A number of the previous chapters have described important black hole space-times that are of algebraic type D – specifically Chapters 8, 9, 11, 12 and 14. In fact these are all members of a larger family of solutions that can be expressed in a common form. The purpose of this chapter is to present this wider class, particularly showing the relation between these and related space-times, and to indicate their further generalisations.

A general family of type D space-times with an aligned non-null electromagnetic field and a possibly non-zero cosmological constant can be represented by a metric that was given originally by Debever (1971), and in a more convenient form by Plebański and Demiański (1976). These solutions are characterised by two related quartic functions, each of a single coordinate, whose coefficients are determined by seven arbitrary parameters which include Λ and both electric and magnetic charges. Together with cases that can be derived from it by certain transformations and limiting procedures, this gives the complete family of such solutions.

Non-accelerating solutions of this class were obtained by Carter (1968b). For the vacuum case with no cosmological constant, they include all the particular solutions identified by Kinnersley (1969a). Metrics with an expanding repeated principal null congruence were analysed further by Debever (1969) and Weir and Kerr (1977), where the relations between the different forms of the line element can be deduced. They have also been studied by Debever and Kamran (1980) and Ishikawa and Miyashita (1982). The most general metric form which covers both expanding and non-expanding cases was given by Debever, Kamran and McLenaghan (1984) and García D. (1984).