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9 - Global Behavior of Solutions to Einstein's Equations

from Part Three - Gravity is Geometry, after all

Summary

Introduction

From a purely analytical perspective, Einstein's equations constitute a formidable PDE system. They mix constraint equations with evolution equations, their manifest character (hyperbolic or not) depends on the choice of coordinates, they are defined and studied on a spacetime manifold which is field-dependent and therefore not fixed, and the system is nonlinear in a serious way. These features make it challenging to study Einstein's equations using the analytical techniques and ideas which have been successfully applied to other nonlinear PDE systems. This is especially true of those analyses concerned with global, evolutionary aspects of solutions of Einstein's equations, which are the focus of interest in this chapter.

During the past thirty years, it has become apparent that the most successful way to meet these challenges and understand the behavior of solutions of Einstein's equations is to recognize the fundamental role played by spacetime geometry in general relativity and exploit some of its structures. Indeed, the Christodoulou–Klainerman proof of the stability of Minkowski spacetime [1] provides a good example of this: It relies strongly on the use of spacetime geometric structures such as null foliations, maximal hypersurfaces, “almost Killing fields”, and the Bel–Robinson tensor, combined with sophisticated use of standard analytical tools such as the control of energy functionals and hyperbolic radiation estimates. The more recent work of Christodoulou and others, which has discovered sufficient conditions for the formation of trapped surfaces and black holes, also relies on a strong alliance between geometric insight and the mastery of analytical technique.

As we saw in Chapters 3 and 4, gravitational effects play an important role both in astrophysics and cosmology. However, the two areas feature two fairly distinct branches of the mathematical analysis of solutions of Einstein's equations: One works with asymptotically flat solutions to Einstein's equations and often focuses on issues related to black holes, while the other works predominantly with spatially closed solutions and is more focused on the nature of cosmological singularities and possible mechanisms for isotropization and long-distance correlation.

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