Extensions of the Schwarzschild solution into regions of non-zero energy density and pressure

Understand

We present solutions of the Einstein equations that extend the static Schwarzschild solution in empty space into regions of non-zero energy density $\rho$ and radial pressure $p= w \rho$, where $w$ is a constant equation of state parameter.

• For simplicity we focus mainly on solutions with constant $\rho$.
• For $w=0$ we find solutions both with and without a singularity at the origin.
• Possible applications to galaxies are considered, where we find enhanced velocity rotation curves towards the edge of a galaxy.