Cosmological multi-black-hole solutions

Abstract

We present simple, analytic solutions to the Einstein-Maxwell equation, which describe an arbitrary number of charged black holes in a spacetime with a positive cosmological constant $\Lambda$. In the limit $\Lambda =0\mathrm{}$, these solutions reduce to the well-known Majumdar-Papapetrou (MP) solutions. Like the MP solutions, each black hole in a $\Lambda >0\mathrm{}$ solution has charge $Q$ equal to its mass $M$, up to a possible overall sign. Unlike the $\Lambda =0\mathrm{}$ limit, however, solutions with $\Lambda >0\mathrm{}$ are highly dynamical. The black holes move with respect to one another, following natural trajectories in the background de Sitter spacetime. Black holes moving apart eventually go out of causal contact. Black holes on approaching trajectories ultimately merge. To our knowledge, these solutions give the first analytic description of coalescing black holes. Likewise, the thermodynamics of the $\Lambda >0\mathrm{}$ solutions is quite interesting. Taken individually, a $\mathrm{}\left|Q\right|=M$ black hole is in thermal equilibrium with the background de Sitter Hawking radiation. With more than one black hole, because the solutions are not static, no global equilibrium temperature can be defined. In appropriate limits, however, when the black holes are either close together or far apart, approximate equilibrium states are established.