Dynamical Evolutions of Electrically Charged Proca Stars

Abstract

In a previous work we constructed different families of stationary electrically charged Proca stars characterized by a charge parameter $q$, by solving the Einstein--Maxwell--Proca system in spherical symmetry, and imposing a harmonic time dependence ansatz for the Proca field (Mio and Alcubierre, 2025). We showed that there is a critical value for the charge $q_c$ that corresponds to the value for which the Coulomb repulsion of the charged Proca field exactly cancels the Newtonian gravitational attraction, and we found that supercritical solutions can only exist for a limited range of charges above this critical value $q>q_c$. Here we study the dynamical stability properties of these charged Proca stars by adding a small but finite perturbation to the original stationary configurations, and then performing numerical evolutions while keeping the spherical symmetry. We show that, for any given family, the parameter space can be separated into three regions corresponding to gravitationally bound stable configurations, gravitationally bound unstable configurations, and gravitationally unbound unstable configurations. For the unstable configurations we follow the evolution in time in order to determine their final state, and find that this final state can be collapse to a charged Reissner--Nordstrom black hole, migration to a new state in the stable branch, or dispersion to infinity, depending on the value of the binding energy and the specific form of the perturbation.

Figures (7)

• Figure 1: Different families of charged Proca star configurations for $q=0.3, 0.6, 0.9, 1.0, 1.02$. The red curve connects the points of maximum mass (minimum binding energy) for each family, while the blue curve connects the points where the binding energy becomes zero. The black dots correspond to the specific configurations considered in Table \ref{['resultado_final']} below.
• Figure 2: Time evolution of the central value of the lapse funtcion $\alpha$ for charged Proca star configurations with $q=0.6$, and six different values of the initial central potential $\varphi_0$. The upper panel corresponds to a positive perturbation with amplitude $\epsilon=+0.02$, while the lower panel corresponds to a negative perturbation with $\epsilon=-0.02$.
• Figure 3: Left: Time evolution of the real part of the Proca scalar potential $\varphi_0(t)$ evaluated at the origin, for two unperturbed $(\epsilon=0)$ stable configurations with charge $q=0.6$ and scalar potential $\varphi_0(0)=0.056$ (top panel) and $\varphi_0(0)=0.085$ (bottom panel). Right: Fast Fourier Transform (FFT) of the Proca scalar potential at the origin for the same two configurations.
• Figure 4: Time evolution of the total Reissner--Nordström mass evaluated at the grid boundary for five migrating solutions with charge $q=0.6$ and $\varphi_0 = 0.14, 0.22, 0.26, 0.30, 0.36$. These configurations correspond to a negative perturbation with $\epsilon=-0.02$.
• Figure 5: Initial and final states of migrating configuration corresponding to a charge $q=0.6$, perturbation with negative amplitude $\epsilon=-0.02$, and $\varphi_0 = 0.14, 0.22, 0.26, 0.30, 0.36$. The red square indicates the maximum mass configuration, while the black dots represent the initial and final states of our evolving solutions.