Dynamical and observational properties of weakly Proca-charged black holes

TL;DR

This work analyzes weakly Proca-charged black holes within the Einstein–Proca framework, showing that in a weak-charge regime the spacetime remains Schwarzschild while the vector potential departs from the Coulomb form. The authors derive a closed-form perturbative solution, discuss the dynamics of charged and neutral particles, and evaluate observational signatures in gravitational lensing, black-hole shadows, and GRAVITY flare orbits, using these to constrain the dimensionless Proca parameter $\mu$ and the interaction parameter $\mathcal{Q}$. They find that photon-mass effects on the shadow are negligible for realistic wavelengths, but the Proca coupling can noticeably affect orbital dynamics and can be probed with high-precision observations of supermassive black holes, with constraints like $-1.1<\mathcal{Q}<0.5$ and $0\leq \mu\leq 0.125$ under stability assumptions. The results highlight SMBHs as promising arenas to test photon mass effects and emphasize the need for non-perturbative treatment near the horizon and future observational campaigns to tighten bounds.

Abstract

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.

Figures (8)

• Figure 1: Behaviour of the specific angular momentum, $\mathcal{L}_{\rm c}$, and the redshift factor, $-u_{t}=\mathcal{E}_{\rm c}+q_{\rm{s}}A_{t}=\mathcal{E}_{\rm c}-\mathcal{Q}P$ of the circling charged particle. The numbers above the curves indicate the charge parameter $\mathcal{Q}$ values; the dashed curves correspond to $\mathcal{Q}<0$ cases, while the solid curves to $\mathcal{Q}>0$ cases.
• Figure 2: Behavior of the functions $\mathcal{L}_{\rm{ISCO}}$ and $\mathcal{E}_{\rm{ISCO}}$.
• Figure 3: Locations of radii $r_{\rm{ISCO}}$ (black curve), $r_{\rm{LUCO}}$ (red curve) $r_{\rm{SSP}}$ (blue curve), $r_{\rm{USP}}$ (orange curve), $r_{\rm{LSSP}}$ (purple curve) and $r_{\rm{LUSP}}$ (green curve) depending on the charge interaction parameter $\mathcal{Q}$ for different values of $\mu$. In the shaded area, there are no circular orbits of charged massive particles.
• Figure 4: Schematic representation of gravitational lensing effect. The motion of a photon from point $A$ to $B$ is presented in two ways. The black curve represents motion at $\phi_{tot}=\pi$ and red when it makes one revolution around a black hole $\phi_{tot}=3\pi$. The last unstable photon circular orbit is depicted by solid dashed curve
• Figure 5: Upper row: deflection angle, $\phi_{\rm tot}$, and total time dilation, $t_{\rm tot}$, in dependence on the impact parameter, $l$. Lower row: the critical impact parameter, $l_{\rm cr}$, versus energy of massive photon, $\mathcal{E}$ (lower left) and critical impact parameter versus photon's velocity (lower right).