  Classical and semi-classical energy states of relativistic electrons bounded
by a massive and charged core with the charge-mass-radio Q/M and macroscopic
radius R_c are discussed. We show that the energies of semi-classical (bound)
states can be much smaller than the negative electron mass-energy (-mc^2), and
energy-level crossing to negative energy continuum occurs. Electron-positron
pair production takes place by quantum tunneling, if these bound states are not
occupied. Electrons fill into these bound states and positrons go to infinity.
We explicitly calculate the rate of pair-production, and compare it with the
rates of electron-positron production by the Sauter-Euler-Heisenberg-Schwinger
in a constant electric field. In addition, the pair-production rate for the
electro-gravitational balance ratio Q/M = 10^{-19} is much larger than the
pair-production rate due to the Hawking processes.
