Effective gravity and effective quantum equations in a system inspired by walking droplets experiments

In this paper we suggest a macroscopic toy system in which a potential-like energy is generated by a non-uniform pulsation of the medium (i.e. pulsation of transverse standing oscillations that the elastic medium of the system tends to support at each point). This system is inspired by walking droplets experiments with submerged barriers. We first show that a Poincar\'e-Lorentz covariant formalization of the system causes inconsistency and contradiction. The contradiction is solved by using a general covariant formulation and by assuming a relation between the metric associated with the elastic medium and the pulsation of the medium. (Calculations are performed in a Newtonian-like metric, constant in time). We find ($i$) an effective Schr\"odinger equation with external potential, ($ii$) an effective de Broglie-Bohm guidance formula and ($iii$) an energy of the `particle' which has a direct counterpart in general relativity as well as in quantum mechanics. We analyze the wave and the `particle' in an effective free fall and with a harmonic potential. This potential-like energy is an effective gravitational potential, rooted in the pulsation of the medium at each point. The latter, also conceivable as a natural clock, makes easy to understand why proper time varies from place to place.