In this part of the book we deal with exact solutions to the Einstein topologically massive gravity equations. However, since the material to be included only represents twenty per cent of the whole book's subject matter, we prefer to present this content in the form of chapters devoted to a concise but, we hope, complete (in the range of the possibilities) exposition of the exact solutions in topologically massive gravity (TMG) in three dimensions in the case of vacuum in the presence of a cosmological constant Λ of both signs. Thus, this chapter has an introductory character, while the next three chapters deal with very specific families of the existing Petrov-type Cotton solutions in TMG.

The extension of the 3D Einstein gravity to other field theories to provide them with certain degrees of freedom (a massive spin 2 graviton) was proposed more than thirty-five years ago by Deser, Jackiw and Templeton; see Deser et al. (1982a): “Three-dimensional massive gauge theories,” which is known as the TMG. It includes a Chern–Simons term constructed from connections (Cotton tensor) with broken parity invariance; see also its extended version, with a detailed analysis, in Deser et al. (1982b). A cosmological constant was introduced in these three-dimensional theories by Deser (1984). The sign in front of the curvature scalar has been chosen opposite to the standard one of the Einstein gravity to yield, in the limit of the linearized theory, to the existence of a spin 2 graviton with positive energy. A modern treatment of these aspects in cosmological massive gravity appeared recently in Carlip et al. (2009).

As far as the determination of exact solutions to vacuum equations with a cosmological constant in TMG is concerned, through this long period, various classes of solutions have been reported apart from the trivial Minkowski flat spacetime, the de Sitter cosmology, and the BTZ black hole – conformally flat (zero Cotton tensor) solutions – for any value of the coupling mass parameter μ.