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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 theEinstein-Maxwell equations. The impossibility of imposing the condition oflocal charge neutrality on such systems is proved, consequently overcoming thetraditional Tolman-Oppenheimer-Volkoff treatment. We emphasize the crucial roleof imposing the constancy of the generalized Fermi energies. A new approachbased on the coupled system of the general relativisticThomas-Fermi-Einstein-Maxwell equations is presented and solved. We obtain anexplicit solution fulfilling global and not local charge neutrality by solvinga sophisticated eigenvalue problem of the general relativistic Thomas-Fermiequation. The value of the Coulomb potential at the center of the configurationis $eV(0)\simeq m_\pi c^2$ and the system is intrinsically stable againstCoulomb repulsion in the proton component. This approach is necessary, but notsufficient, when strong interactions are introduced.

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