Self-gravitating anisotropic fluids. I: context and overview

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.