Self-gravitating anisotropic fluid. III: relativistic theory

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.