Black holes in the generalized Proca theory

We investigate static and spherically symmetric black hole solutions in the generalized Proca theory which corresponds to the generalization of the shift-symmetric scalar–tensor Horndeski theory to the vector–tensor theory. Any solution obtained in this paper possesses a constant spacetime norm of the vector field, \(X:=-\frac{1}{2}g^{\mu \nu }A_\mu A_\nu =X_0=\mathrm{constant}\). The solutions in the theory with generalized quartic coupling \(G_4(X)\) generalize the stealth Schwarzschild and the Schwarzschild- (anti-) de Sitter solutions obtained in the theory with the nonminimal coupling to the Einstein tensor \(G^{\mu \nu } A_\mu A_\nu \). While in the vector–tensor theory with the coupling \(G^{\mu \nu }A_\mu A_\nu \) the electric charge does not explicitly affect the spacetime geometry, in more general cases with nonzero \(G_{4XX}(X_0)\ne 0\) this property does not hold in general. The solutions in the theory with generalized cubic coupling \(G_3(X)\) are given by the Schwarzschild- (anti-) de Sitter spacetime, where the dependence on \(G_3(X)\) does not appear in the metric function.