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The Cauchy Problem in General Relativity
Hans Ringström, KTH Royal Institute of Technology, Stockholm, Sweden
A publication of the European Mathematical Society.
cover
ESI Lectures in Mathematics and Physics
2009; 307 pp; softcover
Volume: 6
ISBN-10: 3-03719-053-1
ISBN-13: 978-3-03719-053-1
List Price: US$58
Member Price: US$46.40
Order Code: EMSESILEC/6
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The general theory of relativity is a theory of manifolds equipped with Lorentz metrics and fields which describe the matter content. Einstein's equations equate the Einstein tensor (a curvature quantity associated with the Lorentz metric) with the stress energy tensor (an object constructed using the matter fields). In addition, there are equations describing the evolution of the matter. Using symmetry as a guiding principle, one is naturally led to the Schwarzschild and Friedmann-Lemaître-Robertson-Walker solutions, modelling an isolated system and the entire universe respectively. In a different approach, formulating Einstein's equations as an initial value problem allows a closer study of their solutions.

This book first provides a definition of the concept of initial data and a proof of the correspondence between initial data and development. It turns out that some initial data allow non-isometric maximal developments, complicating the uniqueness issue. The second half of the book is concerned with this and related problems, such as strong cosmic censorship.

The book presents complete proofs of several classical results that play a central role in mathematical relativity but are not easily accessible to those without prior background in the subject. Prerequisites are a good knowledge of basic measure and integration theory as well as the fundamentals of Lorentz geometry. The necessary background from the theory of partial differential equations and Lorentz geometry is included.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

Readership

Graduate students and research mathematicians interested in mathematical physics.

Table of Contents

  • Introduction
  • Outline
Part I. Background from the theory of partial differential equations
  • Functional analysis
  • The Fourier transform
  • Sobolev spaces
  • Sobolev embedding
  • Symmetric hyperbolic systems
  • Linear wave equations
  • Local existence, non-linear wave equations
Part II. Background in geometry, global hyperbolicity and uniqueness
  • Basic Lorentz geometry
  • Characterizations of global hyperbolicity
  • Uniqueness of solutions to linear wave equations
Part III. General relativity
  • The constraint equations
  • Local existence
  • Cauchy stability
  • Existence of a maximal globally hyperbolic development
Part IV. Pathologies, strong cosmic censorship
  • Preliminaries
  • Constant mean curvature
  • Initial data
  • Einstein's vacuum equations
  • Closed universe recollapse
  • Asymptotic behaviour
  • LRS Bianchi class A solutions
  • Existence of extensions
  • Existence of inequivalent extensions
  • Appendices
  • Bibliography
  • Index
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