Dimensionally continued black holes

Abstract

Static, spherically symmetric solutions of the field equations for a particular dimensional continuation of general relativity with a negative cosmological constant are studied. The action is, in odd dimensions, the Chern-Simons form for the anti-de Sitter group and, in even dimensions, the Euler density constructed with the Lorentz part of the anti-de Sitter curvature tensor. Both actions are special cases of the Lovelock action, and they reduce to the Hilbert action (with a negative cosmological constant) in the lower dimensional cases $\mathcal{D}=3$ and $\mathcal{D}=4$. Exact black hole solutions characterized by mass ($M$) and electric charge ($Q$) are found. In odd dimensions a negative cosmological constant is necessary to obtain a black hole, while in even dimensions both asymptotically flat and asymptotically anti-de Sitter black holes exist. The causal structure is analyzed and the Penrose diagrams are exhibited. The curvature tensor is singular at the origin for all dimensions greater than three. In dimensions of the form $\mathcal{D}=4k,4k\ensuremath{-}1$, the number of horizons may be zero, one, or two, depending on the relative values of $M$ and $Q$, while for a negative mass there is no horizon for any real value of $Q$. In the other cases, $\mathcal{D}=4k+1,4k+2$, both naked and dressed singularities with a positive mass exist. As in three dimensions, in all odd dimensions anti-de Sitter space appears as a "bound state" of mass $M=\ensuremath{-}1$, separated from the continuous spectrum ($M\ensuremath{\ge}0$) by a gap of naked curvature singularities. In even dimensions anti-de Sitter space has zero mass. The analysis is Hamiltonian throughout, considerably simplifying the discussion of the boundary terms in the action and the thermodynamics. The Euclidean black hole has the topology ${R}^{2}\ifmmode\times\else\texttimes\fi{}{S}^{\mathcal{D}\ensuremath{-}2}$. Evaluation of the Euclidean action gives explicit expressions for all the relevant thermodynamical parameters of the system. The entropy, defined as a surface term in the action coming from the horizon, is shown to be a monotonically increasing function of the black hole radius, different from the area for $\mathcal{D}>4$.