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
8 - Space-time singularities
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 §8.1, we discuss the problem of defining singularities in spacetime. We adopt b-incompleteness as an indication that singular points have been cut out of spacetime, and characterize two ways in which b-incompleteness can be associated with some form of curvature singularity. In §8.2, four theorems are given to prove the existence of incompleteness under a wide variety of situations. In §8.3 we give Schmidt’s construction of the b-boundary which represents the singular points of spacetime. In §8.4 we prove that the singularities predicted by at least one of the the theorems cannot be just a discontinuity in the curvature tensor. We also show that there is not only one incomplete geodesic, but a three-parameter family of them. In §8.5 we discuss the situation in which the incomplete curves are totally or partially imprisoned in a compact region of spacetime, shown to be related to non-Hausdorff behaviour of the b-boundary. We show that in a generic spacetime, an observer travelling on one of these incomplete curves would experience infinite curvature forces. We also show that the kind of behaviour which occurs in Taub–NUT space cannot happen if there is some matter present.
- Type
- Chapter
- Information
- The Large Scale Structure of Space-Time50th Anniversary Edition, pp. 256 - 298Publisher: Cambridge University PressPrint publication year: 2023