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

Abstract

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. We propose that our explicit non-singular solution with $w=-1$ describes the interior of a black hole, which is a form of vacuum energy, and verify that its entropy is consistent with the Bekenstein-Hawking entropy. We propose that this idea can perhaps be applied to dark energy, if one views the latter as arising from black holes as pockets of vacuum energy. We estimate the density of the resulting dark energy to be $\rho_\Lambda \approx 10^{-30} g/{\rm cm}^3$, which is close to the measured value for the observable universe. We also present solutions with non-constant $\rho \propto 1/r^2$.