All "static" spherically symmetric perfect fluid solutions of Einstein's equations with constant equation of state parameter and finite-polynomial "mass function"

TL;DR

This paper classifies all finite-polynomial solutions for static spherically symmetric perfect fluid spacetimes with any constant equation of state parameter, revealing new black hole-like and phantom solutions, including those with tachyonic matter.

Contribution

It provides a complete classification of finite-polynomial solutions for Einstein's equations with perfect fluids for all equation of state parameters, including dark energy and phantom energy.

Findings

Identified all finite-polynomial solutions for the mass function.

Discovered new black hole-like solutions with segregated matter.

Found static and dynamic phantom and tachyonic solutions.

Abstract

We look for "static" spherically symmetric solutions of Einstein's Equations for perfect fluid source with equation of state $p=w\rho$. In order to include the possibilities of recently popularized dark energy and phantom energy possibly pervading the spacetime, we put no constraints on the constant $w$. We consider all four cases compatible with the standard ansatz for the line element, discussed in previous work. For each case we derive the equation obeyed by the mass function or its analogs. For these equations, we find {\em all} finite-polynomial solutions, including possible negative powers. For the standard case, we find no significantly new solutions, but show that one solution is a static phantom solution, another a black hole-like solution. For the dynamic and/or tachyonic cases we find, among others, dynamic and static tachyonic solutions, a Kantowski-Sachs (KS) class phantom solution, another KS-class solution for dark energy, and a second black hole-like solution. The black hole-like solutions feature segregated normal and tachyonic matter, consistent with the assertion of previous work. In the first black hole-like solution, tachyonic matter is inside the horizon, in the second, outside. The static phantom solution, a limit of an old one, is surprising at first, since phantom energy is usually associated with super-exponential expansion. The KS-phantom solution stands out since its "mass function" is a ninth order polynomial.