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 $V^\mu$ and gravitational field tensor $F^{\mu\nu}$ that is associated with gravitational interaction. Next we show that gravitational four-current $J^\mu$ derived for introduced four-potential produce energy-stress tensor and reproduce main General Relativity formula. Next we introduce valid Lagrangian and equations of motion that explains obtained results. At the end we introduce new approach to quantization of gravity that results in proper quantum values and is open to further generalization.