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Equivalent quantum equations with effective gravity in a system inspired by bouncing droplets experiments

Christian Borghesi

Published Jun 18, 2017• 0 citations
OpenAlex

Abstract

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 Poincare-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 Schrodinger 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.

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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 Poincaré-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 Schrödinger 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.

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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 Poincare-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 (iii) an effective Schrodinger equation with external potential, (iiiiii) an effective de Broglie-Bohm guidance formula and (iiiiiiiii) 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.
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 Poincaré-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 (iii) an effective Schrödinger equation with external potential, (iiiiii) an effective de Broglie-Bohm guidance formula and (iiiiiiiii) 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.
In this paper we study a system which consists in an elastic medium carrying transverse waves and one punctual high elastic medium density, called concretion. We compute the equation of motion for the concretion as well as the wave equation of this classical and theoretical system. Afterwards we always consider the case in which the concretion is not the wave source any more. Then, the velocity of the concretion obeys an equivalent de Broglie guidance formula and moves as if exists an equivalent Bohm quantum potential. A strictly equivalent free Schr\odinger equation is retrieved; and we analyse this system in a linear or spherical cavity. We compute the energy (and momentum) of the concretion, naturally defined from the energy (and momentum) density of the vibrating elastic medium. Provided one condition about the amplitude of oscillation, it strikingly appears that the energy and momentum of the concretion not only write in the same form as in quantum mechanics, but also encapsulate equivalent relativistic formulas.
We introduce a dual wave-particle macroscopic system, where a bead oscillator oscillates in an elastic medium which obeys the Klein-Gordon equation. This theoretical system is mostly inspired by bouncing droplets experiments and bead sliding on a vibrating string experiments. This system is studied using a common and simple mathematical formalism. We compute the motion equation of the bead as well as the wave equation of the system. We introduce the effective velocity of the bead with respect to the elastic medium and the wave ψψψ, created by the bead, which modulates the natural wave of the medium. Provided some conditions, ψψψ obeys an equation analogous to the free Schrödinger equation. In the case of linear and spherical cavities, the particle-like characteristics of the bead, expressed with its effective velocity, are proportional to the corresponding wave-like characteristics of the system. This paper is a translation of Dualité onde-corpuscule formée par une masselotte oscillante dans un milieu élastique : étude théorique et similitudes quantiques.
We introduce a dual wave-particle macroscopic system, where a bead oscillator oscillates in an elastic medium which obeys the Klein-Gordon equation. This theoretical system is mostly inspired by bouncing droplets experiments and bead sliding on a vibrating string experiments. This system is studied using a common and simple mathematical formalism. We compute the motion equation of the bead as well as the wave equation of the system. We introduce the effective velocity of the bead with respect to the elastic medium and the wave ψ\psiψ, created by the bead, which modulates the natural wave of the medium. Provided some conditions, ψ\psiψ obeys an equation analogous to the free Schrodinger equation. In the case of linear and spherical cavities, the particle-like characteristics of the bead, expressed with its effective velocity, are proportional to the corresponding wave-like characteristics of the system. This paper is a translation of Dualite onde-corpuscule formee par une masselotte oscillante dans un milieu elastique : etude theorique et similitudes quantiques.
It is known from quantum mechanics that particles are associated with wave functions, and that the probability of observing a particle at some future location is proportional to the squared modulus of the amplitude of its wave function. Although this statistical relationship is well quantified, the interpretations have remained controversial, with many split between the classical Copenhagen interpretation and some variation of the de Broglie-Bohm pilot wave models. Recent experiments with Hydrodynamic Quantum Analogs (HQAs) have demonstrated that droplets of real fluid may achieve stable dynamical states, where interaction of the droplets with their own ripples results in motion analogous to the motion of particles subject to the guiding equation under the pilot wave models. Indeed, many effects previously thought to be exclusively quantum have now been observed as emergent phenomena in these macroscopic HQAs. This has motivated us to explore the possibility that quantum mechanics may actually be the result of fluid dynamics on some real quantum scale fluid. In this paper, we show that if there is a real quantum scale fluid having dynamics analogous to the fluid in HQAs, then this fluid must be a superfluid, and the dynamics of that fluid must take place on the surface of a 4-dimensional hypersphere. Under the influence of cosmological inflation, we further show that these bouncing droplets would have the illusion of a property analogous to rest mass, and that the principles of inertia, momentum, mass-energy equivalence, general relativity, the uncertainty principle and the appearance of a time-like dimension can all be derived for droplets as purely emergent phenomena from the fluid dynamics of this system. As such, we believe that this model merits consideration as a potential foundation for a new unifying theory of physics at all scales.