Self-gravitating anisotropic fluid. II: Newtonian theory

This paper is the second in a sequence of three devoted to the formulation of a theory of self-gravitating anisotropic fluids in both Newtonian gravity and general relativity. In the first paper we set the stage, placed our work in context, and provided an overview of the results obtained in this paper and the next. In this second paper we develop the Newtonian theory, inspired by a real-life example of an anisotropic fluid, the (nematic) liquid crystal. We apply the theory to the construction of static and spherical stellar models. In the third paper we port the theory to general relativity, and exploit it to build relativistic stellar models. In addition to the usual fluid variables (mass density, velocity field), the Newtonian theory features a director vector field \(\varvec{c}(t,\varvec{x})\), whose length provides a local measure of the size of the anisotropy, and whose direction gives the local direction of anisotropy. The theory is defined in terms of a Lagrangian which implicates all the relevant forms of energy: kinetic energy (with contributions from the velocity field and the time derivative of the director vector), internal energy (with isotropic and anisotropic contributions), gravitational interaction energy, and gravitational-field energy. This Lagrangian is easy to motivate, and it provides an excellent starting point for a relativistic generalization in the third paper. The equations of motion for the fluid, and Poisson’s equation for the gravitational potential, follow from a variation of the action functional, given by the time integral of the Lagrangian. Because our stellar models feature a transition from an anisotropic phase at high density to an isotropic phase at low density, a substantial part of the paper is devoted to the development of a mechanics for the interface fluid, which mediates the phase transition.