Contents

606 38 The interconnections between the classification schemesFor perfect fluid solutions, the connection between the kinematicalproperties of the four-velocity (see §6.1) and groups of motions was discussede.g. by Ehlers (1961) and Wainwright (1979).The reader who is interested in an introductory survey of the solutionsas classified by some invariant properties should consult (besides the subjectindex and the table of contents):Table 11.1 for metrics with isometries,Tables 11.2–11.4 for metrics with homothetiesTable 12.1 for homogeneous solutions,Tables 13.2–13.4 for hypersurface-homogeneous solutions§§26.1–26.4 for algebraically special solutions,Table 32.1 for Kerr-Schild solutions,Table 37.2 for solutions of embeddding class one,Table 37.3 for solutions of embedding class two.38.2 The connection between Petrov typesand groups of motionsIn Parts II and III of this book, two methods for the invariant classificationof gravitational fields, namely groups of motions and Petrov types, weretreated quite independently. We have seen that many solutions admittingan isometry group are algebraically special and vice versa. Occasionally, ifit was known to us, we mentioned the Petrov type of a solution classifiedaccording to the underlying group structure or referred to the group ofmotions of a solution of a certain Petrov type. In this section and inTables 38.3–38.10 we want to collect the known results on the connectionbetween these two invariant classifications.If one knows the group of motions and asks for the possible Petrovtypes, the following facts impose some restrictions.(i) The existence of an isotropy group I s (see §11.2) implies that theWeyl tensor is degenerate, i.e. the Petrov type is N, D or O. In particular,a group G 3 on non-null orbits V 2 implies type D or O (Theorem 15.1).(ii) The static solutions are of type I, D or O (§18.6.1).(iii) The stationary axisymmetric vacuum solutions cannot be of typeIII (see §19.5). For all admissible Petrov types, the subclasses admittinga group G r ,r≥ 3, were determined by Collinson and Dodd (1971). Thoseof type II are given by (20.32).The hypersurface-homogeneous (Chapter 13) algebraically specialEinstein spaces (R ab =Λg ab ) were determined by Siklos (1981). Apartfrom special plane waves (§24.5) and the Λ-term solutions of (12.8),(13.48), (13.65) and (13.67), Siklos obtained the Petrov type III