Gravitational Schrödinger Equation from Ginzburg-Landau Equation, and Its Noncommutative Spacetime Coordinate Representation

Despite known analogy between condensed matter physics and various cosmological phenomena, a neat linkage between low-energy superfluid and celestial quantization is not yet widely accepted in literature. In the present article we argue that gravitational Schr�dinger equation could be derived from time-dependent Ginzburg-Landau (or Gross-Pitaevskii) that is commonly used to describe superfluid dynamics. The solution for celestial quantization takes the same form with Nottale equation. Provided this proposed solution corresponds to the facts, and then it could be used as alternative solution to predict celestial orbits from quantized superfluid vortice dynamics. Furthermore, we also discuss a representation of the wavefunction solution using noncommutative spacetime coordinate. Some implications of this solution were discussed particularly in the context of offering a plausible explanation of the physical origin of quantization of motion of celestial objects.