ESI Lectures in Mathematics and Physics 2009; 307 pp; softcover Volume: 6 ISBN10: 3037190531 ISBN13: 9783037190531 List Price: US$58 Member Price: US$46.40 Order Code: EMSESILEC/6
Temporarily out of stock. Expected date of availability is September 21, 2016.
 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 FriedmannLemaîtreRobertsonWalker 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 nonisometric 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 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, nonlinear 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
