In the theory of black holes, an important place is assigned to the problem of their interior: understanding it as the matter that generates the external gravitational field. Commonly, the matter is modeled through a perfect fluid. It is worth recalling that in (3 + 1) gravity the interior solution to the Schwarzschild black hole is modeled by an interior Schwarzschild perfect fluid solution with constant energy density, but the question still remains open as to what is the interior solution for the rotating Kerr black hole? In (2+1) gravity one may ask the same question with respect to the rotating BTZ black hole. As we shall see in this chapter, there is an interior solution to the BTZ black hole, modeled by a perfect fluid with constant energy density. Moreover, we succeeded in deriving perfect fluid solutions, such that their fluid velocity possesses differential rotation. On the other hand, rigidly rotating exact perfect fluid solutions derived by Rooman and Spindel (1998) and Lubo et al. (1999) are also reported. Those papers deal with stationary circularly symmetric gravitational sources of the perfect fluid type, with a cosmological constant with focus on restrictions on the physical parameters of the solutions due to the matching conditions between the interior and exterior geometries. In particular, it is established there that finite sources and absence of closed timelike curves privilege negative values of the cosmological constant. Moreover, for stationary configurations, the field equations for constant energy densities have been explicitly solved; if, additionally, the pressure vanishes, interior Gödel-like stars arise.

It is worth pointing out that the literature on stationary perfect fluid solutions of (2+1) gravity is rather scarce; among it one may cite Rooman and Spindel (1998); Lubo et al. (1999); García (2004); Cataldo (2004); Gürses (1994); Obukhov (2003). From Section 8.1 to Section 8.2.5 I follow the presentation given in García (2004), while for the remaining two sections I follow Rooman and Spindel (1998), and Lubo et al. (1999).

I have been working on this book project over a period of many years in order to create a concise but comprehensive account of exact solutions in the threedimensional Einstein theory of gravity. This theory, although tangentially related to the real gravity world, is a good model from which to extract some relevant conclusions about that world, taking advantage of simplifications due to the reduction in dimensions. Of course, after fifty years of existence, (2 + 1) gravity has been approached from many perspectives, but, to my mind, there is still no satisfactory account of solutions of physical interest. I therefore wrote the present book on exact solutions in (2 + 1) gravity following closely the pattern of the classic book, Exact solutions in Einstein's field equations with the aim of bringing to each class of solutions presented its theoretical framework, its general physical and geometrical characteristics, and references to the researchers who discovered or studied them.

I am greatly indebted to my late friends and collaborators J.F. Plebański, N. R. Sibgatullin and S. del Campo for stimulating talks. Many thanks are due to my collaborators E. Ayón-Beato, C. Campuzano, M. Cataldo, G. Gutierrez, F. Hehl, A. Macías, V. Manko, N. Mitskievich, and C. Terrero for many enlightening talks, and advice on various problems I faced when dealing with this task. This work could not have come about without a fruitful sabbatical stay at the Physics Department of the California University at Davis under the sponsorship of Professor S. Carlip, to whom I am indebted, and to UC MEXUS-CONACYT for financial support. I cordially thank my friends, particularly Y. Gurievich and M. Lopez, and colleagues from the Departamento de Física, Centro de Investigación y de Estudios Avanzados del I.P.N., for their encouragement. Many thanks to my assistant E.Vargas-Díaz for her daily support and typographical help in the final preparation of the manuscript of this book. The support of the Consejo Nacional de Ciencia y Tecnología (CONACYT), México, through various grants (at present: Grant CONACyT 178346) is acknowledged.

I really do appreciate the patience and support of my daughters, Ana Alicia and Alexandra Sofia, and sons, Sergey, Albert, Alberto-Tiko, and Alex, over all these years of writing this book.

Please do not hesitate to let me know about inconsistencies, unevenness, omissions or errors to be found in this text. Constructive comments are welcome: [email protected].

In (3+1) gravity it is well known that the vacuum plus a cosmological constant Λ, i.e., the (anti) de Sitter–Kottler (1918) solution, is a regular non-asymptotically flat solution (the scalar curvature is equal to 4Λ and all the invariants of the conformal Weyl tensor are zero.) On the other hand, Einstein–Maxwell electrovacuum asymptotically flat metrics endowed with timelike and spacelike symmetries do not allow for the existence of regular black hole solutions. In order to be able to derive regular (black hole) solutions one has to enlarge the class of electrodynamics to nonlinear ones; as an example, the regular solution by Ayón–Beato and García (1998), which is a solution to gravitational–nonlinear electromagnetic fields; the first examples belonging to this class are the Borde (1997) model and the Ayón–García regular charged static black hole.

In (2+1) gravity, in the vacuum case, all solutions are locally Minkowski (the Riemann tensor is zero); the extension to the vacuum plus cosmological constant allows for the existence of the static and the rotating anti-de Sitter regular black holes; see Bañados et al. (1992). The static (2+1)-charged black hole with cosmological constant is singular (when radial coordinate goes to zero the curvature and the Ricci square invariants blow up). Similarly, as in (3+1) gravity, one may search for regular solutions in (2+1) gravity incorporating nonlinear electromagnetic fields to which one imposes the weak energy conditions in order to have physically plausible matter-field distributions. One may look for regular solutions with nonlinear electromagnetic fields of the Born–Infeld type (Born and Infeld, 1934); Salazar et al. (1987); Salazar et. al (1984); Gibbons and Rasheed (1995); Fradkin and Tseytlin (1985); Deser and Gibbons (1998), and/or electrodynamics of wider spectra (Cataldo and García, 1999, 2000; Cataldo et al., 2000).

The Born–Infeld electrodynamics (Born and Infeld, 1934) is free from certain singularities appearing in the classical Maxwell electromagnetic field theory. Nonlinear electromagnetic Lagrangians, in particular the Born–Infeld Lagrangian, arise in open string theory (the low-energy effective action for an electromagnetic field is precisely the Born–Infeld action; see Fradkin and Tseytlin, 1985); string theory has emerged as the most promising candidate for the consistent quantization of gravity. In particular, the open string theory has Born–Infeld coupled vector fields, but it is not clear that this remains the case after compactification to a 3D space with negative cosmological constant Λ.

In this chapter a systematic derivation of the Cotton tensor is presented following García et al. (2004). In 3D spaces the Cotton tensor is prominent as the substitute for the Weyl tensor. It is conformally invariant and its vanishing is equivalent to conformal flatness. However, the Cotton tensor arises in the context of the Bianchi identities and is present in any dimension n. In this text, its irreducible decomposition is performed and the number of independent components as n(n2−4)/3 is determined. Subsequently, its characteristic properties are exhibited. An algebraic classification of the Cotton tensor in three dimensions is accomplished. This classification is used throughout this text; for each class of the derived here solutions the Cotton tensor is evaluated and classified.

The nonlinear coupling of gravity to matter in general relativity presents difficult technical problems in attempts to understand the gravitational interaction of elementary particles and strings or to investigate details of the gravitational collapse; see Witten (1991). Progress in the former area has come mainly from treating quantum fields as propagating on fixed background geometries: Polchinski (1998), whereas much of the progress in the latter has come from detailed numerical works: Choptuik (1993); Abrahams and Evans (1993); Gundlach (1999).

Exact solutions of the relevant matter–gravity equations can play an important role by shedding light on questions of interest in both general relativity and string theory. One is often interested in certain classes of solutions with specified asymptotic properties; the most common of them are the asymptotically flat spacetimes. Recent work in string theory has, via the AdS/CFT conjecture, highlighted the importance of the asymptotically AdS spacetimes, see Maldacena (1998). The AdS/CFT correspondence relates a quantum field theory in d dimensions to a theory in d + 1 dimensions that includes gravity Gubser et al. (1998), and Witten (1998). This is the motivation for looking at the conformally flat spaces and at the spaces of constant curvature. For this reason we decided to review the subject and to collect some old and new results that are nowadays important in the context of anti-de Sitter spacetimes and to present them in modern language. These results seem presently not to be too well known in the community.