Hairy black-hole solutions in generalized Proca theories

We present a family of exact black-hole solutions on a static spherically symmetric background in second-order generalized Proca theories with derivative vector-field interactions coupled to gravity. We also derive nonexact solutions in power-law coupling models including vector Galileons and numerically show the existence of regular black holes with a primary hair associated with the longitudinal propagation. The intrinsic vector-field derivative interactions generally give rise to a secondary hair induced by nontrivial field profiles. The deviation from General Relativity is most significant around the horizon and hence there is a golden opportunity for probing the Proca hair by the measurements of gravitational waves (GWs) in the regime of strong gravity.

Figure 1

Numerical solutions of $f$, $h$, $A_0$, $A_1$, $h-f$ outside the horizon for the cubic vector-Galileon model $G_3(X)={\beta }_{3}X$ with ${\beta }_3={10}^{-3}/(r_h{M}_{\mathrm{pl}})$. The boundary conditions around the horizon are chosen to satisfy Eqs. (17)–(19) with $\mu =0.1$ and $a_0=0.6M_{\mathrm{pl}}$ at $r=1.001r_h$.