Book contents
- Frontmatter
- Contents
- Preface
- List of tables
- Notation
- 1 Introduction
- Part I General methods
- Part II Solutions with groups of motions
- 11 Classification of solutions with isometries or homotheties
- 12 Homogeneous space-times
- 13 Hypersurface-homogeneous space-times
- 14 Spatially-homogeneous perfect fluid cosmologies
- 15 Groups G3 on non-null orbits V2. Spherical and plane symmetry
- 16 Spherically-symmetric perfect fluid solutions
- 17 Groups G2 and G1 on non-null orbits
- 18 Stationary gravitational fields
- 19 Stationary axisymmetric fields: basic concepts and field equations
- 20 Stationary axisymmetric vacuum solutions
- 21 Non-empty stationary axisymmetric solutions
- 22 Groups G2I on spacelike orbits: cylindrical symmetry
- 23 Inhomogeneous perfect fluid solutions with symmetry
- 24 Groups on null orbits. Plane waves
- 25 Collision of plane waves
- Part III Algebraically special solutions
- Part IV Special methods
- Part V Tables
- References
- Index
12 - Homogeneous space-times
from Part II - Solutions with groups of motions
Published online by Cambridge University Press: 10 November 2009
- Frontmatter
- Contents
- Preface
- List of tables
- Notation
- 1 Introduction
- Part I General methods
- Part II Solutions with groups of motions
- 11 Classification of solutions with isometries or homotheties
- 12 Homogeneous space-times
- 13 Hypersurface-homogeneous space-times
- 14 Spatially-homogeneous perfect fluid cosmologies
- 15 Groups G3 on non-null orbits V2. Spherical and plane symmetry
- 16 Spherically-symmetric perfect fluid solutions
- 17 Groups G2 and G1 on non-null orbits
- 18 Stationary gravitational fields
- 19 Stationary axisymmetric fields: basic concepts and field equations
- 20 Stationary axisymmetric vacuum solutions
- 21 Non-empty stationary axisymmetric solutions
- 22 Groups G2I on spacelike orbits: cylindrical symmetry
- 23 Inhomogeneous perfect fluid solutions with symmetry
- 24 Groups on null orbits. Plane waves
- 25 Collision of plane waves
- Part III Algebraically special solutions
- Part IV Special methods
- Part V Tables
- References
- Index
Summary
The possible metrics
A homogeneous space-time is one which admits a transitive group of motions. It is quite easy to write down all possible metrics for the case where the group is or contains a simply-transitive G4; see §8.6 and below. Difficulties may arise when there is a multiply-transitive group Gr, r > 4, not containing a simply-transitive subgroup, and we shall consider such possibilities first. In such space-times, there is an isotropy group at each point. From the remarks in §11.2 we see that there are only a limited number of cases to consider, and we take each possible isotropy group in turn.
For Gr, r ≥ 8, we have only the metrics (8.33) with constant curvature admitting an I6 and a G10.
If the space-time admits a G6 or G7, and its isotropy group contains the two-parameter group of null rotations (3.15), but its metric is not of constant curvature, then it is either of Petrov type N, in which case we can find a complex null tetrad such that (4.10) holds, or it is conformally flat, with a pure radiation energy-momentum tensor, and we can choose a null tetrad such that (5.8) holds with Φ2 = 1. In either case the tetrad is fixed up to null rotations (together with a spatial rotation in the latter case).
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- Exact Solutions of Einstein's Field Equations , pp. 171 - 182Publisher: Cambridge University PressPrint publication year: 2003