Black holes in the quadratic-order extended vector-tensor theories

We investigate the static and spherically black hole solutions in the quadratic-order extended vector-tensor theories without suffering from the Ostrogradsky instabilities, which include the quartic-order (beyond-)generalized Proca theories as the subclass. We start from the most general action of the vector-tensor theories constructed with up to the quadratic-order terms of the first-order covariant derivatives of the vector field, and derive the Euler-Lagrange equations for the metric and vector field variables in the static and spherically symmetric backgrounds. We then substitute the spacetime metric functions of the Schwarzschild, Schwarzschild-de Sitter/ anti-de Sitter, Reissner-Nordstr\"{o}m-type, and Reissner-Nordstr\"{o}m-de Sitter/ anti-de Sitter-type solutions and the vector field with the constant spacetime norm into the Euler-Lagrange equations, and obtain the conditions for the existence of these black hole solutions. These solutions are classified into the two cases 1) the solutions with the vanishing vector field strength; the stealth Schwarzschild and the Schwarzschild de Sitter/ anti- de Sitter solutions, and 2) those with the nonvanishing vector field strength; the charged stealth Schwarzschild and the charged Schwarzschild de Sitter/ anti- de Sitter solutions, in the case that the tuning relation among the coupling functions is satisfied. In the latter case, if this tuning relation is violated, the solution becomes the Reissner-Nordstr\"{o}m-type solution. We show that the conditions for the existence of these solutions are compatible with the degeneracy conditions for the Class-A theories, and recover the black hole solutions in the generalized Proca theories as the particular cases.