GEOMETRIC RELATIVITY

GEOMETRIC RELATIVITY

Editorial:
AMS (AMERICAN MATHEMATICAL SOCIETY)
Año de edición:
Materia
Matematicas
ISBN:
978-1-4704-5081-6
Páginas:
361
N. de edición:
1
Idioma:
Inglés
Disponibilidad:
Disponibilidad inmediata

Descuento:

-5%

Antes:

110,00 €

Despues:

104,50 €

Cover1
Title page4
Preface10
Part 1 . Riemannian geometry14

Chapter 1. Scalar curvature16

1.1. Notation and review of Riemannian geometry16
1.2. A survey of scalar curvature results30
Chapter 2. Minimal hypersurfaces36

2.1. Basic definitions and the Gauss-Codazzi equations36
2.2. First and second variation of volume39
2.3. Minimizing hypersurfaces and positive scalar curvature51
2.4. More scalar curvature rigidity theorems67
Chapter 3. The Riemannian positive mass theorem76

3.1. Background76
3.2. Special cases of the positive mass theorem89
3.3. Reduction to Theorem 1.3099
3.4. A few words on Ricci flow117
Chapter 4. The Riemannian Penrose inequality120

4.1. Riemannian apparent horizons120
4.2. Inverse mean curvature flow134
4.3. Bray’s conformal flow155
Chapter 5. Spin geometry172

5.1. Background172
5.2. The Dirac operator179
5.3. Witten’s proof of the positive mass theorem182
5.4. Related results188
Chapter 6. Quasi-local mass194

6.1. Bartnik mass and static metrics194
6.2. Bartnik minimizers200
6.3. Brown-York mass206
6.4. Bartnik data with ??=0212
Part 2 . Initial data sets218

Chapter 7. Introduction to general relativity220

7.1. Spacetime geometry220
7.2. The Einstein field equations227
7.3. The Einstein constraint equations234
7.4. Black holes and Penrose incompleteness241
7.5. Marginally outer trapped surfaces253
7.6. The Penrose inequality262
Chapter 8. The spacetime positive mass theorem268

8.1. Proof for ??<8269
8.2. Spacetime positive mass rigidity288
8.3. Proof for spin manifolds288
Chapter 9. Density theorems for the constraint equations298

9.1. The constraint operator298
9.2. The density theorem for vacuum constraints305
9.3. The density theorem for DEC (Theorem 8.3)308
Appendix A. Some facts about second-order linear elliptic operators314

A.1. Basics314
A.2. Weighted spaces on asymptotically flat manifolds331
A.3. Inverse function theorem and Lagrange multipliers350
Bibliography356
Index372
Back Cover377
Preview Material
Preface
Table of Contents
Index

Many problems in general relativity are essentially geometric in nature, in the sense that they can be understood in terms of Riemannian geometry and partial differential equations. This book is centered around the study of mass in general relativity using the techniques of geometric analysis. Specifically, it provides a comprehensive treatment of the positive mass theorem and closely related results, such as the Penrose inequality, drawing on a variety of tools used in this area of research, including minimal hypersurfaces, conformal geometry, inverse mean curvature flow, conformal flow, spinors and the Dirac operator, marginally outer trapped surfaces, and density theorems. This is the first time these topics have been gathered into a single place and presented with an advanced graduate student audience in mind; several dozen exercises are also included.

The main prerequisite for this book is a working understanding of Riemannian geometry and basic knowledge of elliptic linear partial differential equations, with only minimal prior knowledge of physics required. The second part of the book includes a short crash course on general relativity, which provides background for the study of asymptotically flat initial data sets satisfying the dominant energy condition.

Author
Dan A. Lee: CUNY Graduate Center and Queens College, New York, NY

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