## 1. Introduction

A dramatic situation in physical understanding of the nature emerged in the late of 19th century. Observed phenomena on micro scales came into contradiction with the general positions of classical physics. It was a time of the origination of new physical ideas explaining these phenomena. Actually, in a very short period, postulates of the new science, quantum mechanics were formulated. The Copenhagen interpretation was first who proposed an ontological basis of quantum mechanics [1]. These positions can be stated in the following points: (a) the description of nature is essentially probabilistic; (b) a quantum system is completely described by a wave function; (c) the system manifests wave–particle duality; (d) it is not possible to measure all variables of the system at the same time; (e) each measurement of the quantum system entails the collapse of the wave function.

Can one imagine a passage of a quantum particle (the heavy fullerene molecule [2], for example) through all slits, in once, at the interference experiment? Following the Copenhagen interpretation, the particle does not exists until it is registered. Instead, the wave function represents it existence within an experimental scene [3].

Another interpretation was proposed by Louis de Broglie [4], which permits to explain such an experiment. In de Broglie's wave mechanics and the double solution theory there are two waves. There is the wave function that is a mathematical construct. It does not physically exist and is used to determine the probabilistic results of experiments. There is also a physical wave guiding the particle from its creation to detection. As the particle moves from a source to a detector, the particle perturbs the wave field and gets a reverse effect from it. As a result, the physical wave guides the particle along some optimal trajectory up to its detection.

A question arises, what is the de Broglie physical wave? Recently, Couder and Fort [5] has executed the experiment with the classical oil droplets bouncing on the oil surface. A remarkable observation is that an ensemble of the droplets passing through the barrier having two gates shows the interference fringes typical for the two slit experiment. Their explanation is that the droplet while moving on the surface induces on this surface the weak Faraday waves. The latter provide the guidance conditions for the droplets. In this perspective, we can draw conclusion that the de Broglie physical wave can be represented by perturbations of the ether when the particle moves through it. In order to describe behavior of such an unusual medium we shall use the Navier-Stokes equation with slightly modified some terms. As the final result we shall get the Schrödinger equation.

In physical science of the New time the assumption for the existence of the ether medium was originally used to explain propagation of light and the long-range interactions. As for the propagation of light, the wave ideas of Huygens and Fresnel require the existence of a continuous intermediate environment between a source and a receiver of the light - the light-bearing ether. It is instructive to compare here the two opposite doctrines about the nature of light belonging to Sir Isaac Newton and Christian Huygens. Newton maintained the theory that the light was made up of tiny particles, corpuscles. They spread through an empty space in accordance with the law of the classical mechanics. Christian Huygens (a contemporary of Newton), believed that the light was made up of waves vibrating up and down perpendicularly to the direction of its propagation, as waves on a water surface. One can imagine all space populated everywhere densely by Huygens's vibrators. All vibrators are silent until a wave reaches them. As soon as a wave front reaches them, the vibrators begin to radiate waves on the frequency of the incident wave. So, the infinitesimal volume

In order to come to idea about existence of the intermediate medium (ether) that penetrates overall material world, we begin from the fundamental laws of classical physics. Three Newton's laws first published in Mathematical Principles of Natural Philosophy in 1687 [6] we recognize as basic laws of physics. Namely: (a) the first law postulates existence of inertial reference frames: an object that is at rest will stay at rest unless an external force acts upon it; an object that is in motion will not change its velocity unless an external force acts upon it. The inertia is a property of the bodies to resist to changing their velocity; (b) the second law states: the net force applied to a body with a mass *M* is equal to the rate of change of its linear momentum in an inertial reference frame

(c) the third law states: for every action there is an equal and opposite reaction.

Leonard Euler had generalized the second Newton's laws on motion of deformable bodies [7]. We rewrite this law for such media. Let the deformable body be in volume *M*. We divide Eq. (1) by

The total derivatives in the right side can be written through partial derivatives:

Eq. (3) equated to zero is seen to be the continuity equation. As for Eq. (4) we may rewrite the rightmost term in detail

As follows from this formula, the first term, multiplied by the mass, is gradient of the kinetic energy. It represents a force applied to the fluid element for its shifting on the unit of length, *vorticity*. The vector product

The term (5) entering in the Navier-Stokes equation [9, 10] is responsible for emergence of vortex structures. The Navier-Stokes equation stems from Eq. (2) if we omit the rightmost term, representing the continuity equation, and specify forces in this equation in detail:

This equation contains two modifications represented in the two last terms from the right side: the dynamic viscosity *μ* depends on time and the rightmost term has a slightly modified view, namely

## 2. Vortex dynamics

The second term from the right in Eq. (6) represents the viscosity of the fluid (^{2}=kg/(m⋅s)). Let us suppose that the fluid is ideal, barotropic, and the mass forces are conservative [10]. At assuming that the external force is conservative, we apply to this equation the operator curl. We get right away the equation for the vorticity:

Here ^{2}/s what corresponds to dimensionality of the diffusion coefficient. The rightmost term describes dissipation of the energy stored in the vortex. As a result, the vortex with the lapse of time will disappear.

With omitted the term from the right (i.e.,

Assuming that the fluid is a physical vacuum, which meets the requirements specified earlier, we must say that the viscosity vanishes. In that case, the vorticity

We shall not remove the viscosity. Instead of that, we hypothesize that even if there is an arbitrary small viscosity, because of the zero-point oscillations in the vacuum, the vortex does not disappear completely. The vortex can be a long-lived object. The foundation for that hypothesis is the observation (performed by French scientific team [5, 11, 12]) of behavior of the droplets moving on the oil surface, on which the waves Faraday exist. Here an important moment is that the Faraday waves are supported slightly below the super-critical threshold. Due to this trick the droplets can live on the oil surface arbitrary long, before they disappear in the oil. The Faraday waves that are supported near the super-critical threshold may play a role analogous to the zero-point oscillations of the vacuum.

Observe that the bouncing droplet simulates some aspects of quantum mechanics, stimulating theoretical investigations in this area [13-20]. It is interesting to note in this place that Grössing considers a quantum particle as a dissipative phase-locked steady state, where an amount of zero-point energy of the wave-like environment is absorbed by the particle, and then during a characteristic relaxation time is dissipated into the environment again [14].

Here we shall give a simple model of such a picture. Let us look on the vortex tube in its cross-section which is oriented along the axis *z* and its center is placed in the coordinate origin of the plane (*x*, *y*). Eq. (7), written down in the cross-section of the vortex, is as follows

We do not write a sign of vector on top of *z*. We introduce time-dependent the kinematic viscosity. For the sake of simplicity, let it be looked as

where

Solution of the equation (8) in this case is as follows

Here ^{2}/s. An extra number *n* > 1. It prevents appearance of singularity in the cases when *n*=16 is shown in Fig. 1(a).

The velocity of the fluid matter around the vortex results from the integration of the vorticity function

Fig. 1(b) shows behavior of this function at the same input parameters.

In particular, for *n*=0 and

As seen from here, the Lamb–Oseen solution decays with time since the viscosity

One can see from the solutions (10) and (11), depending on the distance to the center the functions *v*(*r,t*) show typical behavior for the vortices. The both functions do not decay with time, however. Instead of that, they demonstrate pulsations on the frequency *n*. At *n* tending to infinity the amplitude of the pulsations tends to zero. At the same time the vortex disappears entirely.

The undamped solution was obtained thanks to assumption, that the kinematic viscosity is a periodic function of time, namely,

Qualitative view of the vortex tube in its cross-section is shown in Fig. 3. Values of the velocity *v* are shown by grey color ranging from light grey (minimal velocities) to dark grey (maximal ones). A visual image of this picture can be a hurricane (tropical cyclone [23]) shown from the top. In the center of the vortex, a so-called eye of the hurricane (the vortex core) is well viewed. Here it looks as a small light grey disk, where the velocities have small values. In the very center of the disk, in particular, the velocity vanishes. Observe that in the region of the hurricane eye a wind is really very weak, especially near the center. This is in stark contrast to conditions in the region of the eyewall, where the strongest winds exist (in Fig. 3 it looks as a dark grey annular region enclosing the light grey inner area). The eyewall of the vortex tube (a zone where the velocity reaches maximal values) has the nonzero radius.

Let us find the radius of the vortex core. In order to evaluate this radius we equate to zero the first derivative by *r* of equation (11)

The radius is a root of this equation

Here *n*. Let us evaluate the radius ^{-5} m^{2}/s. As for Ω, let it be equal to 2mc^{2}/ℏ, or approximately 1.6∙10^{21} radians per second for electron. Here *c* is the speed of light and *m* is the electron mass. Then we have (ν/Ω)^{1/2} ≈ 1.93∙10^{-13} m. This length is seen to be smaller then the Compton wavelength, λ_{C} = 2.426∙10^{-12} m, in about 12 times. So, for choosing *n* ≈ 31 we find from Eq. (15) that the radius of the vortex is about the Compton wavelength. From the above one can see that, on a distance about the Compton wavelength, virtual particles can be involved into a vorticity dancing around the electron core, by polarizing the electron charge. This dancing happens at trembling motion of the electron with the frequency Ω = 2mc^{2}/ℏ. That oscillating motion has a deep relation to the so-called “Zitterbewengung” [25].

One can give a general solution of Eq. (8) which has the following presentation

The viscosity function

### 2.1. Vortex rings and vortex balls

If we roll up the vortex tube in a ring and glue together its opposite ends we obtain a vortex ring. A result of such an operation put into the (*x*, *y*) plane is shown in Fig 4. Position of points on the helicoidal vortex ring in the Cartesian coordinate system is given by

Here *r*_{0} is the radius of the tube. And *r*_{1} represents the distance from the center of the tube (pointed in the figure by arrow c) to the center of the torus located in the origin of coordinates (*x,y,z*). A body of the tube, for the sake of visualization is colored in cyan. Eq. (18) parametrized by *t* gives a helicoidal vortex ring shown in this figure. Parameters *z*) and rotation along the arrow b about the center of the tube (about the axis pointed by arrow c), respectively. Phases

Let the radius *r*_{1} in Eq. (18) tends to zero. The helicoidal vortex ring in this case will transform into a vortex ring enveloping a spherical ball. The vortex ring for the case *r*_{ 0}=4, *r*_{ 1} ≈ 0,

The velocity of the clot at the initial time is *x, y, z*)=(4, 0, 0) is on the top of the ball). We designate this velocity as *k*=1, 2, ⋯) the clot travels through the positions 1 and 2 both in the forward and in backward directions, respectively. In the vicinity of these points the velocities *x, y*).

The ball can be filled everywhere densely by other rings at adding them with other phases *z*. We see a dense ball that rolls along the axis *y*, Fig. 5(b). Observe that the ball pulsates on the frequency Ω as it rolls along its path, as it follows from the above computations. Perfect modes describing the rolling ball are spherical harmonics [26].

## 3.3. Derivation of the Schrödinger equation

The third term in the right side of Eq. (6) deals with the pressure gradient. One can see, however, it is slightly differ from the pressure gradient presented in the customary Navier-Stokes equation [9, 10]. One can rewrite this term in detail

The first term, *P*. It may mean that change of the pressure is induced by change of the entropy per length, or else by change of the information flow [27, 28] per length. This term has signs typical of the osmotic pressure, mentioned by Nelson [24].

Let us consider in this respect the pressure *P* in more detail. We shall represent the pressure consisting of two parts, *P*_{1} and *P*_{2}. We begin from the Fick’s law [14]. The law says that the diffusion flux, *J*, is proportional to the negative value of the density gradient *J*=*D* is the diffusion coefficient *P*_{1} as the pressure having diffusion nature

Observe that the kinetic energy of the diffusion flux is

Now we can see that sum of the two pressures, *P*_{1}+*P*_{2}, divided by

One can see that accurate to the divisor *m* this term represents the quantum potential.

To bring the expression (23) to a form of the quantum potential, we need to introduce instead of the mass density

Here the mass *M* is a product of an elementary mass *m* by the number of these masses, *N*, filling the volume *m* by the density of quasi-particles

Here *g* (*t)* is the dimensionless time dependent function. The function *Q* here is the real quantum potential

Grössing noticed that the term

Since the pressure provides a basis of the quantum potential, as was shown above, it would be interesting to interpret an osmotic nature of the pressure [24]. The interpretation can be the following (see Appendix A): a semipermeable membrane where the osmotic pressure manifests itself is an instant, which divides the past and the future (that is, the 3D brane of our being represents the semipermeable membrane in the 4D world). In other words, the thermalized fluctuating force field described by Grössing [13, 14] is asymmetric with respect to the time arrow.

### 3.1. Transition to the Schrödinger equation

The current velocity *v* contains two component – irrotational and solenoidal [10] that relate to vortex-free and vortex motions of the medium, respectively. The basis for the latter is the Kelvin-Stokes theorem. Scalar and vector fields underlie of manifestation of the irrotational and solenoidal velocities

Here subscripts *S* and *R* hint to scalar and vector (rotational) potentials underlying emergence of these two components of the velocity. These velocities are submitted by the following equations

The scalar field is represented by the scalar function *S* – action in classical mechanics. Both velocities are perpendicular to each other. We may define the momentum and the kinetic energy

Now we may rewrite the Navier-Stokes equation (25) in the more detailed form

Note that the term embraced by the curly bracket (*a*) stems from *U* is the potential energy relating to the single quasi-particle. The term embraced by the curly bracket (*b*) describes the viscosity of the medium. As was said above the viscosity coefficient in the average is equal to zero.

Let us rewrite Eq. (30) by regrouping the terms

We assume that fluctuations of the viscosity about zero occur much more frequent, than characteristic time of displacements of the quasi-particles. For that reason, we omit the term

The modification is due to adding the quantum potential (26). In this equation, *C* is an integration constant. We see that the third term in this equation represents energy of the vortex. On the other hand, we can see that the vortex given by Eq. (7) is replenished by the kinetic energy coming from the scalar field *S*, namely via the term

Both the continuity equation

which stems from Eq. (3), and the quantum Hamilton-Jacobi equation (32) can be extracted from the following Schrödinger equation

The kinetic momentum operator

By substituting into Eq. (34) the wave function *Ψ* represented in a polar form

and separating on real and imaginary parts we come to Eqs. (32) and (33). So, the Navier-Stokes equation (6) with the slightly expanded the pressure gradient term can be reduced to the Schrödinger equation if we take into consideration also the continuity equation.

There are confirmations that the Schrödinger equation is deduced from the Feynman path integral [30, 31]. Therefore, for searching solutions of the Schrödinger equation we may apply the path integral. The solution of the Schrödinger equation (34) with the potential that simulates a grating with *N* slits has the following view [32]

Here *λ* is the de Broglie wavelength, *d* is the distance between slits, *b* is the slit width, and **i**=√-1. In this calculation we have used *N*=9 slits, for example, we find the interference pattern shown in Fig. 6 as the density distribution function [32]. This function is a scalar product of the wave function

A useful unit of length at observation of the interference patterns is the Talbot length:

This length bears name of Henry Fox Talbot who discovered in 1836 [33] a beautiful interference pattern, named further the Talbot carpet [34, 35].

The particles, incident on the slit grating, come from a distant coherent source. The de Broglie wavelength of the particle, *h* is the Planck constant, and *p* is the particle momentum) is a main characteristics binding the corpuscular Newtonian physics with the wave Huygens’ physics. It is that we call now the wave-particle dualism. The de Broglie pilot wave being represented by the complex-valued wave function

Fig. 7 shows in lilac color Bohmian trajectories divergent from the slit grating. The probability density distribution is shown here in grey color ranging from white for *p=*0 to light grey for max *p*. Bundle of the Bohmian trajectories imitates a fluid flow through the obstacle, containing slits, relatively well. One can see that characteristic streamlets are formed in the flow, along which particles move. Such a vision of hydrodynamical behaviors of quantum systems is typical for many scientists since the formation of the quantum mechanics up to our days [38-42]. Principal moment is that the Schrödinger equation describes the expiration of the superfluid medium, which depend on the boundary conditions and other devices perturbing it (as, for example, the slit gratings, collimators and others). The vortex balls move along optimal directions of the flows – along the Bohmian paths.

## 4. Physical vacuum as a superfluid medium

The Schrödinger equation (34) describes a flow of the peculiar fluid that is the physical vacuum. The vacuum contains pairs of particle-antiparticles. The pair, in itself, is the Bose particle that stays at a temperature close to zero. In aggregate, the pairs make up Bose-Einstein condensate. It means that the vacuum represents a superfluid medium [43]. A 'fluidic' nature of the space itself is exhibited through this medium. Another name of such an 'ideal fluid' is the ether [29].

The physical vacuum is a strongly correlated system with dominating collective effects [44] and the viscosity equal to zero. Nearest analogue of such a medium is the superfluid helium [22], which will serve us as an example for further consideration of this medium. The vacuum is defined as a state with the lowest possible energy. We shall consider a simple vacuum consisting of electron-positron pairs. The pairs fluctuate within the first Bohr orbit having energy about *m*_{p}, is equal to doubled mass of the electron, *m*. The charge of the pair is zero. The total spin of the pair is equal to 0. The angular momentum, *L*, is nonzero, however. For the first Bohr orbit

We may evaluate the dispersion relation between the energy,

Here *f*(*p*-*p*_{ R}) is a form-factor relating to the electron-positron pairs rotating about the center of their mass *m*_{p}. The form-factor describes dispersion of the momentum *p* around *p*_{ R} conditioned by fluctuations about the ground state with the lowest energy. The form-factor is similar to the Gaussian curve

Here *σ* is the variance of this form-factor. It is smaller or close to *p*_{ R}. The dispersion relation (39) is shown in Fig. 8. The hump on the curve is due to the contribution of the rotating electron-positron pair about the center of their masses. These rotating objects are named rotons [45].

Rotons are ubiquitous in vacuum because of a huge availability of pairs of particle-antiparticle. The movement of the roton in the free space is described by the Schrödinger equation

The constant *C* determines an uncertain phase shift of the wave function, and most possible this phase relates to the chemical potential of a boson (the electron-positron pair) [45]. We shall not take into account contribution of this term in the dispersion diagram because of its smallness. As follows from the above consideration of the Navier-Stokes equation, Eq. (41) can be reduced to the Euler equation

that describes a flow of the inviscid incompressible fluid under the pressure field *P*. One can see from here that the Coriolis force appears as a restoring force, forcing the displaced fluid particles to move in circles. The Coriolis force is the generating force of waves called inertial waves [45]. The Euler equation admits a stationary solution for uniform swirling flow under the pressure gradient along *z*.

Formation of the swirling flow, the twisted vortex state, has been studied in the superfluid ^{3}He-B [46]. These observations give us a possibility to suppose the existence of such phenomena in the physical vacuum. The twisted vortex states observed in the superfluid ^{3}He-B are closely related to the inertial waves in rotating classical fluids. The superfluid initially is at rest [46]. The vortices are nucleated at a bottom disk platform rotating with the angular velocity *z*. As the platform rotates they propagate upward by creating the twisted vortex state spontaneously, Fig. 9. The Coriolis forces take part in this twisting. The twisted vortices grow upward along the cylinder axis [47].

Analogous experiment with nucleating vortices can be realized when the lower disk *A* rotates in the vacuum, Fig. 10. In this case, the vortices are viewed as the dancing electron-positron pairs on the first Bohr orbit. As the vortices grow upward the spontaneous twisted vortex states arise. The latter by reaching upper fixed disk *B* can capture it into rotation.

Pr. V. Samohvalov has shown through the experiment [48], that the vortex bundle induced by rotating the bottom non-ferromagnetic disk *A* leads to rotation of the upper fixed initially non-ferromagnetic disk *B*, Fig. 11. Both disks at room temperature have been placed in the container with technical vacuum at 0.02 Torr, The utmost number of the vortices that may be placed on the square of the disk *A* is ^{18}, where *R*=82.5 mm is the radius of the disk and *N*, is considerably smaller. It can be evaluated by multiplying *N*_{max} by a factor

at the angular rate *V*_{D}=13.2 m/s. Now we can evaluate the kinetic energy of the vortex bundle induced by the rotating disk *A*. This kinetic energy is *B*. Measured in the experiment [48] the torque is about 0.01 N∙m. So, the disk *B* can be captured by the twisted vortex.

The formation of the growing twisted vortices can be confirmed with attraction of modern methods of interference of light rays passing through the gap between the disks. Light traveling along two paths through the space between the disks undergoes a phase shift manifested in the interference pattern [29] as it was shown in the famous experiment of Aharonov and Bohm [49].

## 5. Conclusion

The Schrödinger equation is deduced from two equations, the continuity equation and the Navier-Stokes equation. At that, the latter contains slightly modified the gradient pressure term, namely, *m* is mass of the particle.

We have shown that a vortex arising in a fluid can exist infinitely long if the viscosity undergoes periodic oscillations between positive and negative values. At that, the viscosity, in average on time, stays equal to zero. It can mean that the fluid is superfluid. In our case, the superfluid consists of pairs of particle-antiparticle representing the Bose-Einstein condensate.

As for the quantum reality, such a periodic regime can be interpreted as exchange of the energy quanta of the vortex with the vacuum through the zero-point vacuum fluctuations. In reality, these fluctuations are random, covering a wide range of frequencies from zero to infinity. Based on this observation we have assumed that the fluctuations of the vacuum ground state can support long-lived existence of vortex quantum objects. The core of such a vortex has nonzero radius inside of which the velocity tends to zero. In the center of the vortex, the velocity vanishes. The velocity reaches maximal values on boundary of the core, and then it decreases to zero as the distance to the vortex goes to infinity.

The experimental observations of the Couder’s team [5, 11, 12] can have far-reaching ontological perspectives in regard of studying our universe. Really, we can imagine that our world is represented by myriad of baryonic and lepton “droplets” bouncing on a super-surface of some unknown dark matter. A layer that divides these “droplets”, i.e., particles, and the dark matter is the superfluid vacuum medium. This medium, called also the ether [24], is populated by the particles of matter (“droplets”), which exist in it and move through it [29, 50, 51]. The particle traveling through this medium perturbs virtual particle-antiparticle pairs, which, in turn, create both constructive and destructive interference at the forefront of the particle [30]. Thus, the virtual pairs interfering each other provide an optimal, Bohmian, path for the particle.

Assume next, that the baryonic matter is similar, say, on “hydrophobic” fluid, whereas the dark matter, say, is similar to “hydrophilic” fluid. Then the baryonic matter will diverge each from other on cosmological scale owing to repulsive properties of the dark matter, like soap spots diverge on the water surface. Observe that this phenomenon exhibits itself through existence of the short-range repulsive gravitational force that maintains the incompatibility between the dark matter and the baryonic matter [52, 53]. At that, the dark matter stays invisible. One can imagine that the zero-point vacuum fluctuations are nothing as weak ripples on a surface of the dark matter.

## Appendix A: Nelson’s derivation of the Schrödinger equation

Nelson proclaim that the medium through which a particle moves contains myriad sub-particles that accomplish Brownian motions by colliding with each other chaotically. The Brownian motions is described by the Wiener process with the diffusion coefficient

Here *m* is mass of the particle and *ν* with the upper bar in order to avoid confusion with the kinematic viscosity adopted in hydrodynamics. As seen this motion has a quantum nature [24] in contrast to the macroscopic Brownian motions where the diffusion coefficient has a view *k* is Boltzmann constant, *T* is a temperature, and

Two equations are main in the article [24]. The position **x** (*t)* of the Brownian particle, being subjected either by external forces or by currents in the medium, can be written by two equivalent equations:

Here

Here *E*_{t} denotes the conditional expectation (average) given the state of the system at time *t*, and 0_{+}means that

One can see that these calculations are symmetrical with respect to the time arrow, whereas the calculations (46) and (47) are not, in general (see below).

It should be noted that

There is a one more velocity, which is represented via difference of

According to Einstein's theory of Brownian motion,

where **x**(*t*) and *S* called the action

The both equations, (44) and (45), introduced above are important for derivation of the Schrödinger equation. The derivation of the equation is provided by the use of the wave function presented in the polar form

by replacing the velocities *R+***i***S*} instead of the generally accepted

Observe that the wave function represented in the polar form (53) is used for getting equations underlying the Bohmian mechanics [27]. These two equations are the continuity equation and the Hamilton-Jacobi equation containing an extra term known as the Bohmian quantum potential. The quantum potential has the following view:

One can see that the quantum potential depends only on the osmotic velocity, which is expressed through difference of the forward and backward averaged quantities (46) and (47). These forward and backward quantities can be interpreted as uncompensated flows through a “semipermeable membrane” which represents an instant dividing the past and the future. Following to Licata and Fiscaletti, who have shown that the quantum potential has relation to the Bell length indicating a non-local correlation [28], one can add that the non-local correlation exists also between the past and the future. E. Nelson as one can see has considered a particle motion through the ether populated by sub-particles experiencing accidental collisions with each other. The Brownian motions of the sub-particles submits to the Wiener process with the diffusion coefficient ν proportional to the Plank constant as shown in Eq. (43). The ether behaves itself as a free-friction fluid.

## Acknowledgments

The author thanks Mike Cavedon for useful and valuable remarks and offers. The author thanks also Miss Pipa (quantum portal administrator) for preparing a program drawing Fig. 7.