Book contents
- Frontmatter
- Dedication
- Contents
- Foreword
- Preface to 50th Anniversary Edition
- Preface
- 1 The role of gravity
- 2 Differential geometry
- 3 General relativity
- 4 The physical significance of curvature
- 5 Exact solutions
- 6 Causal structure
- 7 The Cauchy problem in General Relativity
- 8 Space-time singularities
- 9 Gravitational collapse and black holes
- 10 The initial singularity in the universe
- Appendix A: Translation of an essay by Peter Simon Laplace
- Appendix B: Spherically symmetric solutions and Birkhoff’s theorem
- References
- Notation
- Index
7 - The Cauchy problem in General Relativity
Published online by Cambridge University Press: 17 February 2023
- Frontmatter
- Dedication
- Contents
- Foreword
- Preface to 50th Anniversary Edition
- Preface
- 1 The role of gravity
- 2 Differential geometry
- 3 General relativity
- 4 The physical significance of curvature
- 5 Exact solutions
- 6 Causal structure
- 7 The Cauchy problem in General Relativity
- 8 Space-time singularities
- 9 Gravitational collapse and black holes
- 10 The initial singularity in the universe
- Appendix A: Translation of an essay by Peter Simon Laplace
- Appendix B: Spherically symmetric solutions and Birkhoff’s theorem
- References
- Notation
- Index
Summary
In this chapter we give an outline of the Cauchy problem in general relativity and show that, given certain data on a space like three-surface there is a unique maximal future Cauchy development D+() and that the metric on a subset of D+() depends only on the initial data on J–() ∩. We also show that this dependence is continuous if has a compact closure in D+(). This discussion is included here because of its intrinsic interest, its use of some of the results of the previous chapter, and its demonstration that the Einstein field equations do indeed satisfy postulate (a) of §3.2 that signals can only be sent between points that can be joined by a non-spacelike curve.
In §7.1, we discuss the various difficulties and give a precise formulation of the problem. In §7.2 we introduce a global background metric ĝ to generalize the relation which holds between the Ricci tensor and the metric in each coordinate patch to a single relation which holds over the whole manifold. We impose four gauge conditions on the covariant derivatives of the physical metric g with respect to the background metric ĝ.
- Type
- Chapter
- Information
- The Large Scale Structure of Space-Time50th Anniversary Edition, pp. 226 - 255Publisher: Cambridge University PressPrint publication year: 2023