Phases of Kaluza-Klein Black Holes: A Brief Review

TL;DR

The paper surveys the phase structure of static, neutral Kaluza-Klein black holes in $D=d+1$ dimensions, using the $(\mu,n)$ phase diagram to classify solutions into two sectors: those without KK bubbles with local $SO(d-1)$ symmetry and those with KK bubbles. It analyzes the three main branches—uniform black strings, non-uniform black strings, and localized black holes—and discusses their connections, including the Gregory–Laflamme instability and possible horizon-topology-changing endpoints. It also reviews bubble–black hole sequences, their non-uniqueness, and dualities (notably equal-temperature families and 5D–6D maps), highlighting how bubbles provide a mechanism to realize a wide range of horizon topologies and tensions. The review further connects these gravitational phase structures to brane configurations and gauge/gravity dualities in string theory, and outlines open questions and directions for future work, including analytic progress and stability analyses across dimensions.

Abstract

We review the latest progress in understanding the phase structure of static and neutral Kaluza-Klein black holes, i.e. static and neutral solutions of pure gravity with an event horizon that asymptote to a d-dimensional Minkowski-space times a circle. We start by reviewing the (mu,n) phase diagram and the split-up of the phase structure into solutions with an internal SO(d-1) symmetry and solutions with Kaluza-Klein bubbles. We then discuss the uniform black string, non-uniform black string and localized black hole phases, and how those three phases are connected, involving issues such as classical instability and horizon-topology changing transitions. Finally, we review the bubble-black hole sequences, their place in the phase structure and interesting aspects such as the continuously infinite non-uniqueness of solutions for a given mass and relative tension.

Figures (5)

• Figure 1: Illustration of the Intersection Rule.
• Figure 2: Black hole and string phases for $d=4$ and $d=5$, drawn in the $(\mu,n)$ phase diagram. The horizontal (red) line at $n=1/2$ and 1/3 respectively is the uniform string branch. The (blue) branch emanating from this at the Gregory-Laflamme mass is the non-uniform string branch. For $d=4$ only the linear behavior close to the Gregory-Laflamme mass is known, while for $d=5$ the entire behavior has been obtained numerically by Wiseman Wiseman:2002zc. The (purple) branch starting in the point $(\mu,n)=(0,0)$ is the black hole branch which was numerically obtained by Kudoh and Wiseman Kudoh:2004hs. In particular for $d=5$ we observe the remarkable result that the black hole and non-uniform black string branch meet.
• Figure 3: Entropy $\mathfrak{s}/\mathfrak{s}_{\rm u}$ versus the mass $\mu/\mu_{GL}$ diagram for the uniform string (red), non-uniform string (blue) and localized black hole (purple) branches.
• Figure 4: Sketch of the $(\mu,n)$ phase diagram for large $d$.
• Figure 5: $(\mu,n)$ phase diagrams for five (left figure) and six (right figure) dimensions. We have drawn the $(p,q)=(1,1)$, $(1,2){}_{\mathfrak{t}}$ and $(2,1)$ solutions. These curves lie in the region $1/2 < n \leq 2$ for the five-dimensional case and $1/3 < n \leq 3$ for the six-dimensional case. The lowest (red) curve corresponds to the $(1,1)$ solution. The (blue) curve that has highest $n$ for high values of $\mu$ is the equal temperature $(1,2){}_{\mathfrak{t}}$ solution. The (green) curve that has highest $n$ for small values of $\mu$ is the $(2,1)$ solution. The entire phase space of the $(1,2)$ configuration is the wedge bounded by the equal temperature $(1,2){}_{\mathfrak{t}}$ curve and the $(1,1)$ curve. For completeness we have also included the uniform (orange) and non-uniform (cyan) black string branch, and the small black hole branch (magenta) displayed in Figure \ref{['fig1']}.