We compute the properties of a class of charged black holes in anti-de Sitter space-time, in diverse dimensions. These black holes are solutions of consistent Einstein-Maxwell truncations of gauged supergravities, which are shown to arise from the inclusion of rotation in the transverse space. We uncover rich thermodynamic phase structures for these systems, which display classic critical phenomena, including structures isomorphic to the van der Waals-Maxwell liquid-gas system. In that case, the phases are controlled by the universal ''cusp'' and ''swallowtail'' shapes familiar from catastrophe theory. All of the thermodynamics is consistent with field theory interpretations via holography, where the dual field theories can sometimes be found on the world volumes of coincident rotating branes. ͓S0556-2821͑99͒02316-4͔
The vacuum Einstein equations in five dimensions are shown to admit a solution describing a stationary asymptotically flat spacetime regular on and outside an event horizon of topology S 1 3 S 2 . It describes a rotating "black ring." This is the first example of a stationary asymptotically flat vacuum solution with an event horizon of nonspherical topology. The existence of this solution implies that the uniqueness theorems valid in four dimensions do not have simple five-dimensional generalizations. It is suggested that increasing the spin of a spherical black hole beyond a critical value results in a transition to a black ring, which can have an arbitrarily large angular momentum for a given mass. Black holes in four spacetime dimensions are highly constrained objects. A number of classical theorems show that a stationary, asymptotically flat, vacuum black hole is completely characterized by its mass and spin [1], and event horizons of nonspherical topology are forbidden [2].In this Letter we show explicitly that in five dimensions the situation cannot be so simple by exhibiting an asymptotically flat, stationary, vacuum solution with a horizon of topology S 1 3 S 2 : a black ring. The ring rotates along the S 1 and this balances its gravitational self-attraction. The solution is characterized by its mass M and spin J. The black hole of [3] with rotation in a single plane (and horizon of topology S 3 ) can be obtained as a branch of the same family of solutions. We show that there exist black holes and black rings with the same values of M and J. They can be distinguished by their topology and by their mass dipole measured at infinity. This shows that there is no obvious fivedimensional analog of the uniqueness theorems.S 1 3 S 2 is one of the few possible topologies for the event horizon in five dimensions that was not ruled out by the analysis in [4] (although this argument does not apply directly to our black ring because it assumes time symmetry). An explicit solution with a regular (but degenerate) horizon of topology S 1 3 S 2 and spacelike infinity with S 3 topology has been built recently in [5]. An uncharged static black ring solution is presented in [6], but it contains conical singularities. Our solution is the first asymptotically flat vacuum solution that is completely regular on and outside an event horizon of nonspherical topology.Our starting point is the following metric, constructed as a Wick-rotated version of a solution in [7]:
We review black-hole solutions of higher-dimensional vacuum gravity and higher-dimensional supergravity theories. The discussion of vacuum gravity is pedagogical, with detailed reviews of Myers-Perry solutions, black rings, and solution-generating techniques. We discuss black-hole solutions of maximal supergravity theories, including black holes in anti-de Sitter space. General results and open problems are discussed throughout.
We examine the recently proposed technique of adding boundary counterterms to the gravitational action for spacetimes which are locally asymptotic to anti-de Sitter spacetimes. In particular, we explicitly identify higher order counterterms, which allow us to consider spacetimes of dimensions dр7. As the counterterms eliminate the need of ''background subtraction'' in calculating the action, we apply this technique to study examples where the appropriate background was ambiguous or unknown: topological black holes, Taub-NUTAdS and Taub-Bolt-AdS. We also identify certain cases where the covariant counterterms fail to render the action finite, and we comment on the dual field theory interpretation of this result. In some examples, the case of a vanishing cosmological constant may be recovered in a limit, which allows us to check results and resolve ambiguities in certain asymptotically flat spacetime computations in the literature. ͓S0556-2821͑99͒07318-X͔
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