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In this article, we investigate the stationary, soliton-like solutions in the model of the Einstein gravity coupled to a free and complex scalar field, and extend chains of mini-boson stars to the rotating case. These solutions manifest as multiple rotating mini-boson stars uniformly arranged along the rotation axis. Through numerical methods, we obtain chains of rotating mini-boson stars with one to five constituents. We show the distribution of the field functions for these chain solutions. Additionally, we also study the effect of the frequency of the complex scalar field on the ADM mass $M$ and angular momentum $J$. By comparing the conclusions of the rotating case with the non-rotating case, there are some intriguing differences. Furthermore, we observe that there exist two ergospheres for some of these solutions., Comment: 17 pages, 8 figures

In this paper, we study solutions of a static spherically symmetric system, which is composed of the coupling with the Bardeen action and two Dirac fields. For the case where only the Bardeen action is present, the magnetic charge $q$ can be infinite, then when the magnetic charge is greater than a certain value $q_b$, there exists a black hole solution, which is called the Bardeen black hole (BBH). However, if the Dirac field is introduced, we find that the magnetic charge can only be smaller than the critical value $q_b$, in which there is no black hole solution. Moreover, in the region $q

We investigate the tidal deformability of spherically symmetric axion stars on the stable branches, including the Newtonian and relativistic branches. The results suggest that on the stable branch, the electric Love numbers of axion star are positive, while the magnetic Love numbers are negative. On the Newtonian stable branch, the electric tidal Love numbers are much larger than the magnetic ones, while on the relativistic stable branch, they are slightly larger. Furthermore, the relativistic stable branch has much smaller tidal Love numbers than the Newtonian stable branch, indicating weaker deformability of axion stars on the relativistic stable branch. This could be attributed to the fact that on the relativistic branch, axion stars are more compact, resulting hardly distorted by tidal forces., Comment: 22 pages, 5 figures

In this paper, we construct a Dirac star model composed of $|\kappa|$ pairs of spinor fields. The azimuthal harmonic indeces $m$ of these spinor fields are half-integers, and they satisfiy $-(|\kappa|-\frac{1}{2})\leq m \leq |\kappa|-\frac{1}{2}$. When $\kappa=1$, it corresponds to the conventional Dirac star model, formed by two spinor fields with $m=\frac{1}{2}$ and $m=-\frac{1}{2}$. When $|\kappa|>1$, among these $|\kappa|$ pairs of spinor fields, there exist spinor fields with azimuthal harmonic indices $m>\frac{1}{2}$, and all spinor fields still conform to the same Dirac field equation. Different families of solutions are distinguished by the value of $\kappa$, so we named these solutions $\kappa$-Dirac stars. We obtain solutions for $\kappa=\pm1,\pm2,\pm3,\pm4,\pm5,\pm6$ by using numerical methods. Additionally, we compute their ADM mass $M$, Noether charge $Q$, and binding energy $E$, and illustrate how these quantities change with the spinor field's frequency $\omega$ for different $\kappa$. We observe significant differences between solutions for $|\kappa|>1$ and the $|\kappa|=1$ case. Furthermore, we provide the energy density distribution of the Dirac stars, wherein for $|\kappa|>1$ scenarios, the Dirac stars exhibit a spherical shell-like structure. Moreover, we employ three-dimensional diagrams to intuitively depict how $\kappa$ influences the combination of spinor fields to form a spherically symmetric configuration., Comment: 19 pages, 5 figures. The case of negative value $\kappa$ added, and some discussion expanded

Boson stars are potential sources of gravitational radiation. To improve our comprehension of these exotic objects, we present the study of geodesics within the space-times of stable, collapsing, and migrating boson stars. We focus on timelike geodesics that are initially circular or reciprocating. We verify that orbits initially bound within a stable boson stars persist in their bound states. For a collapsing boson star, we show that orbits initially bound and reciprocating finally either become unbound or plunge into the newly formed black hole, depending on their initial maximal radii. For initially circular geodesics, orbits with small radii plunge into the black hole; orbits with intermediate radii become unbound; and large radii orbits linger around the black hole with nonvanishing eccentricities. For the migrating case, a black hole does not form. In this case, the reciprocating orbits span a wider radial range. For initially circular geodesics, orbits with small radii become unbound, and orbits with large radii remain bound with nonvanishing eccentricities. This geodesic study provides a novel approach to investigate the gravitational collapse and migration of boson stars., Comment: 6 pages, 5 figures, comments are welcome

In this paper, we investigate the properties of the first excited state Dirac stars (DSs) with higher azimuthal harmonic index (specifically, the azimuthal harmonic indexes $m_D$ = $3/2$, $5/2$, $7/2$), as well as the relationship between the ADM mass and angular momentum of Dirac stars with respect to frequency. Moreover, We find that the ergospheres of DSs appear at lower spinor field frequencies, and both the ergospheres and the distribution of the spinor field functions are asymmetric about the equatorial plane. Furthermore, we introduce the ground state scalar field and examine its impact on this system, which is known as the multi-state Dirac-boson stars (DBSs) model. We show various types of solution families for DBSs under both synchronized frequency $\omega$ and nonsynchronized frequencies and find that similar to DSs, the spinor field and the ergospheres of DBSs are also asymmetric about the equatorial plane, but the ergospheres appear at higher spinor field frequencies., Comment: 22 pages, 8 figures

In this paper, we study the spherically symmetric Dirac star model in the presence of a phantom field, obtaining a traversable wormhole solution in non-trivial topological spacetime. This solution exhibits asymmetry in both the field configuration and the metric and possesses a finite ADM mass $M$ and Noether charge $Q$. Furthermore, we find that due to the presence of a wormhole at the center, this solution exhibits many differences from the Dirac star under trivial spacetime. Notably, when the wormhole throat size is small, our numerical calculations indicate the emergence of an extremely approximate black hole solution on one side of the wormhole spacetime, a phenomenon unexplored. At this time, the Kretschmann scalar near the throat tends to infinity, indicating the wormhole becomes untraversable., Comment: 32 pages, 23 figures

We consider excited configuration and multistate configuration of rotating axion boson stars~(RABSs).RABSs are asymptotically flat, stationary, spinning, horizonless solutions of Einstein-Klein-Gordon theory in which the scalar potential depends on scalar field mass $\mu$ and axion decay constant $f_a$. The excited RABSs have two types of solutions, including $^2S$ state and $^2P$ state. The rotating multistate axion boson stars~(RMABSs) consist of coupled fundamental configuration and excited configuration, and also include two types of solutions: $^1S^2S$ state and $^1S^2P$ state. Some differences between RABSs models and rotating mini-boson stars models are discussed. We show the solution space of these models for different values of decay constant $f_a$. We found fundamental RABSs have a higher maximum mass than the excited state in the low decay constant region. Moreover, the multistate configuration allows a higher mass than both the fundamental configuration and the excited configuration at the same frequency. We also found the RMABSs have the second branch in which the ergo-region emerges. This means the second branch may be superradiant unstable., Comment: 18 pages, 9 figures

In this paper, we construct the multi-state Dirac stars (MSDSs) consisting of two pairs of Dirac fields. The two pairs of Dirac fields are in the ground state and the first excited state, respectively. Each pair consists of two fields with opposite spins, ensuring spherical symmetry of the system. We discuss the solutions of the MSDSs under synchronized and nonsynchronized frequencies. By varying the mass $\tilde{\mu}_1$ of the excited state Dirac field and the frequency $\tilde{\omega}_0$ of the ground state Dirac field, we obtain different types of solutions, including single-branch and double-branch solutions. These two types of solutions do not smoothly transition into each other as the parameters $\tilde{\mu}_1$ and $\tilde{\omega}_0$ continuously change, but undergo a sudden transition when $\tilde{\mu}_1$ ($\tilde{\omega}_0$) is greater than or less than the threshold value of $0.7694$ ($0.733$). Furthermore, we analyze the characteristics of the various MSDSs solutions and analyze the relationship between the ADM mass $M$ of the MSDSs and the synchronized and nonsynchronized frequencies. Subsequently, we calculate the binding energy $E_B$ of the MSDSs and discuss the stability of the solutions. Finally, we discuss the feasibility of simulating the dark matter halos using MSDSs., Comment: 26 pages, 11 figures

In this paper, we re-investigate the stationary, soliton-like solutions in the model of the Einstein gravity coupled to a free and complex scalar field, which have been known as mini-boson stars. With numerical method, we find that in addition to the usual single mini-boson star solution, there exist a novel family of solutions interpreted as chains of boson stars, which is made of some boson stars along the symmetry axis. We show the configuration of two types of chains, including an even number of constituents and an odd number of constituents. Furthermore, we also study the effect of the frequency of the complex scalar field on the ADM mass $M$ and the $U(1)$ scalar charge $Q$. It is interesting to note that the existence of chains of boson stars does not require the introduction of a complex scalar field with self-interacting potential., Comment: 15 pages, 5 figures

In this paper, we construct \textit{Dirac-boson stars} (DBSs) model composed of a scalar field and two Dirac fields. The scalar field and both Dirac fields are in the ground state. We consider the solution families of the DBSs for the synchronized frequency $\tilde{\omega}$ and the nonsynchronized frequency $\tilde{\omega}_D$ cases, respectively. We find several different solutions when the Dirac mass $\tilde{\mu}_D$ and scalar field frequency $\tilde{\omega}_S$ are taken in some particular ranges. In contrast, no similar case has been found in previous studies of multistate boson stars. Moreover, we discuss the characteristics of each type of solution family of the DBSs and present the relationship between the ADM mass $M$ of the DBSs and the synchronized frequency $\tilde{\omega}$ or the nonsynchronized frequency $\tilde{\omega}_D$. Finally, we calculate the binding energy $E_B$ of the DBSs and investigate the relationship of $E_B$ with the synchronized frequency $\tilde{\omega}$ or the nonsynchronized frequency $\tilde{\omega}_D$., Comment: 26 pages, 12 figures

We construct rotating hybrid axion-miniboson stars (RHABSs), which are asymptotically flat, stationary, axially symmetric solutions of (3+1)-dimensional Einstein-Klein-Gordon theory. RHABSs consist of a axion field (ground state) and a free complex scalar field (first excited state). The solutions of the RHABSs have two types of nodes, including $^1S^2S$ state and $^1S^2P$ state. For different axion decay constants $f_a$, we present the mass $M$ of RHABSs as a function of the synchronized frequency $\omega$, as well as the nonsynchronized frequency $\omega_2$, and explore the mass $M$ versus the angular momentum $J$ for the synchronized frequency $\omega$ and the nonsynchronized frequency $\omega_2$ respectively. Furthermore, we study the effect of axion decay constant $f_a$ and scalar mass $\mu_2$ on the existence domain of the synchronized frequency $\omega$., Comment: 19 pages, 8 figures