Vol. 8 (2005) > lrr-2005-6

doi: 10.12942/lrr-2005-6
Living Rev. Relativity 8 (2005), 6

Theorems on Existence and Global Dynamics for the Einstein Equations

1 Max-Planck-Institut für Gravitationsphysik, Am Mühlenberg 1, D-14424 Golm, Germany

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Article Abstract

This article is a guide to theorems on existence and global dynamics of solutions of the Einstein equations. It draws attention to open questions in the field. The local-in-time Cauchy problem, which is relatively well understood, is surveyed. Global results for solutions with various types of symmetry are discussed. A selection of results from Newtonian theory and special relativity that offer useful comparisons is presented. Treatments of global results in the case of small data and results on constructing spacetimes with prescribed singularity structure or late-time asymptotics are given. A conjectural picture of the asymptotic behaviour of general cosmological solutions of the Einstein equations is built up. Some miscellaneous topics connected with the main theme are collected in a separate section.

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Article Citation

Since a Living Reviews in Relativity article may evolve over time, please cite the access <date>, which uniquely identifies the version of the article you are referring to:

Alan D. Rendall,
"Theorems on Existence and Global Dynamics for the Einstein Equations",
Living Rev. Relativity 8,  (2005),  6. URL (cited on <date>):
http://www.livingreviews.org/lrr-2005-6

Article History

ORIGINAL http://www.livingreviews.org/lrr-1998-4
Title Local and Global Existence Theorems for the Einstein Equations
Author Alan D. Rendall
Date accepted 15 January 1998, published 26 January 1998
UPDATE http://www.livingreviews.org/lrr-2000-1
Title Local and Global Existence Theorems for the Einstein Equations
Author Alan D. Rendall
Date accepted 4 January 2000, published 7 January 2000
Changes Article revision that included substantial additions to various sections, the number of references nearly doubled
UPDATE http://www.livingreviews.org/lrr-2002-6
Title Theorems on Existence and Global Dynamics for the Einstein Equations
Author Alan D. Rendall
Date accepted 2 July 2002, published 27 September 2002
Changes Article revision. New sections (7 and 8) and subsections (2.4, 4.3 and 5.4) have been added, almost all sections have been updated. The number of references grew again considerably
UPDATE http://www.livingreviews.org/lrr-2005-6
Title Theorems on Existence and Global Dynamics for the Einstein Equations
Author Alan D. Rendall
Date accepted 28 September 2005, published 18 October 2005
Changes 1. All sections have been updated and new references added with the result that the total number of references has increased by 40 per cent in comparison with the previous version
2. Section 3.4 (weak null singularities and Price's law) is new
3. Section 3.6 (spatially compact solutions) includes a new discussion of Gowdy spacetimes
4. Section 6.3 (asymptotics for a phase of accelerated expansion) is new
5. The old section 7.3 has been much expanded to constitute the new sections 7.3-7.6.
6. Section 8.4 (cosmic censorship in Gowdy spacetimes) is new.
7. Section 9.6 (the geodesic hypothesis) is new
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