Charged Proca Stars

We consider self-gravitating stationary configurations of a charged massive complex Proca field, also known as "charged Proca stars", in the particular case of spherical symmetry. We first present a general 3+1 decomposition of the Einstein-Maxwell-Proca system, starting from the action and field equations. We then restrict our system to the case of spherical symmetry and, after imposing a harmonic time dependence ansatz for the Proca field, we construct families of charged Proca stars for different values of the charge parameter $q$, and different values of the central Proca scalar potential $\varphi$. In a similar way to the case of scalar boson stars, one can define a critical charge $q=q_c$ that corresponds to the value for which the Coulomb repulsion of the charged Proca field exactly cancels their newtonian gravitational attraction. Just as in the case of boson stars studied in Pugliese et al. (2013) and López et al (2023), we find that supercritical solutions can exist for a limited range of charges above the critical value $q>q_c$. We also consider the binding energy $E_B$ for the different families of solutions, and find that gravitationally bound solutions such that $E_B<0$ can only exist for subcritical charges such that $q < q_c $indicating that our supercritical solutions are probably dynamically unstable against perturbations. PACS numbers: 04.20.Ex, 04.25.Dm, 95.30.Sf