General relativistic polytropes in anisotropic stars

TL;DR

This paper derives generalized equations for relativistic anisotropic stars with polytropic EoS, finds analytical solutions, and studies how pressure anisotropy affects their stability and physical properties.

Contribution

It introduces new generalized Lane-Emden equations for anisotropic relativistic stars and analyzes their solutions and stability, extending previous isotropic models.

Findings

Anisotropy can destabilize incompressible fluid stars.

Analytical solutions for specific anisotropy ansatz are obtained.

Pressure anisotropy influences mass-radius relations and stability.

Abstract

Spherically symmetric relativistic stars with the polytropic equation of state (EoS), which possess the local pressure anisotropy, are considered within the framework of general relativity. The generalized Lane-Emden equations are derived for the arbitrary anisotropy parameter $\Delta=p_t-p_r$ ($p_t$ and $p_r$ being the transverse and radial pressure, respectively). They are then applied to some special ansatz for the anisotropy parameter in the form of the differential relation between the anisotropy parameter $\Delta$ and the metric function $\nu$. The analytical solutions of the obtained equations are found for incompressible fluid stars and then used for getting their mass-radius relation, gravitational and binding energy. Also, following the Chandrasekhar variational approach, the dynamical stability of incompressible anisotropic fluid stars with the polytropic EoS against radial oscillations is studied. It is shown that the local pressure anisotropy with $p_t>p_r$ can make the incompressible fluid stars unstable with respect to radial oscillations, in contrast to incompressible isotropic fluid stars with the polytropic EoS which are dynamically stable.