Dynamical and observational properties of weakly Proca-charged black holes

The simplest approach to include a mass into the electromagnetic vector potential is to modify the Einstein-Maxwell action to the Einstein-Proca form. There are currently no exact analytical solutions for this scenario. However, by using perturbation theory, where both the Proca mass and the black hole charge are small parameters, it is possible to find an exact analytical solution. In this solution, the metric tensor remains unchanged, but the vector potential deviates from the Coulomb potential. In particular, even if the Proca mass is limited by the value $m_γ<10^{-48}\text{g}$, which is the current experimental upper limit for photon mass, it makes a significant contribution to the dynamical equations. In this paper, we study the motion of neutral and charged particles in the vicinity of a weakly Proca-charged black hole, and test the observational implications of the solution of the Einstein-Proca equations for gravitational bending, the black hole shadow, and the fit to the orbits of the Galactic center flares observed by the near-infrared GRAVITY instrument. We find that only extremely cold photons, which are likely scattered before reaching a distant observer, could reveal the non-zero photon mass effect through the black hole shadow. For the Galactic center flare analysis we obtained constraints on the dimensionless Proca parameter to $μ\leq 0.125$ for the electric interaction parameter in the range $-1.1 < \mathrm{Q} < 0.5$, which can be potentially tested by future GRAVITY flare astrometry. Since the Proca parameter is coupled to the black hole mass, the effect of the Proca charge becomes more pronounced for supermassive black holes compared to stellar-mass objects. Our perturbative treatment remains valid essentially up to the horizon, with divergences appearing only in the immediate near-horizon region, where a fully non-perturbative analysis would be required.

💡 Research Summary

The paper investigates the consequences of endowing the electromagnetic field with a tiny mass, i.e. the Proca theory, within the framework of general relativity. Because an exact solution of the Einstein‑Proca equations for a charged black hole does not exist, the authors adopt a perturbative approach in which both the Proca mass μ (related to the photon mass mγ) and the electric charge Q are treated as small, dimension‑less parameters. In this “weak‑charge” regime the spacetime metric remains exactly Schwarzschild (f = h = 1 − 2M/r), while the vector potential deviates from the Coulomb law. Solving the Proca field equation yields a closed‑form expression for the temporal component of the potential, \