Multi-black-hole configurations on the cylinder

Abstract

We construct the metric of new multi-black-hole configurations on a $d$-dimensional cylinder ${\mathbb{R}}^{d\ensuremath{-}1}\ifmmode\times\else\texttimes\fi{}{S}^{1}$, 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 nonuniqueness in the phase diagram. The new multi-black-hole configurations raise the question of existence of new nonuniform 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.