The spin connections of the Dirac field have three ingredients that are connected with the Ricci rotations, the Maxwell field, and an axial field which is coupled to the axial current. I demonstrate that the axial field provides an effective mechanism of auto-localization of the Dirac field into compact objects. A non-linear system of Dirac and sine-Gordon equations that has a potential to yield a mass spectrum is derived. The condition that the compact objects are stable (the energy-momentum is self-adjoint) leads to the Einstein's field equations. The Dirac field with its spin connection seem to be a natural material carrier of the space-time continuum in which the compact objects are moving along the geodesic lines. The long distance effect of the axial field is indistinguishable from the Newton's gravity which reveals the microscopic nature of gravity and the origin of the gravitational mass.