Global Studies of Einstein Spacetimes

Yale University, New Haven CT

Investigators

Abstract

One of the main aims of this project is to continue the development of light-cone estimates for the Einstein equations that parallel those that have already been developed and successfully used to prove a global existence theorem for the Yang-Mills-Higgs equations. Though strict global existence is not valid for general relativity the Penrose cosmic censorship conjecture provides its natural, gravitational analogue and the resolution of this conjecture is considered to be one of the main open mathematical problems in classical general relativity. Recent analysis has succeeded to derive an explicit, exact integral formula for the curvature of a vacuum spacetime, at an arbitrary point. This integral formula should also admit applications to other fundamental, open mathematical problems such as the rigorous proof of black hole stability. An additional, principal aim of this project is to continue work on the development of improved numerical methods for the analysis of gravitational waves. A primary goal is to determine how best to integrate this discovery with standard, unconstrained, numerical evolution techniques. A final goal of this project is to adapt recently developed "Euclidean signature semi-classical" approximation methods to the study of the Wheeler-DeWitt equation of formal, quantum general relativity. Much of the research involved in this project is of interest to mathematicians working in geometry and partial differential equations. Techniques developed for studying, numerically, the propagation of waves that literally extend all the way to infinity could well prove to be useful in other areas of science and engineering. It is anticipated that these same methods, when applied to the study of gravitational radiation, will have a significant influence on the ongoing worldwide efforts to model, numerically, interesting sources of such waves and thereby, ultimately, to impact the experimental search efforts. The proposal to further develop "Euclidean-signature semi-classical" methods for the study of quantum field theories, especially gauge theories, should have far-reaching implications for the understanding of such systems. It is expected that enlisting the contributions of students and younger researchers in these endeavors should have significant implications for the development of human resources in the mathematical and physical sciences and this is an additional, principal aim of the project.