On the Topology and Future Stability of the Universe
Series: Oxford Mathematical Monographs;
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Product details:
- Publisher OUP Oxford
- Date of Publication 23 May 2013
- ISBN 9780199680290
- Binding Hardback
- No. of pages734 pages
- Size 235x163x44 mm
- Weight 1208 g
- Language English
- Illustrations 27 b/w line drawings, 2 b/w halftones, and 1 colour halftone 0
Categories
Short description:
A general introduction to the initial value problem for Einstein's equations coupled to collisionless matter. The book contains a proof of future stability of models of the universe consistent with the current observational data and a discussion of the restrictions on the possible shapes of the universe imposed by observations.
MoreLong description:
The standard starting point in cosmology is the cosmological principle; the assumption that the universe is spatially homogeneous and isotropic. After imposing this assumption, the only freedom left, as far as the geometry is concerned, is the choice of one out of three permissible spatial geometries, and one scalar function of time. Combining the cosmological principle with an appropriate description of the matter leads to the standard models. It is worth noting that these models yield quite a successful description of our universe.
However, even though the universe may, or may not, be almost spatially homogeneous and isotropic, it is clear that the cosmological principle is not exactly satisfied. This leads to several questions. The most natural one concerns stability: given initial data corresponding to an expanding model of the standard type, do small perturbations give rise to solutions that are similar to the future? Another question concerns the shape of the universe: what are the restrictions if we only assume the universe to appear almost spatially homogeneous and isotropic to every observer?
The main purpose of the book is to address these questions. However, to begin with, it is necessary to develop the general theory of the Cauchy problem for the Einstein-Vlasov equations. In order to to make the results accessible to researchers who are not mathematicians, but who are familiar with general relativity, the book contains an extensive prologue putting the results into a more general context.
This impressive new book is first and foremost an original and thought-provoking contribution to the study of cosmology in research monograph form, in the best tradition of the kind of deep mathematical work which has played a crucial role in the development of the subject.
Table of Contents:
I Prologue
Introduction
The Initial Value Problem
The Topology of the Universe
Notions of Proximity
Observational Support
Concluding Remarks
II Introductory Material
Main Results
Outline, General Theory
Outline, Main Results
References and Outlook
III Background and Basic Constructions
Basic Analysis Estimates
Linear Algebra
Coordinates
IV Function Spaces, Estimates
Function Spaces, Distribution Functions
Function Spaces on Manifolds
Main Weighted Estimate
Concepts of Convergence
V Local Theory
Uniqueness
Local Existence
Stability
VI The Cauchy Problem in General Relativity
The Vlasov Equation
The Initial Value Problem
Existence of an MGHD
Cauchy Stability
VII Spatial Homogeneity
Spatially Homogeneous Metrics
Criteria Ensuring Global Existence
A Positive Non-Degenerate Minimum
Approximating Fluids
VIII Future Global Non-Linear Stability
Background Material
Estimates for the Vlasov Matter
Global Existence
Asymptotics
Proof of the Stability Results
Models with Arbitrary Spatial Topology
IX Appendices
Pathologies
Quotients and Universal Covering Spaces
Spatially Homogeneous and Isotropic Metrics
Auxiliary Computations in Low Regularity
Curvature, Left Invariant Metrics
Comments, Einstein-Boltzmann
