Conformally Flat Metric, Position-Dependent Mass and Cold Dark Matter

The maximal acceleration (MA) problem associated with the position-dependent rest mass concept is considered. New arguments in favor of the mass-dependent maximal acceleration (MDMA) are put forward. The hypothesis that there exists a maximal force with the numerical value equal to the inverse Einstein's gravitation constant is advanced. The Lagrangian and Hamiltonian classical dynamics of a point-like particle with the coordinate-dependent mass is given. The effective Lagrangian for the pure gravitational interaction of a test particle is proposed. Within the scope of this model the typical spiral galaxy rotation is described. It is shown that by this model the peculiar form of the corresponding rotation curve is as a whole reproduced without recourse to the dark matter concept. Also, it is demonstrated that the canonical quantization of this model leads directly to the Dirac oscillator model for a particle with Plank's mass.