  The Feynman, Metropolis and Teller treatment of compressed atoms is extended
to the relativistic regimes. Each atomic configuration is confined by a
Wigner-Seitz cell and is characterized by a positive electron Fermi energy. The
non-relativistic treatment assumes a point-like nucleus and infinite values of
the electron Fermi energy can be attained. In the relativistic treatment there
exists a limiting configuration, reached when the Wigner-Seitz cell radius
equals the radius of the nucleus, with a maximum value of the electron Fermi
energy $(E_e^F)_{max}$, here expressed analytically in the ultra-relativistic
approximation. The corrections given by the relativistic Thomas-Fermi-Dirac
exchange term are also evaluated and shown to be generally small and negligible
in the relativistic high density regime. The dependence of the relativistic
electron Fermi energies by compression for selected nuclei are compared and
contrasted to the non-relativistic ones and to the ones obtained in the uniform
approximation. The relativistic Feynman, Metropolis, Teller approach here
presented overcomes some difficulties in the Salpeter approximation generally
adopted for compressed matter in physics and astrophysics. The treatment is
then extrapolated to compressed nuclear matter cores of stellar dimensions with
$A\simeq (m_{\rm Planck}/m_n)^3 \sim 10^{57}$ or $M_{core}\sim M_{\odot}$. A
new family of equilibrium configurations exists for selected values of the
electron Fermi energy varying in the range $0 < E_e^F \leq (E_e^F)_{max}$. Such
configurations fulfill global but not local charge neutrality. They have
electric fields on the core surface, increasing for decreasing values of the
electron Fermi energy reaching values much larger than the critical value $E_c
= m_e^2c^3/(e\hbar)$, for $E_e^F=0$. We compare and contrast our results with
the ones of Thomas-Fermi model in strange stars.
