Sequences of Bubbles and Holes: New Phases of Kaluza-Klein Black Holes

Abstract

We construct and analyze a large class of exact five-and six-dimensional regular and static solutions of the vacuum Einstein equations. These solutions describe sequences of KaluzaKlein bubbles and black holes, placed alternately so that the black holes are held apart by the bubbles. Asymptotically the solutions are Minkowski-space times a circle, i.e. KaluzaKlein space, so they are part of the (µ, n) phase diagram introduced in hep-th/0309116. In particular, they occupy a hitherto unexplored region of the phase diagram, since their relative tension exceeds that of the uniform black string. The solutions contain bubbles and black holes of various topologies, including six-dimensional black holes with ring topology S 3 × S 1 and tuboid topology S 2 × S 1 × S 1 . The bubbles support the S 1 's of the horizons against gravitational collapse. We find two maps between solutions, one that relates fiveand six-dimensional solutions, and another that relates solutions in the same dimension by interchanging bubbles and black holes. To illustrate the richness of the phase structure and the non-uniqueness in the (µ, n) phase diagram, we consider in detail particular examples of the general class of solutions.

Stationary and axisymmetric solutions of higher-dimensional general relativity

“…For example one can look for new five-dimensional black ring solutions with two angular momenta, or for new solutions with a rotating black hole attached to a Kaluza-Klein bubble, as advocated in [8].…”

“…Since we just found thatG 1i andG ii are finite for z = z * and r → 0 (with i = 2, ..., D − 2) and since we require thatG ij (r, z) and its derivatives are finite as a necessary condition for regular solutions, 8 we see that the LHS and the second and third term on the RHS of (3.2) are finite for z = z * and r → 0. Sincẽ G 11 → ∞ we see therefore that we need ∇G 11 → 0 for z = z * and r → 0.…”

“…Using the integrability condition (H.6) we see that 8) where z 0 is a constant. Now, if we take G(ξ) to be of the form G(ξ) = a 0 + a 1 ξ + a 2 ξ 2 + a 3 ξ 3 + a 4 ξ 4 , (H.9) one can check that (H.6) is obeyed.…”

“…where we have defined R 1 , R 2 and R 3 by 8) as also defined in (H.15). For simplicity, we do not write G 22 explicitly here, but note that it is given implicitly as a function of (r, z) by…”

Thermodynamics of the near-extremal NS5-brane

“…As described in [14,27] there are two classes of Kaluza-Klein black holes in six dimensions: Solutions with a local S0(4) symmetry, and solutions without a local SO(4) symmetry. The latter we shall briefly discuss in the Conclusions, and these are ones that contain Kaluza-Klein bubbles.…”