Anisotropic spheres in general relativity reexamined

In this paper we review and study several different features of anisotropic spherical star models. We first show that if the radial pressure is larger than the tangential pressure, then the radial pressure is also larger than the corresponding pressure for a fiducial isotropic model with the same mass function and total mass, while the opposite holds if the tangential pressure is larger than the radial one. If, in the first case, we further impose an energy condition, then using previous results (maximum mass for an isotropic spherically symmetric star) we obtain a value for the maximum possible redshift at the surface of the star. These results are then applied in the analysis of a specific example given by the Bower-Liang model. Finally, imposing energy conditions, we study the maximum possible value for the redshift, independently of whether the radial or the tangential pressure is larger than the other. We first reobtain Ivanov’s bound for the ratio of mass to radius, but using a different approach, and then we find a refinement of this bound, expressing it as a function of the ratio between the central and the average energy density.