Charged $Q$-balls and boson stars and dynamics of charged test particles

We construct electrically charged $Q$-balls and boson stars in a model with a scalar self-interaction potential resulting from gauge mediated supersymmetry breaking. We discuss the properties of these solutions in detail and emphasize the differences to the uncharged case. We observe that $Q$-balls can only be constructed up to a maximal value of the charge of the scalar field, while for boson stars the interplay between the attractive gravitational force and the repulsive electromagnetic force determines their behavior. We find that the vacuum is stable with respect to pair production in the presence of our charged boson stars. We also study the motion of charged, massive test particles in the space-time of boson stars. We find that in contrast to charged black holes the motion of charged test particles in charged boson star space-times is planar, but that the presence of the scalar field plays a crucial role for the qualitative features of the trajectories. Applications of this test particle motion can be made in the study of extreme-mass ratio inspirals as well as astrophysical plasmas relevant e.g. in the formation of accretion discs and polar jets of compact objects.

Figure 1

(Left): The mass $M$ (red) and particle number $N$ (black) of uncharged $Q$-balls (solid, $e=0.0$) and charged $Q$-balls (dashed, $e=0.1$), respectively, as a function of $\omega$. (Right): The mass $M$ as function of the particle number $N$ for uncharged $Q$-balls (blue, $e=0.0$) and two differently charged $Q$-balls (red for $e=0.05$, black for $e=0.1$). The insert shows the region around the point where $M=N$ for $Q$-balls with $e=0.1$.

Figure 2

(Left): We give the value of $\mathrm{\Omega }≔\sqrt{1-(e\mu -\omega {)}^2}$ (solid) and $\omega$ (dashed) as functions of $\varphi (0)$ for uncharged and charged $Q$-balls. From top to bottom, the curves correspond to $e=0.0$, $e=0.03$, $e=0.05$ and $e=0.1$, respectively. (Right): We give the corresponding values of the charge $Q$ as function of $\varphi (0)$ for $e\ne 0$. From bottom to top, the values of $e$ are $e=0.03$, $e=0.05$ and $e=0.1$, respectively. The insert shows the value of $\omega -e\mu$ as a function of $\varphi (0)$.

Figure 3

We give the minimal and maximal value of $\varphi (0)$ of $Q$-balls in dependence on the charge $e$. Charged $Q$-balls exist only in the region in between these curves. In particular, for $e\gtrsim 0.1262$ no charged $Q$-balls exist at all.

Figure 4

The metric functions $f$ and $l$, the scalar field function $\varphi$ and the derivative of the gauge field function $A^{\prime }$, which is equivalent the radial electric field, are shown for a typical charged boson star solution.

Figure 5

We show the mass $M$ and the particle number $N$ of boson stars in dependence on $\omega$ for $e=0$ and $e=0.2$, respectively (upper figure). We also present $M$ and $N$ as functions of $\varphi (0)$ (lower left) as well as $\omega$, $f(0)$ and $l(0)$ as functions of $\varphi (0)$ (lower right). Here, $\alpha =0.1$.

Figure 6

We show the effective potential (62) for massive, charged test particles with charge $q=0.1$, angular momentum $L=0.5$ and separation constant $K=1$ (corresponding to $\theta \in [\frac{\pi }{6}:\frac{5\pi }{6}]$) in the space-time of a boson star with $\alpha =0.1$, $e=0.1$ and different values of $\varphi (0)$. The grey region is forbidden. (a). $\varphi (0)=2.25$. (b). $\varphi (0)=2.42$. (c) $\varphi (0)=0.405$. (d) $\varphi (0)=4.2$.

Figure 7

We show the effective potential (62) for massive, charged test particles with charge $q=0.1$, angular momentum $L=0.5$ and varying values of the separation constant $K$ in the space-time of a boson star with $\alpha =0.1$, $e=0.1$ and different values of $\varphi (0)$. The grey region is forbidden for the given value of $K$. We choose $K=0.01$, $K=1.5$ and $K=10$, respectively. (a) $\varphi (0)=2.25$. (b) $\varphi (0)=0.405$. (c) $\varphi (0)=4.2$.

Figure 8

We show the effective potential (62) for massive, charged test particles with angular momentum $L=0.5$ and separation constant $K=1$ in the space-time of a boson star with $\alpha =0.1$, $e=0.1$ and $\varphi (0)=4.2$. In Fig. (a) the charge of the test particle is $q=1$ and in Fig. (b) the charge is $q=3$. The grey region is forbidden for test particle motion.

Figure 9

We show the effective potential (62) for massive, charged test particles with angular momentum $L=0.5$ and separation constant $K=1$ in the space-time of a boson star with $\alpha =0.1$, $e=0.1$ and $\varphi (0)=2.25$. In the plot (a) the charge of a test particle $q=-2$ and in the plot (b) the charge $q=-4$. The grey region is forbidden.

Figure 10

Motion of a charged, massive test particle with $q=0.1$, $L=0.5$ and $K=1.0$ in the space-time of a boson star with $\alpha =0.1$, $e=0.1$ and $\varphi (0)=2.25$. (a) BO for $E=0.25$. (b) $r\text{−}\theta$ plot for figure (a). (c) BO for $E=0.13$. (d) $r\text{−}\theta$ plot for figure (c). (e) EO for $E=0.33$. (f) EO for $E=0.5$.

Figure 11

Motion of a charged, massive test particle with $L=0.5$ and $K=1.0$ in the space-time of a boson star with $\alpha =0.1$, $e=0.1$ and $\varphi (0)=2.25$. (a) BO for $E=14$, $q=-2$. (b) BO for $E=28$, $q=-4$.

Figure 12

Motion of a charged, massive test particle with $q=0.1$, $L=0.5$ and $K=1.0$ in the space-time of a boson star with $\alpha =0.1$, $e=0.1$ and $\varphi (0)=2.42$. (a) BO for $E=0.25$. (b) BO for $E=0.13$. (c) EO for $E=0.5$. (d) EO for $E=0.33$.

Figure 13

Motion of a charged, massive test particle with $q=0.1$, $L=0.5$ and $K=1.0$ in the space-time of a boson star with $\alpha =0.1$, $e=0.1$ and $\varphi (0)=0.405$. (a) BO for $E=0.33$. (b) $r\text{−}\theta$ plot for figure (a). (c) BO for $E=0.008$. (d) $r\text{−}\theta$ plot for figure (c). (e) EO for $E=0.5$. (f) EO for $E=0.065$.

Figure 14

Motion of a charged, massive test particle with $q=0.1$, $L=0.5$ and $K=1.0$ in the space-time of a boson star with $\alpha =0.1$, $e=0.1$ and $\varphi (0)=4.2$. (a) BO for $E=0.19$. (b) $r\text{−}\theta$ plot for figure (a). (c) BO for $E=0.1635$. (d) $r\text{−}\theta$ plot for figure (c). (e) EO for $E=0.4$. (f) EO for $E=0.3$.

Figure 15

Motion of a charged, massive test particle with $L=0.5$, $K=1.0$ in the space-time of a boson star with $\alpha =0.1$, $e=0.1$ and $\varphi (0)=4.2$. (a) EO for $E=0.001$, $q=1$. (b) EO for $E=0.001$, $q=3$.