The self-consistent general relativistic solution for a system of degenerate neutrons, protons and electrons in beta-equilibrium

TL;DR

This paper develops a self-consistent relativistic model for a degenerate system of neutrons, protons, and electrons in beta-equilibrium, demonstrating the importance of global charge neutrality and solving complex coupled equations.

Contribution

It introduces a new approach using coupled relativistic Thomas-Fermi-Einstein-Maxwell equations to model such systems without assuming local charge neutrality.

Findings

Global charge neutrality is achievable without local neutrality.

The Coulomb potential at the center is approximately equal to the pion rest energy.

The system remains stable against Coulomb repulsion in the proton component.

Abstract

We present the self-consistent treatment of the simplest, nontrivial, self-gravitating system of degenerate neutrons, protons and electrons in $\beta$-equilibrium within relativistic quantum statistics and the Einstein-Maxwell equations. The impossibility of imposing the condition of local charge neutrality on such systems is proved, consequently overcoming the traditional Tolman-Oppenheimer-Volkoff treatment. We emphasize the crucial role of imposing the constancy of the generalized Fermi energies. A new approach based on the coupled system of the general relativistic Thomas-Fermi-Einstein-Maxwell equations is presented and solved. We obtain an explicit solution fulfilling global and not local charge neutrality by solving a sophisticated eigenvalue problem of the general relativistic Thomas-Fermi equation. The value of the Coulomb potential at the center of the configuration is $eV(0)\simeq m_\pi c^2$ and the system is intrinsically stable against Coulomb repulsion in the proton component. This approach is necessary, but not sufficient, when strong interactions are introduced.