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

TL;DR

This work constructs and analyzes a broad class of exact static vacuum solutions in five and six dimensions describing sequences of Kaluza-Klein bubbles and black holes, revealing rich phase-structure and non-uniqueness in the (μ,n) diagram. By employing generalized Weyl solutions, the authors build (p,q) families with varied bubble and horizon topologies, study their regularity, thermodynamics, and asymptotics, and establish mappings between 5D and 6D quantities. A key result is the equal-temperature sector (p,q)_{t}, which is self-dual under double Wick rotation and extremizes entropy for fixed mass, highlighting a duality between bubble-dominated and horizon-dominated configurations. The findings show that bubble-black hole sequences occupy the upper region of the phase diagram and typically have lower entropy than the uniform black string, underscoring non-uniqueness and potential instability across a wide class of KK spacetimes.

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 Kaluza-Klein 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. Kaluza-Klein 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 x S^1 and tuboid topology S^2 x S^1 x S^1. The bubbles support the S^1's of the horizons against gravitational collapse. We find two maps between solutions, one that relates five- and 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.

Figures (22)

• Figure 1: The $(\mu,n)$ phase diagram for six dimensions.
• Figure 2: Phase diagram for $d=5$ with $n \leq 1/3$. To the left we have the diagram without copies while we included the copies to the right.
• Figure 3: Plot of the static Kaluza-Klein bubble mass $\mu_{\rm b}$ and the Gregory-Laflamme mass $\mu_{\rm GL}$ versus $d$.
• Figure 4: Rod configurations for the four-dimensional (left) and five-dimensional (right) Schwarzschild black holes. The ends of the figures are supposed to represent $z=\pm\infty$, so that the semi-infinite rods for $U_2$ and $U_3$ extend all the way out to infinity.
• Figure 5: Rod configurations for the uniform black string (left) and the static Kaluza-Klein bubble (right) in five dimensions ($d=4$).