[...]... perfect fluid solutions of embedding class one 37.4.3 Type D perfect fluid solutions of embedding class one 37.4.4 Pure radiation field solutions of embedding class one Exact solutions of embedding class two 37.5.1 The Gauss–Codazzi–Ricci equations 37.5.2 Vacuum solutions of embedding class two 37.5.3 Conformally flat solutions Exact solutions of embedding class p > 2 Part V: Tables 571 571 573 573 573 577 579... 29 Twisting vacuum solutions 29.1 Twisting vacuum solutions – the field equations 29.1.1 The structure of the field equations 29.1.2 The integration of the main equations 29.1.3 The remaining field equations 29.1.4 Coordinate freedom and transformation properties 437 437 437 438 440 436 441 Contents 29.2 Some general classes of solutions 29.2.1 Characterization of the known classes of solutions 2 29.2.2... enormous number of known exact solutions Those who do not work in the field often suppose that the 1.2 The development of the subject 3 intractability of the full Einstein equations means that very few solutions are known In a certain sense this is true: we know relatively few exact solutions for real physical problems In most solutions, for example, there is no complete description of the relation of the field... Robinson–Trautman solutions 28.1 Robinson–Trautman vacuum solutions 28.1.1 The field equations and their solutions 28.1.2 Special cases and explicit solutions 28.2 Robinson–Trautman Einstein–Maxwell fields 28.2.1 Line element and field equations 28.2.2 Solutions of type III, N and O 28.2.3 Solutions of type D 28.2.4 Type II solutions 28.3 Robinson–Trautman pure radiation fields 28.4 Robinson–Trautman solutions. .. Expanding solutions without shear 15.6.5 Solutions with nonvanishing shear Plane-symmetric perfect fluid solutions 15.7.1 Static solutions 15.7.2 Non-static solutions xi 226 226 228 229 230 230 231 232 233 235 237 237 238 238 239 240 243 243 244 16 Spherically-symmetric perfect fluid solutions 16.1 Static solutions 16.1.1 Field equations and first integrals 16.1.2 Solutions 16.2 Non-static solutions 16.2.1... between Petrov types and groups of motions 606 38.3 Tables 609 References 615 Index 690 Preface When, in 1975, two of the authors (D.K and H.S.) proposed to change their field of research back to the subject of exact solutions of Einstein’s field equations, they of course felt it necessary to make a careful study of the papers published in the meantime, so as to avoid duplication of known results A fairly... 29.2.6 Solutions independent of ζ and ζ 29.3 Solutions of type N (Ψ2 = 0 = Ψ3 ) 29.4 Solutions of type III (Ψ2 = 0, Ψ3 = 0) 29.5 Solutions of type D (3Ψ2 Ψ4 = 2Ψ2 , Ψ2 = 0) 3 29.6 Solutions of type II 30 30.1 30.2 30.3 30.4 30.5 30.6 30.7 31 31.1 31.2 31.3 31.4 31.5 31.6 31.7 31.8 xv 442 442 445 446 447 449 450 451 452 452 454 Twisting Einstein–Maxwell and pure radiation fields 455 The structure of the... manifolds The why of embedding The basic formulae governing embedding Some theorems on local isometric embedding 37.3.1 General theorems 37.3.2 Vector and tensor fields and embedding class 37.3.3 Groups of motions and embedding class Exact solutions of embedding class one 37.4.1 The Gauss and Codazzi equations and the possible types of Ωab 37.4.2 Conformally flat perfect fluid solutions of embedding class... classes of algebraically special solutions Some algebraically general solutions Solutions of Petrov type II, D, III or N Petrov type D solutions Conformally flat solutions Algebraically general vacuum solutions with geodesic and non-twisting rays The line element for metrics with κ = σ = 0 = R11 = R14 = R44 , Θ + i ω = 0 27.1 The line element in the case with twisting rays (ω = 0) 27.1.1 The choice of the... solutions (Weyl’s class) 20.3 The class of solutions U = U (ω) (Papapetrou’s class) 20.4 The class of solutions S = S(A) 20.5 The Kerr solution and the Tomimatsu–Sato class 20.6 Other solutions 20.7 Solutions with factor structure 304 304 21 Non-empty stationary axisymmetric solutions 21.1 Einstein–Maxwell fields 21.1.1 Electrostatic and magnetostatic solutions 21.1.2 Type D solutions: A general metric and . information, which would have made the job akin to the proverbial painting of the Forth bridge? How could we decide which topics to include and which to omit? How could we check the calculations, put the. Professor of Theoretical Physics at the Friedrich-Schiller- Universit¨at Jena. He graduated from this university, where he also finished his Ph.D. (1966) and habilitation (1970). His current research. of general relativity. hans stephani gained his Diploma, Ph.D. and Habilitation at the Friedrich- Schiller-Universit¨at Jena. He became Professor of Theoretical Physics in 199 2, before retiring