The general static spherical perfect fluid solution with EoS parameter w=-1/5

TL;DR

This paper derives an analytical solution for static spherically symmetric perfect fluid configurations with an isothermal equation of state parameter w=-1/5, filling a gap in known solutions for such models.

Contribution

It presents the first known analytical solution for the case w=-1/5 using Buchdahl coordinates, expanding the set of exact solutions for perfect fluid spheres.

Findings

Derived the analytical solution for w=-1/5 in static spherical symmetry.

Connected the w=-1/5 case to the known w=-1 case.

Discussed the properties and implications of the solution.

Abstract

The general analytical solution for the static spherically symmetric metric supported by a perfect fluid with isothermal (proportional) equation-of-state $p = w \rho$ is not known at the time of this writing, except for the trivial cases $w=0$ and $w=-1$; and for $w=-1/3$. We show that if Buchdahl coordinates are used, the problem becomes analytically solvable for $w=-1/5$; display and discuss the solution(s), and exhibit the connection of this case to the $w=-1$ case.