This is the third and final entry in a sequence of papers devoted to the formulation of a theory of self-gravitating anisotropic fluids in Newtonian gravity and general relativity. In the first paper we placed our work in context and provided an overview of the results obtained in the second and third papers. In the second paper we took the necessary step of elaborating a Newtonian theory, and exploited it to build anisotropic stellar models. In this third paper we elevate the theory to general relativity, and apply it to the construction of relativistic stellar models. The relativistic theory is crafted by promoting the fluid variables to a curved spacetime, and promoting the gravitational potential to the spacetime metric. Thus, the director vector, which measures the local magnitude and direction of the anisotropy, is now a four-dimensional vector, and to keep the number of independent degrees of freedom at three, it is required to be orthogonal to the fluid’s velocity vector. The Newtonian action is then generalized in a direct and natural way, and dynamical equations for all the relevant variables are once more obtained through a variational principle. We specialize our relativistic theory of a self-gravitating anisotropic fluid to static and spherically symmetric configurations, and thus obtain models of anisotropic stars in general relativity. As in the Newtonian setting, the models feature a transition from an anisotropic phase at high density to an isotropic phase at low density. Our survey of stellar models reveals that for the same equations of state and the same central density, anisotropic stars are always less compact than isotropic stars.

这是一系列论文中的第三篇也是最后一篇文章，这些论文专门讨论在牛顿引力和广义相对论中建立自引力各向异性流体理论。在第一篇论文中，我们将我们的工作放在上下文中，并概述了第二篇和第三篇论文中获得的结果。在第二篇论文中，我们采取了必要的步骤来阐述牛顿理论，并利用它来构建各向异性恒星模型。在这第三篇论文中，我们将该理论提升为广义相对论，并将其应用于相对论恒星模型的构建。相对论理论是通过将流体变量提升到弯曲的时空，并将引力势提升到时空度量来构建的。因此，测量各向异性的局部大小和方向的导向向量现在是四维向量，为了将独立自由度的数量保持在 3，它需要与流体的速度矢量正交。然后，牛顿作用以直接和自然的方式进行推广，并且所有相关变量的动力学方程再次通过变分原理获得。我们将自引力各向异性流体的相对论理论专门化为静态和球对称构型，从而获得广义相对论中各向异性恒星的模型。与牛顿设置一样，这些模型具有从高密度的各向异性相到低密度的各向同性相的转变。我们对恒星模型的调查表明，对于相同的状态方程和相同的中心密度，各向异性恒星总是比各向同性恒星更紧凑。