Self-gravitating anisotropic fluid. III: Relativistic theory

TL;DR

This paper develops a relativistic theory of self-gravitating anisotropic fluids, extending previous Newtonian models to construct and analyze anisotropic stellar models within general relativity.

Contribution

It generalizes Newtonian anisotropic fluid models to a relativistic framework and constructs static, spherically symmetric stellar models with anisotropic phases.

Findings

Anisotropic stars are less compact than isotropic stars with the same central density.

The models feature a transition from anisotropic to isotropic phases as density decreases.

The relativistic theory is derived via a variational principle, ensuring consistency with general relativity.

Abstract

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 this third paper we elevate the Newtonian theory of the second paper 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. 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.