Black hole solutions in dark photon models

We construct static, spherically symmetric black hole solutions in Einstein gravity coupled to a hidden U (1) vector (dark photon), treating both a minimal massive-vector sector and the phenomenologically standard kinetically mixed Einstein-Maxwell-Proca theory. A key ingredient is a clean separation between (i) the source density that enters the Helmholtz (Proca) equation and (ii) the field energy density and pressures that gravitate in Einstein’s equations. In the Einstein-Proca setup, the exterior Yukawa profile ∝ e − m X r / r implies a Yukawa-squared correction to the metric, yielding a closed-form deformation of the Schwarzschild/Reissner-Nordström lapse proportional to ( 1 + m X r ) e − 2 m X r / r 2 at leading order in the dark-sector backreaction. From this geometry we obtain compact analytic expressions for horizon quantities and curvature diagnostics, and we compute first-order shifts of the photon-sphere radius and shadow size. The resulting fractional corrections are controlled by a small dimensionless backreaction parameter and are exponentially suppressed for m X M ≳ O ( 1 ) . In the kinetically mixed theory, after diagonalizing the kinetic terms we show that the asymptotic exterior is fixed by the Gauss-law charge of the massless (Maxwell) combination, while any massive-mode contribution is necessarily short-ranged and Yukawa suppressed unless a conserved hidden charge is present. We further emphasize that magnetic-dipole/Breit operators are intrinsically angle dependent and do not generate isotropic exterior modifications for an unpolarized spherically symmetric configuration at leading order. Overall, our results provide a minimal, self-consistent analytic framework for how a massive hidden vector can induce short-ranged deformations of Schwarzschild/RN observables relevant to horizon-scale imaging and precision strong-field tests.