This paper is the first in a sequence of three devoted to the formulation of a theory of self-gravitating anisotropic fluids in both Newtonian and relativistic gravity. In this first paper we set the stage, place our work in the context of a vast literature on anisotropic stars in general relativity, and provide an overview of the results obtained in the remaining two papers. In the second paper we develop the Newtonian theory, inspired by a familiar example of an anisotropic fluid, the (nematic) liquid crystal, and apply the theory to the construction of Newtonian stellar models. In the third paper we port the theory to general relativity, and exploit it to obtain relativistic stellar models. In both cases, Newtonian and relativistic, the state of the fluid is described by the familiar variables of an isotropic fluid (such as mass density and velocity field), to which we adjoin a director vector, which defines a locally preferred direction within the fluid. The director field contributes to the kinetic and potential energies of the fluid, and therefore to its dynamics. Both the Newtonian and relativistic theories are defined in terms of an action functional; variation of the action gives rise to dynamical equations for the fluid and gravitational field. While each theory is formulated in complete generality, in these papers we apply them to the construction of stellar models by restricting the fluid configurations to be static and spherically symmetric. We find that the equations of anisotropic stellar structure are generically singular at the stellar surface. To avoid a singularity, we postulate the existence of a phase transition at a critical value of the mass density; the fluid is anisotropic at high densities, and goes to an isotropic phase at low densities. In the case of Newtonian stars, we find that sequences of equilibrium configurations terminate at a maximum value of the central density; beyond this maximum the density profile becomes multi-valued within the star, and the model therefore becomes unphysical. In the case of relativistic stars, this phenomenon typically occurs beyond the point at which the stellar mass achieves a maximum, and we conjecture that this point marks the onset of a dynamical instability to radial perturbations (as it does for isotropic stars). Also in the case of relativistic stars, we find that for a given equation of state and a given assignment of central density, anisotropic stellar models are always less compact than isotropic models.

本文是专门讨论在牛顿引力和相对论引力下建立自引力各向异性流体理论的三篇论文中的第一篇。在第一篇论文中，我们奠定了基础，将我们的工作置于广义相对论中各向异性恒星的大量文献的背景下，并概述了其余两篇论文中获得的结果。在第二篇论文中，我们发展了牛顿理论，受到一个熟悉的各向异性流体示例（向列）液晶的启发，并将该理论应用于牛顿恒星模型的构建。在第三篇论文中，我们将该理论移植到广义相对论中，并利用它来获得相对论恒星模型。在牛顿和相对论这两种情况体的状态都由各向同性流体的熟悉变量（例如质量密度和速度场）描述，我们与该变量相邻一个导向向量，该导向向量定义了流体内的局部首选方向。导向器场有助于流体的动能和势能，因此也有助于流体的动力学。牛顿理论和相对论理论都是根据作用泛函定义的;作用的变化会产生流体场和引力场的动力学方程。虽然每个理论都是完全通用的，但在这些论文中，我们通过将流体配置限制为静态和球对称来将它们应用于恒星模型的构建。我们发现各向异性恒星结构方程在恒星表面通常是奇异的。为了避免奇点，我们假设在质量密度的临界值处存在相变;流体在高密度时是各向异性的，在低密度时进入各向同性相。 在牛顿星的情况下，我们发现平衡构型序列终止于中心密度的最大值;超过这个最大值，密度分布在恒星内变得多值，因此模型变得非物理。在相对论性恒星的情况下，这种现象通常发生在恒星质量达到最大值的点之后，我们推测这一点标志着径向扰动动力学不稳定性的开始（就像各向同性恒星一样）。同样在相对论恒星的情况下，我们发现对于给定的状态方程和给定的中心密度分配，各向异性恒星模型总是比各向同性模型更不紧凑。