Contents

Contentsxi15 Groups G 3 on non-null orbits V 2 . Sphericaland plane symmetry 22615.1 Metric, Killing vectors, and Ricci tensor 22615.2 Some implications of the existence of an isotropygroup I 1 22815.3 Spherical and plane symmetry 22915.4 Vacuum, Einstein–Maxwell and pure radiation fields 23015.4.1 Timelike orbits 23015.4.2 Spacelike orbits 23115.4.3 Generalized Birkhoff theorem 23215.4.4 Spherically- and plane-symmetric fields 23315.5 Dust solutions 23515.6 Perfect fluid solutions with plane, spherical orpseudospherical symmetry 23715.6.1 Some basic properties 23715.6.2 Static solutions 23815.6.3 Solutions without shear and expansion 23815.6.4 Expanding solutions without shear 23915.6.5 Solutions with nonvanishing shear 24015.7 Plane-symmetric perfect fluid solutions 24315.7.1 Static solutions 24315.7.2 Non-static solutions 24416 Spherically-symmetric perfect fluid solutions 24716.1 Static solutions 24716.1.1 Field equations and first integrals 24716.1.2 Solutions 25016.2 Non-static solutions 25116.2.1 The basic equations 25116.2.2 Expanding solutions without shear 25316.2.3 Solutions with non-vanishing shear 26017 Groups G 2 and G 1 on non-null orbits 26417.1 Groups G 2 on non-null orbits 26417.1.1 Subdivisions of the groups G 2 26417.1.2 Groups G 2 I on non-null orbits 26517.1.3 G 2 II on non-null orbits 26717.2 Boost-rotation-symmetric space-times 26817.3 Group G 1 on non-null orbits 27118 Stationary gravitational fields 27518.1 The projection formalism 275