A new theory of non-relativistic gravity and its analysis of galaxy rotation curve

Drawing analogies with general relativity and quantum electrodynamics, we construct a non-relativistic theory of quantum mechanics and gravity that also contains two universal constants and a local symmetry. We postulate that the quantum system is invariant to local modulus transformation, which introduces a purely imaginary connection field that is identified as the gravitational escape velocity. Focusing on the completely classical gravitational field equation, we find that there are two parts in this new escape velocity vector field. One has the same magnitude as the Newtonian escape velocity. The other is proportional to the Hubble constant. Since gravitational potential energy is proportional to the square of escape velocity, one finds a coupled term between the two parts, in addition to the Newtonian gravitational potential energy. Our theory thus shows that in an expanding universe, Newtonian gravity needs modification even in the non-relativistic and non-quantum mechanical limits. The gravitational attractive force between any two objects has an extra term that is inversely proportional to the square root of the distance between them, in addition to the Newtonian inverse square term. Applying our theory to study galaxies, we find the inverse square root term, while negligible in star systems, becomes more dominant over the inverse square term when the length increases to galaxy scale. With just visible mass, we are able to produce rotation curve of galaxy that fits well with observations.