Book contents
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 Basic tools and concepts
- 3 Minkowski space-time
- 4 de Sitter space-time
- 5 Anti-de Sitter space-time
- 6 Friedmann–Lemaître–Robertson–Walker space-times
- 7 Electrovacuum and related background space-times
- 8 Schwarzschild space–time
- 9 Space-times related to Schwarzschild
- 10 Static axially symmetric space-times
- 11 Rotating black holes
- 12 Taub–NUT space-time
- 13 Stationary, axially symmetric space-times
- 14 Accelerating black holes
- 15 Further solutions for uniformly accelerating particles
- 16 Plebański–Demiański solutions
- 17 Plane and pp-waves
- 18 Kundt solutions
- 19 Robinson–Trautman solutions
- 20 Impulsive waves
- 21 Colliding plane waves
- 22 A final miscellany
- Appendix A 2-spaces of constant curvature
- Appendix B 3-spaces of constant curvature
- References
- Index
22 - A final miscellany
Published online by Cambridge University Press: 04 February 2010
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 Basic tools and concepts
- 3 Minkowski space-time
- 4 de Sitter space-time
- 5 Anti-de Sitter space-time
- 6 Friedmann–Lemaître–Robertson–Walker space-times
- 7 Electrovacuum and related background space-times
- 8 Schwarzschild space–time
- 9 Space-times related to Schwarzschild
- 10 Static axially symmetric space-times
- 11 Rotating black holes
- 12 Taub–NUT space-time
- 13 Stationary, axially symmetric space-times
- 14 Accelerating black holes
- 15 Further solutions for uniformly accelerating particles
- 16 Plebański–Demiański solutions
- 17 Plane and pp-waves
- 18 Kundt solutions
- 19 Robinson–Trautman solutions
- 20 Impulsive waves
- 21 Colliding plane waves
- 22 A final miscellany
- Appendix A 2-spaces of constant curvature
- Appendix B 3-spaces of constant curvature
- References
- Index
Summary
The authors are well aware that this book has only described a very limited number of the known exact solutions of Einstein's equations. We have concentrated on those we believe to be the most basic or that have particularly significant interpretations. However, many other interesting solutions have been analysed and have their own extensive published literatures. Indeed, a number of these are highly relevant – either because they represent some easily comprehended idealised physical situation, or because they demonstrate some important property. In this concluding chapter, we will therefore comment on a number of further space-times that fall into either of these categories.
After commenting on some aspects of the Bianchi and Kantowski—Sachs cosmological models, that have (at least) three spatial isometries corresponding to homogeneity, we will briefly review the main properties of some other space-times that possess two commuting Killing vectors which have not been described in previous chapters. In general, such space-times include those with cylindrical symmetry, stationary and axial symmetry, boost-rotation or boost-translation symmetry, as well as the interaction region of colliding plane waves, and the so-called Gowdy (or vacuum G2) cosmologies. Remarkably, for space-times of all these types, which possess 2-surfaces that are orthogonal to the group orbits, the resulting vacuum or electrovacuum field equations happen to be integrable. In fact, the field equations in all these cases can be written as a combination of the Ernst equations (Ernst, 1968a,b) in their elliptic or hyperbolic forms and subsidiary quadratures forthe remaining metric function.
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- Exact Space-Times in Einstein's General Relativity , pp. 438 - 461Publisher: Cambridge University PressPrint publication year: 2009