An Einstein field equations solution predicting large and dwarf galaxies rotation velocity.

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Recently we derived a galactic Machian model from the Einstein field equations. This model predicted the rotation velocity of the large galaxies M31, M101 and M81, without any dark matter halo. However, the predicted total mass of the three galaxies was significantly lower than that commonly observed. This results from the imposed axial symmetry which gave an inner galaxy gravitation attraction in as the inverse of the radial distance. We solved here the Einstein fields equation using spherical coordinates. As a result, the gravitation attraction from the bulge at short radial distance is well as the inverse of its square. However, the motion equation is non-linear versus the density due the presence of a slowing term intrinsically originating from the Einstein field equations. The solution correctly predicts the total mass and rotation velocity of the three large galaxies and of the dwarf galaxy M33 as well. The model predicts the presence of an ultra-high dense central core in all galaxies. Furthermore, it also predicts an extended stellar halo in M33, which was recently observed using the Subaru Telescope / Hyper Suprime-Cam. This model overcomes the core-cusp issue observed in dwarf galaxy when using the dark matter halo thesis in Newtonian dynamics and avoids the introduction of an additional physical constant as in MOND.