Numerical evolutions of spherical Proca stars

Vector boson stars, or Proca stars, have been recently obtained as fully nonlinear numerical solutions of the Einstein-(complex)-Proca system [1]. These are self-gravitating, everywhere nonsingular, horizonless Bose-Einstein condensates of a massive vector field, which resemble in many ways, but not all, their scalar cousins, the well-known (scalar) boson stars. In this paper we report fully nonlinear numerical evolutions of Proca stars, focusing on the spherically symmetric case, with the goal of assessing their stability and the end point of the evolution of the unstable stars. Previous results from linear perturbation theory indicate that the separation between stable and unstable configurations occurs at the solution with maximal ADM mass. Our simulations confirm this result. Evolving numerically unstable solutions, we find, depending on the sign of the binding energy of the solution and on the perturbation, three different outcomes: (i) migration to the stable branch, (ii) total dispersion of the scalar field, or (iii) collapse to a Schwarzschild black hole. In the latter case, a long-lived Proca field remnant—a Proca wig—composed by quasibound states, may be seen outside the horizon after its formation, with a lifetime that scales inversely with the Proca mass. We comment on the similarities/differences with the scalar case as well as with neutron stars.

Figure 1

Domain of existence of the spherical (fundamental) PSs (solid line) and spherical (fundamental) SBSs (dashed line) solutions in an ADM mass vs vector/scalar field frequency diagram. Both fields have been assumed to have mass $\mu$. The five highlighted points correspond to the configurations to be evolved below.

Figure 2

Magnitude of the Proca electric potential at the origin, ${\mathrm{\Phi }}_c(r=0)$, vs the frequency. The five highlighted points correspond to the same as in Fig. 1. The inset shows the ADM mass as a function of ${\mathrm{\Phi }}_c(r=0)$.

Figure 3

Time evolution of the Proca field energy and the apparent horizon mass for the unperturbed models 1–5. In this and remaining time series, the time coordinate is given in units of $\mu$.

Figure 4

Time evolution of the minimal value of the lapse for all unperturbed models. The vertical dashed lines indicate the time of formation of an AH for models 3–5.

Figure 5

Time evolutions of the amplitude of the scalar potential extracted at $r=5$ for models 1 (top panel), 3 (middle panel), and 5 (bottom panel). The oscillation frequencies shown in the insets of the top and middle panel are in good agreement with the corresponding $\omega$, in units of $\mu$, of models 1–3 (see Table 1). In the bottom panel, the initial decay of the wig is fitted by the enveloping function $e^{-{\omega }_{I}t}$, where for model 5, ${\omega }_I/\mu \sim 0.009$.

Figure 6

(Top panel) Time evolution of the perturbed models. The inset shows the minimum value of the lapse of perturbed model 4. (Bottom panel) Time evolution of the magnitude of the Proca electric potential for models 1–3.

Figure 7

Convergence analysis for model PS5 employing three different resolutions, $\mathrm{\Delta }r=0.0125$, black curves, $\mathrm{\Delta }r=0.0125/\sqrt{2}$, blue curves, and $\mathrm{\Delta }r=0.0125/2$, red curves. Top panel: L2 norm of the Hamiltonian constraint computed either at all radial points (solid lines) or outside of the AH (dashed lines). Bottom panel: L2 norm of the Gauss constraint computed at all radial points.

Figure 8

Time evolution of the L2 norm of the constraints for all Proca star models, using our canonical radial resolution of $\mathrm{\Delta }r=0.0125$. Top panel: Hamiltonian constraint. Bottom panel: Gauss constraint.