Vol. 3 (2000) > lrr-2000-1

doi: 10.12942/lrr-2000-1
Living Rev. Relativity 3 (2000), 1

Local and Global Existence Theorems 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 the literature on existence theorems for the Einstein equations which also draws attention to open problems in the field. The local in time Cauchy problem, which is relatively well understood, is treated first. Next global results for solutions with symmetry are discussed. A selection of results from Newtonian theory and special relativity which offer useful comparisons is presented. This is followed by a survey of global results in the case of small data and results on constructing spacetimes with given singularity structure. The article ends with some miscellaneous topics connected with the main theme.

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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,
"Local and Global Existence Theorems for the Einstein Equations",
Living Rev. Relativity 3,  (2000),  1. URL (cited on <date>):
http://www.livingreviews.org/lrr-2000-1

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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