Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Acknowledgements
- List of Symbols
- 1 Introduction
- Part I Geometric tools
- Part II General relativity and conformal geometry
- Part III Methods of the theory of partial differential equations
- Part IV Applications
- 15 De Sitter-like spacetimes
- 16 Minkowski-like spacetimes
- 17 Anti-de Sitter-like spacetimes
- 18 Characteristic problems for the conformal field equations
- 19 Static solutions
- 20 Spatial infinity
- 21 Perspectives
- References
- Index
21 - Perspectives
from Part IV - Applications
- Frontmatter
- Dedication
- Contents
- Preface
- Acknowledgements
- List of Symbols
- 1 Introduction
- Part I Geometric tools
- Part II General relativity and conformal geometry
- Part III Methods of the theory of partial differential equations
- Part IV Applications
- 15 De Sitter-like spacetimes
- 16 Minkowski-like spacetimes
- 17 Anti-de Sitter-like spacetimes
- 18 Characteristic problems for the conformal field equations
- 19 Static solutions
- 20 Spatial infinity
- 21 Perspectives
- References
- Index
Summary
And it seemed as though in a little while the solution would be found, and then a new and splendid life would begin; and it was clear to both of them that they had still a long, long road before them, and that the most complicated and difficult part was only just beginning.
– A. Chekhov, The lady with the dogConformal notions provide valuable tools for the analysis of global properties of spacetimes. In Part IV of this book it has been shown how a conformal point of view leads to proofs of the global existence and non-linear stability of de Sitterlike spacetimes, of the semiglobal existence and non-linear stability of Minkowskilike spacetimes, and how they provide a systematic procedure for the construction of anti-de Sitter-like spacetimes. Moreover, conformal methods provide a robust framework for the analysis of the gravitational field of isolated systems in a neighbourhood of both null and spatial infinity.
The application of conformal methods in general relativity is a mature area of research with a considerable number of open problems. Several of these have been discussed in various places of this book. Unavoidably, there are other problems and aspects of the subject which, for reasons of space, could not be covered in the main text. This last chapter presents a list of ideas and problems which, in the opinion of the author, may play a role in the future development of the subject.
Stability of cosmological models
The global non-linear stability of the de Sitter spacetime was discussed in Chapter 15. This exact solution can be regarded as a basic cosmological model. The analysis of Chapter 15 can be extended to include a non-vacuum matter content with good conformal properties: for example, the Maxwell, Yang-Mills and conformally coupled scalar field; see Friedrich (1991) and Lübbe and Valiente Kroon (2012). More recently, the ideas behind these proofs have been adapted in Lübbe and Valiente Kroon (2013b) to provide an analysis of the future stability of Friedman-Robertson-Walker cosmological models with a perfect fluid with the equation of state of radiation; see Section 9.4. A natural question is whether conformal methods could be adapted to more general matter models, that is, matter models with a non-vanishing trace.
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- Conformal Methods in General Relativity , pp. 560 - 568Publisher: Cambridge University PressPrint publication year: 2016