Regular Spherically Symmetric Interior Solution To Schwarzschid's Solution Which Satisfies The Weak Energy Conditions

Abstract

We present a simple spherically symmetric and regular solution of Einstein's equations with two parameters $k$ and $M$, which matches to Schwarzschild's solution, satisfies the weak energy conditions in the interior region and for small $r$ behaves like the de Sitter solution. Its energy density $\rho$ and its radial pressure $p_r$ satisfy the relation $\rho+p_r=0$. For some values of $k/M$ the solution does not have an event horizon and the event horizon of Schwarzschild's solution is inside the matching surface. Therefore it describes the formation of a gravitational soliton, which is shown to be stable. Gravitational solitons are related to dark matter. For the other values of $k/M$ it is a regular black hole solution.