  We address the existence of globally neutral neutron star configurations in
contrast with the traditional ones constructed by imposing local neutrality.
The equilibrium equations describing this system are the Einstein-Maxwell
equations which must be solved self-consistently with the general relativistic
Thomas-Fermi equation and $\beta$-equilibrium condition. To illustrate the
application of this novel approach we adopt the Baym, Bethe, and Pethick (1971)
strong interaction model of the baryonic matter in the core and of the
white-dwarf-like material of the crust. We illustrate the crucial role played
by the boundary conditions satisfied by the leptonic component of the matter at
the interface between the core and the crust. For every central density an
entire new family of equilibrium configurations exists for selected values of
the Fermi energy of the electrons at the surface of the core. Each such
configuration fulfills global charge neutrality and is characterized by a
non-trivial electrodynamical structure. The electric field extends over a thin
shell of thickness $\sim \hbar/(m_e c)$ between the core and the crust and
becomes largely overcritical in the limit of decreasing values of the crust
mass.
