Multi-black hole configurations on the cylinder

TL;DR

This work constructs analytic, first-order metrics for static configurations of multiple small black holes on a d-dimensional cylinder $\mathbb{R}^{d-1} \times S^1$ in the weak-mield limit, enforcing equilibrium via zero external force on each hole. The authors develop a three-step matching approach—Newtonian region, overlap, and near-horizon region—to obtain the full metric and derive the first-order thermodynamics, which admits a Newtonian interpretation. They show that the multi-black hole configurations induce continuous non-uniqueness in the KK phase diagram and discuss implications for potential new non-uniform or lumpy black string/hole phases, as well as a copying mechanism to generate further equilibria. The results offer a controlled perturbative window into higher-dimensional KK gravity, suggesting rich phase structure and guiding future non-perturbative and microscopic explorations, including possible connections to fluid analogies and gauge theories via dualities.

Abstract

We construct the metric of new multi-black hole configurations on a d-dimensional cylinder R^{d-1} x S^1, in the limit of small total mass (or equivalently in the limit of a large cylinder). These solutions are valid to first order in the total mass and describe configurations with several small black holes located at different points along the circle direction of the cylinder. We explain that a static configuration of black holes is required to be in equilibrium such that the external force on each black hole is zero, and we examine the resulting conditions. The first-order corrected thermodynamics of the solutions is obtained and a Newtonian interpretation of it is given. We then study the consequences of the multi-black hole configurations for the phase structure of static Kaluza-Klein black holes and show that our new solutions imply continuous non-uniqueness in the phase diagram. The new multi-black hole configurations raise the question of existence of new non-uniform black strings. Finally, a further analysis of the three-black hole configuration suggests the possibility of a new class of static lumpy black holes in Kaluza-Klein space.

Figures (4)

• Figure 1: Phase diagram for $d=5$ with $n$ versus $\mu$ for the two-black hole configurations spanning the area in between the single black hole (LBH) and two equal size black holes (LBH$_2$). Moreover, we have drawn the uniform black string phase (UBS), the non-uniform black string phase (NUBS) and its two-copied phase (NUBS$_2$).
• Figure 2: Plot of the total entropy $S$ of an equilibrium two-black hole configuration as a function of its mass distribution $\kappa$, for a fixed total mass $\mu$. This is a schematic plot for $\mu <\mu_{\rm c}$.
• Figure 3: Plot of the total entropy $S$ of a two black hole configuration with fixed total mass $\mu$ and fixed mass distribution (here $\kappa =0$) as a function of the relative distance $z_2^*$ between the two black holes. We use a values of $\mu$ that lies below the critical mass $\mu_{\rm c}$ listed in Table \ref{['tabmuc']}.
• Figure 4: A typical plot of $X-1$ versus the distance $y$ ranging from 0 to $\pi/2$.