Contents

Contentsxiii21.1.4 Complexification and the Newman–Janis ‘complextrick’ 32821.1.5 Other solutions 32921.2 Perfect fluid solutions 33021.2.1 Line element and general properties 33021.2.2 The general dust solution 33121.2.3 Rigidly rotating perfect fluid solutions 33321.2.4 Perfect fluid solutions with differential rotation 33722 Groups G 2 I on spacelike orbits: cylindricalsymmetry 34122.1 General remarks 34122.2 Stationary cylindrically-symmetric fields 34222.3 Vacuum fields 35022.4 Einstein–Maxwell and pure radiation fields 35423 Inhomogeneous perfect fluid solutions withsymmetry 35823.1 Solutions with a maximal H 3 on S 3 35923.2 Solutions with a maximal H 3 on T 3 36123.3 Solutions with a G 2 on S 2 36223.3.1 Diagonal metrics 36323.3.2 Non-diagonal solutions with orthogonal transitivity 37223.3.3 Solutions without orthogonal transitivity 37323.4 Solutions with a G 1 or a H 2 37424 Groups on null orbits. Plane waves 37524.1 Introduction 37524.2 Groups G 3 on N 3 37624.3 Groups G 2 on N 2 37724.4 Null Killing vectors (G 1 on N 1 ) 37924.4.1 Non-twisting null Killing vector 38024.4.2 Twisting null Killing vector 38224.5 The plane-fronted gravitational waves with parallel rays(pp-waves) 38325 Collision of plane waves 38725.1 General features of the collision problem 38725.2 The vacuum field equations 38925.3 Vacuum solutions with collinear polarization 39225.4 Vacuum solutions with non-collinear polarization 39425.5 Einstein–Maxwell fields 397