The purpose of this work is to review the status about stationary solutions of the axially symmetric Einstein–Vlasov system with a focus on open problems of both analytical and numerical nature. For the latter we emphasize that the code used to construct stationary solutions in Ames et al (2016 Class. Quantum Grav. 33 155008; 2019 Phys. Rev. D 99 024012) is open source, see Ames and Logg (2023 J. Open Source Softw. 8 5979). In the analytical setting the open problems include establishing methods for proving existence of axisymmetric stationary solutions which are far from spherically symmetric, both in the general case and for certain special classes of solutions pointed out in the text. In the numerical setting there are intriguing properties of highly relativistic solutions that demand further attention, such as whether a sequence of such stationary solutions can approach a Kerr black hole, or if they necessarily approach the thin ring limit reminiscent of cosmic strings. The question of whether stationary solutions include states with thin-disk like morphologies as seen in many galaxies is also open. Finally, there are opportunities to extend this research to new settings such as the case of massless particles and coupled black hole-matter systems. We believe that some of the open problems highlighted here are of central importance for the understanding of nature.

中文翻译：

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轴对称爱因斯坦-弗拉索夫系统的稳态解：现状和未解决的问题

这项工作的目的是回顾轴对称爱因斯坦-弗拉索夫系统的稳态解的现状，重点关注分析和数值性质的开放问题。对于后者，我们强调用于在 Ames 中构造平稳解的代码等人(2016年班级。量子重力 33155008； 2019年物理。牧师。D99024012）是开源的，请参阅 Ames 和 Logg (2023J. 开源软件。 85979）。在分析环境中，开放问题包括建立方法来证明远离球对称的轴对称稳态解的存在，无论是在一般情况下还是对于文本中指出的某些特殊类别的解。在数值设置中，高度相对论解有一些有趣的特性，需要进一步关注，例如一系列这样的稳态解是否可以接近克尔黑洞，或者它们是否必然接近让人想起宇宙弦的薄环极限。静止解是否包括许多星系中所见的薄盘状形态的状态也是一个悬而未决的问题。最后，有机会将这项研究扩展到新的环境，例如无质量粒子和耦合的黑洞-物质系统。我们相信，这里强调的一些开放性问题对于理解自然至关重要。