On the universality of the split monopole black hole magnetosphere

Black holes can acquire magnetic flux from their magnetized progenitor or via prolonged accretion. We study the evolution of black hole magnetospheres by means of axisymmetric general relativistic magnetohydrodynamic simulations. We show that all simulated initial magnetic field geometries of varying complexity ultimately evolve into a split monopole magnetosphere. The magnetospheric evolution consists of two phases. In the first phase, the magnetosphere evolves toward pressure equilibrium accompanied by a large magnetic flux decrease on the event horizon on a fast Alfv\'enic timescale of $\sim 60$ light-crossing times of the gravitational radius. The second phase proceeds in a pressure balance in which the magnetic flux decays and current sheets shift in polar angle over the event horizon on slower resistive timescales. We present an analytic model for the second phase. Furthermore, we show that in a split monopole magnetosphere the magnetic flux on the event horizon decays exponentially with a timescale that depends on the black hole spin, where higher spin results in slower decay. Our results can have an implications for the timescales of reconnection-powered flares and for multimessenger counterparts to gravitational wave events.