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9 - Global Behavior of Solutions to Einstein's Equations
- from Part Three - Gravity is Geometry, after all
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- By James Isenberg, University of Oregon, Stefanos Aretakis, Institute for Advanced Study and Princeton University, Igor Rodnianski, Massachusetts Institute of Technology and Princeton University, Vincent Moncrief, Yale University
- Edited by Abhay Ashtekar, Pennsylvania State University, Beverly K. Berger, James Isenberg, University of Oregon, Malcolm MacCallum, University of Bristol
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- Book:
- General Relativity and Gravitation
- Published online:
- 05 June 2015
- Print publication:
- 01 June 2015, pp 449-498
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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.
A1 - Exact solutions and exact properties of Einstein equations
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- By Vincent Moncrief, Yale University
- Edited by Neil Ashby, David F. Bartlett, Walter Wyss
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- Book:
- General Relativity and Gravitation, 1989
- Published online:
- 05 March 2012
- Print publication:
- 26 October 1990, pp 85-92
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Summary
The study of exact solutions and exact properties of Einstein's equations is a rather broad mathematical subfield of general relativity. Of the roughly 80 abstracts submitted to the symposium devoted to this topic, time limitations permitted only a small fraction to be presented orally. Table 1 lists the papers given at the two sessions of this symposium. The 16 presented papers fall roughly within the categories of “exact solutions,” “gravitational energy,” “mathematical results” and “symmetry properties of Einstein's equations” and are briefly discussed under those headings in the following. Since most of the oral presentations described extremely recent research results, they did not, for the most part, include references to published papers concerning these results. For this reason the attached reference list is extremely sketchy.
Exact solutions
Virtually all of the known exact solutions of Einstein's equations involve some significant element of idealization. One usually imposes a stringent geometrical symmetry upon the solutions to be considered and, in the non-vacuum case, simplifying assumptions upon the matter sources to be included. Goenner and Sippel discussed several classes of exact solutions which, though highly idealized in the geometrical sense (being in fact pp waves), were nevertheless more realistic in their material content. The sources included both Maxwell fields and a viscous, heat-conducting plasma subject to certain natural energy and entropy inequalities. Several classes of solutions were discussed which represented the generation of a gravitational wave by an electromagnetic wave and a temperature wave propagating in the viscous gas.