Asymptotically anti–de Sitter Proca stars

We show that complex, massive spin-1 fields minimally coupled to Einstein’s gravity with a negative cosmological constant, admit asymptotically anti–de Sitter self-gravitating solutions. Focusing on four-dimensional spacetimes, we start by obtaining analytical solutions in the test-field limit, where the Proca field equations can be solved in a fixed anti–de Sitter background, and then find fully nonlinear solutions numerically. These solutions are a natural extension of the recently found asymptotically flat Proca stars and share similar properties with scalar boson stars. In particular, we show that they are stable against spherically symmetric linear perturbations for a range of fundamental frequencies limited by their point of maximum mass. We finish with an overview of the behavior of Proca stars in five dimensions.

Figure 1

Left panel: ADM mass $M$ and charge $Q$ as a function of the parameter $f_0$ for three different values of $\mathrm{\Lambda }$. The black dots denote the point of maximum mass $M_{\max }$. Right panel: ADM mass $M$ as a function of the fundamental frequency $\omega$ for three different values of $\mathrm{\Lambda }$.

Figure 2

Comparison between the perturbative approximations for the mass (24) and charge (25) and the fully nonlinear results, as a function of $f_0$ for $\mathrm{\Lambda }=-1$.

Figure 3

Energy density $\rho =-T_t^t$ of the star for $f_0=0.165$ and different values of $\mathrm{\Lambda }$. The inset plot shows a zoom of the solutions for large $r$.

Figure 4

The squared characteristic vibrational frequency as a function of the total mass of the Proca star for $\mathrm{\Lambda }=-0.05$. The critical mass at which the star becomes unstable is also its point of maximum mass, denoted by a black dot.

Figure 5

ADM mass and charge of a five-dimensional Proca star as a function of the parameter $f_0$, for $\mathrm{\Lambda }=0$ and $\mathrm{\Lambda }=-0.05$.