A metric solution for rotating black holes embedded in dark matter halos with central spikes

Abstract

We propose an analytic metric describing rotating black holes surrounded by generic dark matter halos. This metric is an exact solution of the field equations that incorporates a dark matter halo with a central density spike in the vicinity of the black hole. Following the construction of the corresponding spherically symmetric solution proposed by Cardoso {\it et al.}, the rotating geometry is seeded by a mass function characterizing the dark matter distribution surrounding the black hole. The dark matter profile is truncated at a radius close to the horizon, in accordance with analyses based on adiabatic invariants, so that the energy density as well as the radial and tangential pressures vanish identically beyond this point. The presence of the spike and the associated metric discontinuity implies that the dark matter is locally anisotropic. The resulting geometry is asymptotically flat and reduces to several well-known cases under suitable limits. In particular, it generalizes the corresponding spherically symmetric solution to the case of rotating black holes. We discuss the physical interpretation of the model parameters and illustrate the metric by applying it to several specific gravitational systems.