An exact isotropic solution

TL;DR

This paper derives a general exact solution for static spherically symmetric gravitational fields with pressure isotropy, providing a model for relativistic stellar interiors with well-behaved physical properties.

Contribution

It solves a third-order recurrence relation to obtain a new class of exact solutions for isotropic pressure in relativistic spheres.

Findings

The solution is expressed in elementary functions.

The model satisfies a barotropic equation of state.

The solution is suitable for modeling relativistic stellar interiors.

Abstract

The condition for pressure isotropy is reduced to a recurrence equation with variable, rational coefficients of order three. We prove that this difference equation can be solved in general. Consequently we can find an exact solution to the field equations corresponding to a static spherically symmetric gravitational potential in terms of elementary functions. The metric functions, the energy density and the pressure are continuous and well behaved which implies that this solution could be used to model the interior of a relativistic sphere. The model satisfies a barotropic equation of state in general which approximates a polytrope close to the stellar centre.