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Following a bi-cylindrical model of geometrical dynamics, in the presentstudy we show that Einstein gravitational equation leads to bi-geodesicdescription in an extended symmetrical time-space which fit Hubble expansion ina "microscopic" cosmological model. As a duality, the geodesic solution ismathematically equivalent to the basic Klein-Gordon-Fock equations of freemassive elementary particles, in particular, as the squared Dirac equations ofleptons and as a sub-solution with pseudo-axion. This result would serve anexplicit approach to consistency between quantum mechanics and generalrelativity.

Derivation of Klein-Gordon-Fock equation from General relativity in a time-space symmetrical model