Maxwell-like picture of General Relativity and its Planck limit

Abstract We show that Geroch decomposition leads us to Maxwell-like representation of gravity in (3 + 1) metrics decomposition that may be perceived as Lorentz invariant version of GEM. For such decomposition we derive four-potential V µ and gravitational field ten-sor Fµν that is associated with gravitational interac-tion. Next we show that gravitational four-current Jµ derived for introduced four-potential produce energy-stress tensor and reproduce main General Relativity formula. Next we introduce valid Lagrangian and equa-tions of motion that explains obtained results. At the end we introduce new approach to quantization of grav-ity that results in proper quantum values and is open to further generalization