In previous chapters we presented the key notions associated with stationary black-hole spacetimes, as well as the minimal set of metrics needed to illustrate the basic features of the world of black holes. In this chapter we present some further black holes, selected because of their physical and mathematical interest. We start, in Section 5.1, with the Kerr–de Sitter/anti-de Sitter metrics, the cosmological counterparts of the Kerr metrics. Section 5.2 contains a description of the Kerr–Newman–de Sitter/anti-de Sitter metrics, which are the charged relatives of the metrics presented in Section 5.1. In Section 5.3 we analyse in detail the global structure of the Emparan–Reall ‘black rings’: these are five-dimensional black-hole spacetimes with R × S
1 × S
2-horizon topology. The Rasheed metrics of Section 5.4 provide an example of black holes arising in Kaluza–Klein theories. The Birmingham family of metrics, presented in Section 5.5, forms the most general class known of explicit static vacuum metrics with cosmological constant in all dimensions, with a wide range of horizon topologies.
de Sitter entropy from a lower dimensional black hole