AI 中文总结
该研究求解耦合爱因斯坦-狄拉克-麦克斯韦方程组，得到带电狄拉克场的稳态自引力解，揭示其与自旋0、1玻色星的共性，还发现超临界q对应的引力未束缚构型。
AI 中文摘要
本研究在广义相对论的3+1形式体系下，求解对应自旋单态的两个费米子的静态球对称构型的耦合爱因斯坦-狄拉克-麦克斯韦（EDM）方程组。针对不同电荷参数q值，通过数值打靶法得到狄拉克场的多类稳态自引力解。进一步探究电荷q对解的结合能、质量、半径及致密性的影响，结果表明仅当q<m时存在引力束缚构型，其中m为狄拉克场质量；质量-频率关系呈现出此前自旋s=0和s=1的玻色场所具有的特征螺旋结构，且部分引力束缚构型的致密性可与中子星相当。基于上述结果得出结论：至少在经典层面，自旋s=0、1/2、1的自引力场与引力耦合时存在共性。与此前玻色星的研究结果类似，还发现部分q略大于m的超临界解，这类构型在牛顿力学框架下库仑斥力会超过引力吸引，本应不存在；但即使在广义相对论框架下，它们仅在有限q>m范围内存在，且始终为引力未束缚态。
英文摘要
In this work we solve the coupled Einstein-Dirac-Maxwell (EDM) system for static spherically symmetric configurations of two fermions in a singlet spinor state within the $3+1$ formalism of general relativity. We find different families of stationary self-gravitating solutions for the Dirac field through a numerical shooting method for different values of the electric charge parameter q. Furthermore, we investigate the effect of the charge q on the binding energy, mass, radius, and compactness of the solutions. We show that gravitationally bound configurations exist only for $q<m$, with m the mass of the Dirac field, and that the mass frequency relation exhibits the characteristic spiral structure previously found for bosonic fields of spin $s=0$ and $s=1$. We are able to show that some of these gravitationally bound configurations have a compactness comparable to that of neutron stars. With these results, we conclude that at least at the classical level, self-gravitating fields with different spins $s=0,1/2,1$ share some common characteristics when they are coupled to gravity. As has been previously shown for the case of bosonic stars, we also find some super-critical solutions with q slightly larger than m. Such super-critical solutions correspond to configurations such that in the Newtonian regime the Coulomb repulsion overcomes the gravitational attraction, and as such they would not be expected to exist. Nevertheless, even if they do exist in the general relativistic case for a limited range of values of $q>m$, we find that they are always gravitationally unbound.
Comments23 pages, 11 figures
