Electrically Charged Proca Stars

TL;DR

This work constructs electrically charged Proca stars (CPS) in a static, spherically symmetric spacetime by formulating the Einstein--Maxwell--Proca system in a 3+1 framework and imposing a harmonic time dependence for the Proca field. The authors derive a reduced set of six radial equations under spherical symmetry and electrostatic conditions, solve them via a shooting method, and explore CPS families across a range of charge values $q$, including supercritical cases slightly above the critical value $q_c=m$. They show that subcritical CPS can be gravitationally bound with negative binding energy $E_B$, while supercritical CPS exist only in narrow ranges of the central potential and are always unbound, implying dynamical instability. The results reveal how charge enlarges the effective radius and the maximum mass, while the total charge satisfies $Q/M<1$ and the frequency remains bounded by $\omega\le m$, with detailed profiles illustrating node structure and energy/charge densities. These findings provide a framework for understanding the stability and observational prospects of charged vector boson stars and motivate future dynamical stability analyses and potential axisymmetric ground states.

Abstract

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. 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.

Figures (12)

• Figure 1: Frequency $\omega$ as a function of the central value of scalar potential $\psi_0$ for families of charged Proca stars with different values of the charge parameter $q$.
• Figure 2: Effective radius $R_{99}$ as a function of the central potential $\psi_0$ for the different CPS families.
• Figure 3: Total mass $M$ as a function of frequency $\omega$ for different CPS families. The left panel corresponds to families with subcritical charges and the right panel to families with supercritical charges (including the critical charge $q=1$).
• Figure 4: Total mass $M$ as a function of the effective radius $R_{99}$ for different CPS families. The left panel corresponds to families with subcritical charges and the right panel to families with supercritical charges (including the critical charge $q=1$).
• Figure 5: Compactness $C_{99}$ as a function of the central potential $\psi_0$ for different CPS families. The left panel corresponds to the subcritical charge regime. The right panel to the supercritical charge regim (including the critical charge $q=1)$.