Exact rotating black holes with vector field hair in five dimensions

Using a Kerr-Schild ansatz that aligns a Proca vector field along null geodesics, the authors reduce the full nonlinear field equations to three manageable master equations. The trick forces the vector field to remain light-like, which pins the theory's free couplings to discrete values and makes exact analytic solutions possible. The resulting black holes carry two independent angular momenta, a cosmological constant, and primary hair encoded in an arbitrary function of one angular coordinate — a genuine physical degree of freedom not removable by gauge or coordinate choice. Several distinct solution branches emerge, each defined by specific relations among the Proca coupling constants.