Exact Solutions for Compact Objects in General Relativity

TL;DR

This paper presents seven new exact solutions to Einstein's field equations for static, spherically symmetric compact objects, extending previous models by including anisotropic pressures and electric charge, with three solutions meeting all physical acceptability criteria.

Contribution

The paper introduces seven new solutions to the Einstein field equations that incorporate anisotropic pressures and electric charge, with three solutions satisfying all physical and stability conditions.

Findings

Three solutions are physically acceptable and stable.

An equation of state is derived for all viable solutions.

A generalized pulsation equation for stability analysis is developed.

Abstract

Seven new solutions to the interior static and spherically symmetric Einstein's field equations (EFE) are found and investigated. These new solutions are a generalisation of the quadratic density fall-off profile of the Tolman VII solution. The generalisation involves the addition of anisotropic pressures and electric charge to the density profile. Of these new solutions three are found to obey all the necessary conditions of physical acceptability, including linear stability under radial perturbations, and causality of the speed of pressure waves inside the object. Additionally an equation of state can be found for all the physically viable solutions. The generalised pulsation equation for interior solutions to the EFE that include both electric charge and pressure anisotropy is derived and used to determine the stability of the solutions. However the pulsation equation found is general and can be used for all new solutions that contain these ingredients.