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

TL;DR

This paper introduces a new spherically symmetric, regular interior solution to Einstein's equations that matches Schwarzschild's solution, satisfies weak energy conditions, and can describe either a gravitational soliton or a regular black hole.

Contribution

A novel, simple interior solution to Einstein's equations with parameters that ensure regularity, energy condition satisfaction, and the ability to model gravitational solitons or black holes.

Findings

Solution matches Schwarzschild exterior

Satisfies weak energy conditions in the interior

Can describe stable gravitational solitons or regular black holes

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.