，同时，开发新的方法，比如上面提到的扩展技术。这项研究将受益于美国和法国投资者在数学广义相对论领域的广泛专业知识。

This award will support collaboration in mathematical physics between Dr. Vincent Moncrief, Yale University and Dr. Yvonne Choquet-Bruhat, University of Paris VI, France. The main objective of the project is to obtain new results on the global behavior of large classes of cosmological solutions of Einstein's equations. The investigators aim to characterize the maximal Cauchy developments of such spacetimes by providing the analogue of global existence theorems for general relativity. They will also study the boundary behavior of singular solutions and hope to derive expansion methods which will allow one to "put a microscope" on the spacetime singularities themselves and to classify them in a geometrically natural way. These aims have been longstanding challenges to general relativists, but now mathematical methods are becoming available which may allow genuine progress on such fundamental problems. Earlier global existence results for spacetimes with high symmetry may now be extendible to very much larger families of solutions and new techniques, such as the application of the method of "multiple scales", may provide the needed tools for the analytical study of spacetime singularities. The basic approach of this research will be to try to extend the range of applicability of the established mathematical methods, such as the use of light-cone and higher order energy estimates, to larger and less symmetric families of spacetimes, and at the same time, to develop new methods, such as the expansion techniques alluded to above. This research will benefit from the extensive expertise of the US and French investi- gators in the area of mathematical general relativity.