General relativity is a theory proposed by Einstein in 1915 as a unified theory of space, time and gravitation. It is based on and extends Newton's theory of gravitation as well as Newton's equations of motion. It is thus fundamentally rooted in classical mechanics. The theory can be seen as a development of Riemannian geometry, itself an extension of Gauss' intrinsic theory of curved surfaces in Euclidean space. The domain of application of the theory is astronomical systems. One of the mathematical methods analyzed and exploited in the present volume is an extension of Noether's fundamental principle connecting symmetries to conserved quantities. This is involved at a most elementary level in the very definition of the notion of hyperbolicity for an Euler-Lagrange system of partial differential equations. Another method, the study and systematic use of foliations by characteristic (null) hypersurfaces, is in the spirit of Roger Penrose's approach in his incompleteness theorem. The methods have applications beyond general relativity to problems in fluid mechanics and, more generally, to the mechanics and electrodynamics of continuous media. The book is intended for advanced students and researchers seeking an introduction to the methods and applications of general relativity. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.
Author(s): Demetrios Christodoulou
Series: Zurich Lectures in Advanced Mathematics 11
Publisher: European Mathematical Society
Year: 2008
Language: English
Commentary: no
Pages: 159
Cover......Page 1
Series Editors......Page 3
Title......Page 4
Copyright......Page 5
Contents......Page 6
General introduction......Page 8
1 Introduction......Page 12
The Einstein equations......Page 20
Regular ellipticity and regular hyperbolicity......Page 27
Cauchy problem for the Einstein equations: local in time, existence and uniqueness of solutions......Page 30
Decomposition of the Einstein equations......Page 42
Conserved quantities......Page 46
Asymptotic flatness......Page 65
The maximal time function......Page 69
Positivity of the energy......Page 71
Angular momentum......Page 81
Conservation of angular momentum......Page 82
The center of mass integrals......Page 83
Conservation of center of mass integrals......Page 87
Statement of the problem......Page 89
Field theories in a given spacetime......Page 90
Resolution of the first difficulty......Page 114
Resolution of the second difficulty......Page 120
The controlling quantity......Page 135
The continuity argument......Page 138
Estimates for the geometric quantities associated to the maximal foliation {H_t}......Page 142
Estimates for the geometric quantities associated to the null foliation {C_u}......Page 143
Decomposition of a Weyl field with respect to the surfaces S_{t,u}......Page 148
The borderline error integrals......Page 150
Bibliography......Page 154
Index......Page 156
Back Cover......Page 159
