Contents

xiiContents18.2 The Ricci tensor on Σ 3 27718.3 Conformal transformation of Σ 3 and the field equations 27818.4 Vacuum and Einstein–Maxwell equations for stationaryfields 27918.5 Geodesic eigenrays 28118.6 Static fields 28318.6.1 Definitions 28318.6.2 Vacuum solutions 28418.6.3 Electrostatic and magnetostatic Einstein–Maxwellfields 28418.6.4 Perfect fluid solutions 28618.7 The conformastationary solutions 28718.7.1 Conformastationary vacuum solutions 28718.7.2 Conformastationary Einstein–Maxwell fields 28818.8 Multipole moments 28919Stationary axisymmetric fields: basic conceptsand field equations 29219.1 The Killing vectors 29219.2 Orthogonal surfaces 29319.3 The metric and the projection formalism 29619.4 The field equations for stationary axisymmetric Einstein–Maxwell fields 29819.5 Various forms of the field equations for stationary axisymmetricvacuum fields 29919.6 Field equations for rotating fluids 30220 Stationary axisymmetric vacuum solutions 30420.1 Introduction 30420.2 Static axisymmetric vacuum solutions (Weyl’sclass) 30420.3 The class of solutions U = U(ω) (Papapetrou’s class) 30920.4 The class of solutions S = S(A) 31020.5 The Kerr solution and the Tomimatsu–Sato class 31120.6 Other solutions 31320.7 Solutions with factor structure 31621 Non-empty stationary axisymmetric solutions 31921.1 Einstein–Maxwell fields 31921.1.1 Electrostatic and magnetostatic solutions 31921.1.2 Type D solutions: A general metric and its limits 32221.1.3 The Kerr–Newman solution 325