Topological black holes in the dimensionally continued gravity

We investigate the topological black holes in a special class of Lovelock gravity. In the odd dimensions, the action is the Chern-Simons form for the anti-de Sitter group. In the even dimensions, it is the Euler density constructed with the Lorentz part of the anti-de Sitter curvature tensor. The Lovelock coefficients are reduced to two independent parameters: cosmological constant and gravitational constant. The event horizons of these topological black holes may have constant positive, zero or negative curvature. Their thermodynamics is analyzed and electrically charged topological black holes are also considered. We emphasize the differences because of the different curvatures of event horizons. As a comparison, we also discuss the topological black holes in the higher dimensional Einstein-Maxwell theory with a negative cosmological constant. PACS numbers: 04.20.Jb, 04.20.Gz, 97.60.Lf Typeset using REVTEX 1 I. INSTRUCTION Over the past few years there has been a lot of interest in black holes in the anti-de Sitter spacetimes. This study initiated from the discovery of Bañados-Teitelboim-Zanelli (BTZ) black holes [1], which are exact solutions in the three-dimensional Einstein gravity with a negative cosmological constant, and are locally equivalent to a three-dimensional anti-de Sitter space. That is, the BTZ black holes can be constructed by identifying some discrete points along a boost Killing vector in the three-dimensional anti-de Sitter space [2]. Using such kind of identification, the so-called constant curvature black holes can also be constructed in the four-dimensional [3] as well as higher dimensional [4] anti-de Sitter spacetimes. The Euclidean manifold topologies of these black holes are R × S, where S is the topology of event horizons, in contrast to the usual topology of black holes R × S. Because of the unusually asymptotic behavior of these constant curvature black holes, however, identifying the globally conserved quantities seems difficult (For a quasilocal formulation see [5]). On the other hand, except for the Kerr-Newmann-anti-de Sitter black hole, whose event horizon has the topology S, in the four-dimensional Einstein-Maxwell theory with a negative cosmological constant, it has been found recently that there exist black hole solutions whose event horizons may have zero or negative constant curvature and their topologies are no longer the two-sphere S. Because of the different topological structures of even horizons, properties of these black holes are quite different from those of usually spherical topology black holes. These black holes have been studies extensively in many aspects such as exact solutions [6–10], thermodynamics [11,12], pair production [13], gravitational collapse [14,15], and others [16–21]. So far, most of the works have been limited in the Einstein gravitational theory. Most recently, Klemm [10] has found topological black hole solutions in theWeyl conformal gravity. In a previous paper, we have investigated the topological black holes [22] in a class of dilaton gravity with a Liouville-type dilaton potential. Differing from the topological black holes in the Einstein-Maxwell theory, which approach asymptotically the anti-de Sitter spaces, the topological dilaton black holes are asymptotically neither the anti-de Sitter spaces nor de Sitter spaces or Minkowski spacetimes. But the negative effective cosmological constant plays a crucial role in the existence of these black hole solutions, as the negative cosmological constant does in the Einstein-Maxwell theory. In the present paper, we would like to investigate the topological black holes in the higher dimensional spacetimes. In the Einstein-Maxwell theory, the higher dimensional, spherically symmetric black holes have been studied by Myers and Perry [23]. And their analogues in the Brans-Dicke theory have been investigated recently in [24]. Therefore, four-dimensional topological black holes have their natural generalization in the higher dimensional EinsteinMaxwell theory with a negative cosmological constant. For example, there are the static topological black holes in four-dimensional spacetimes ds = − ( k − 8πM ω2 r + 16πQ ω 2 r 2 + r l2 )