$κ$-Dirac stars

Abstract

In this paper, we construct a Dirac star model composed of $|κ|$ pairs of spinor fields. The azimuthal harmonic indeces $m$ of these spinor fields are half-integers, and they satisfiy $-(|κ|-\frac{1}{2})\leq m \leq |κ|-\frac{1}{2}$. When $κ=1$, it corresponds to the conventional Dirac star model, formed by two spinor fields with $m=\frac{1}{2}$ and $m=-\frac{1}{2}$. When $|κ|>1$, among these $|κ|$ 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 $κ$, so we named these solutions $κ$-Dirac stars. We obtain solutions for $κ=\pm1,\pm2,\pm3,\pm4,\pm5,\pm6$ 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 $ω$ for different $κ$. We observe significant differences between solutions for $|κ|>1$ and the $|κ|=1$ case. Furthermore, we provide the energy density distribution of the Dirac stars, wherein for $|κ|>1$ scenarios, the Dirac stars exhibit a spherical shell-like structure. Moreover, we employ three-dimensional diagrams to intuitively depict how $κ$ influences the combination of spinor fields to form a spherically symmetric configuration.