Black Holes in Higher-Dimensional Gravity

Abstract

These lectures review some of the recent progress in uncovering the phase structure of black hole solutions in higher-dimensional vacuum Einstein gravity. The two classes on which we focus are Kaluza-Klein black holes, i.e. static solutions with an event horizon in asymptotically flat spaces with compact directions, and stationary solutions with an event horizon in asymptotically flat space. Highlights include the recently constructed multiblack hole configurations on the cylinder and thin rotating black rings in dimensions higher than five. The phase diagram that is emerging for each of the two classes will be discussed, including an intriguing connection that relates the phase structure of Kaluza-Klein black holes with that of asymptotically flat rotating black holes.

Phases of Higher Dimensional Black Holes

“…more examples and applications, we recommend the Refs. [1,2,3,4,5,6]. We would like to conclude with a brief list of interesting open problems related to the issues covered in this review.…”

“…There are already several excellent reviews of complementary aspects of this constantly growing subject. These include [1,2,3] and [4,5,6]. Our goal here will be a more modest one -to provide an illustration of some of the key lessons in this field, outline some of the most interesting future directions and equip the reader with a first guide to the literature.…”

Thermodynamics of nonextremal Kaluza-Klein multiblack holes in five dimensions

“…Following the procedure given in [46,47] one deduces that the rod structure of the general solution is described by 2N turning points that divide the z-axis into 2N + 1 rods as follows. 4 For z < a 2N such that all ζ i < 0 one has a semi-infinite rod with direction l 1 = (0, 2ac…”

“…3 Up to a constant. In general, the dual of 4 We are writing the vectors in the basis {∂/∂t, ∂/∂χ, ∂/∂ϕ}. hole) one has a finite spacelike rod…”

“…Notably, with Emparan and Reall's discovery of the five dimensional black ring solution [1], it was realized that higher dimensional black holes exhibit a much richer structure than their fourdimensional counterparts (for nice reviews of the black ring see [2] and of higher dimensional black holes see [3,4]). In four-dimensional asymptotically flat space-times, as shown geometrically by Hawking [5], a black hole can have only spherical horizon topology; this result also follows under more general conditions as a consequence of topological censorship [6,7].…”