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This book offers a systematic exposition of conformal methods and how they can be used to study the global properties of solutions to the equations of Einstein's theory of gravity. It shows that combining these ideas with differential geometry can elucidate the existence and stability of the basic solutions of the theory. Introducing the differential geometric, spinorial and PDE background required to gain a deep understanding of conformal methods, this text provides an accessible account of key results in mathematical relativity over the last thirty years, including the stability of de Sitter and Minkowski spacetimes. For graduate students and researchers, this self-contained account includes useful visual models to help the reader grasp abstract concepts and a list of further reading, making this an ideal reference companion on the topic. This title, first published in 2016, has been reissued as an Open Access publication on Cambridge Core.
List of contents
List of symbols; Preface; 1. Introduction; Part I. Geometric Tools: 2. Differential geometry; 3. Spacetime spinors; 4. Space spinors; 5. Conformal geometry; Part II. General Relativity and Conformal Geometry: 6. Conformal extensions of exact solutions; 7. Asymptotic simplicity; 8. The conformal Einstein field equations; 9. Matter models; 10. Asymptotics; Part III. Methods of PDE Theory: 11. The conformal constraint equations; 12. Methods of the theory of hyperbolic differential equations; 13. Hyperbolic reductions; 14. Causality and the Cauchy problem in General Relativity; 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.
