  Following a bi-cylindrical model of geometrical dynamics, in the present
study we show that Einstein gravitational equation leads to bi-geodesic
description in an extended symmetrical time-space which fit Hubble expansion in
a "microscopic" cosmological model. As a duality, the geodesic solution is
mathematically equivalent to the basic Klein-Gordon-Fock equations of free
massive elementary particles, in particular, as the squared Dirac equations of
leptons and as a sub-solution with pseudo-axion. This result would serve an
explicit approach to consistency between quantum mechanics and general
relativity.
