The general static spherical perfect fluid solution with EoS parameter w=-1/6

TL;DR

This paper derives an analytical solution for a static spherically symmetric perfect fluid with an equation of state parameter w=-1/6, expanding the known solutions beyond trivial cases.

Contribution

It presents the first known analytical solution for the case w=-1/6, using a Buchdahl transformation of a known solution.

Findings

Derived the general static spherically symmetric solution for w=-1/6.

Extended the set of known solutions for proportional EoS parameters.

Confirmed the solution aligns with predictions from recent work.

Abstract

The general analytical solution for the static spherically symmetric metric supported by a perfect fluid with proportional-equation-of-state $p = w \rho$ is not known at the time of this writing, except for the trivial cases $w=0$ and $w=-1$; for $w=-1/3$, and the recently reported $w=-1/5$. We show that the case $w=-1/6$ is also analytically solvable, as predicted in another recent work. The solution is affected by a Buchdahl transformation of a known solution.