Static equilibrium of multi-black holes in expanding bubbles in five dimensions

We investigate possible configurations for vacuum multi-black holes that maintain static equilibrium in expanding bubbles. Our analysis assumes a five-dimensional Weyl metric to describe the spacetime, facilitating the derivation of solutions based on the provided rod structure. We consider a spacetime having expanding bubbles caused by one or two acceleration horizons, and show that various configurations such as two bubbles, four bubbles devoid of horizons, a black saturn, a black di-ring, a bicycling black ring (orthogonal black di-ring), and a five-dimensional black hole binary can achieve equilibrium within expanding bubbles. Specifically, we demonstrate that equilibrium requires two acceleration horizons on both sides for the bicycling ring and the five-dimensional black hole binary. However, only one acceleration horizon is necessary for achieving equilibrium in the case of the black saturn and the black di-ring.