“Every fruitful hypothesis is the beginning of an amazing eruption of a stream of unexpected discoveries.” (French physicist, Leon Brillouin)

This article will discuss if the motion of physical bodies is "discrete" or "continuous", and it will propose and present a scientific hypothesis – a solution – to this millennia-old conundrum. The issue with motion was first realized and raised in the 5th century BCE by the ancient Greek philosopher Zeno of Elea, and to this day, heated discussions on the subject continue. Or as the Dutch mathematician Dirk Jan Stroyk wrote in his book; "A Brief Sketch of the History of Mathematics" (1948): "... [Zeno's paradoxes] ... caused such excitement that even now some ripples can be observed...".

The Paradox of Motion

Zeno formulated several aporias ("insoluble propositions") questioning how rudimentary things work or even if they work - such as simply moving from a to b. The perhaps most famous one is called, "Dichotomy" (division by two) (see figure1): In order to overcome a path (or distance or movement), you must first overcome half of this path, and, in order to overcome that halved-path, you must overcome half of that half too, and so on, ad infinitum. Thus, any movement is expressed in sub-dividing a path into half paths, and these half paths further into other half paths until we have the smallest paths possible in the end. By these mechanics of ordinary physical movement, as described above, we will never be able to budge or cover the original patch. Any movement, it would appear, is approaching halves, continuously and forever; effectively meaning not moving at all. A half becomes a half, becomes a half, etc., becomes nothing.

Did Zenon just prove physical movement is impossible, and thusly an illusion?

Yes, it would seem he did. He is so confident that he argues any physical motion and the appearance in our consciousness of the sequence of its individual fragments enter into a logical contradiction.

On the one hand, Zeno and his aporias, as a representative of the Eleatic philosophical school, always won philosophical disputes, proving the principles of this school: the world is integral, eternal in time, and any change (motion) is, or must be, impossible. It is an illusion that knowledge of this integral world is possible only through reasonable (logical) reasoning. It must be. The sensory picture of the world, including the observed movements, is deceptive and contradictory.

On the other hand, Zeno posed a very succinct question or challenge to science, which is one of the most rudiment questions even to this day, battled amongst classical physics and quantum mechanics: how should motion and reality be understood? "Discrete" or "continuous"? And the main thing is not if Zeno's paradoxes are solvable or not solvable. The main thing is that a person first thought about the structure and mechanism of how motion is or must be.

It wasn't until the seventeenth century, more than two thousand years later, that we had a break in the discussion about Zeno's aporias. The first modern treatment of Zeno belongs to the French thinker Pierre Bayle (1647-1706). He came to the conclusion that Zeno was indeed correct; the concepts of time, extension, and motion are associated with difficulties that are insurmountable to the human mind. German philosopher Georg Hegel (1770-1831) assessed Zeno as "the father of dialectics", who distinguished sensually perceived and conceivable motion as a combination and conflict of opposites, as a dialectic of concepts.

Today, we have many mathematical models through which the authors are assured to have solved Zeno's paradoxes. Infinity can exist mathematically within finity. But, I repeat, the main thing is not if these Zeno's paradoxes are solvable or not solvable; obviously, we do move. The main thing is to comprehend and fully understand the mechanism of motion of physical bodies - and, there is no doubt that the great Zeno of Elea directed us in this voyage of better understanding our reality.

The story of my hypothesis

The ancient Greek philosopher Epicurus (342 - 271 BCE) opposes Zeno. He asserts that motion does not need division ad infinitum; rather, it consists of indivisible parts, each holding no motion. There is only a result of motion - a jump, in his view. He births a new look at the nature of motion, which is very close to my own. In my hypothesis, however, this "jump" has the properties of "discreteness" and "continuity", thusly representing both sides of this discussion. There is truth in their unity. In my view, the movement is "continuous" and "discrete," with a predominance on "discrete."

The story of my hypothesis is my story. In the Soviet Union, we began to study physics from 6th grade and on. This is when the story of my hypothesis begins. When studying Newton's laws, even on the very elementary level, the question arose: how does the speed of a body increase when a force act on it? Continuously in time, or discretely?

If it is continuous, then the body moves from the previous state of the motion (from the previous speed) to the next state of motion (to the next speed) instantly. But, as Newton taught us, this is impossible due to the inertia of the body itself. Therefore, from the age of 13, I had the assumption that speed changes discretely, and I concocted an experiment to prove it (see figure 2).

In my experiment, the switching on/off of the electric motor (to which the load was attached through a cable) occurs discretely so that the load does not move. But even at a young age, it became clear to me that the frequency of switching on the electric motor should be extremely large, and this could not be done manually by any practical means – not least by a young teenager such as me. Alas, I have not since carried out a high-frequency experiment on the direct determination of the discreteness of the speed change, as this remains practically problematic for me: An ultra-high-frequency generator and fully accurate measuring instruments are requisites, none of which I have current access to. However, I have conducted and continue to carry out the hypothesis's theoretical (and mathematical) research, which I will address later in the article.

To create a more coherent hypothesis, I need to explain the reason for the inertness of the bodies, and I assume this reason is bound to the bodies' own gravitational field (OGF). Until the body's OGF reacts, it cannot enter a new state of motion. The analogy of my reasoning is illustrated in figure 3 below.

It is more difficult to move (rotate) a raw egg, than a hard-boiled egg because the white and yolk of a raw egg rub against its walls, creating a barrier to motion, not unlike OGF (It should be noted that, independently of me, physicist-theorist Alexander Koreysha made the same assumption about the cause of the inertia of physical bodies (for reference see attribution [1]).

Later in life, after arriving in Israel, I cooperated with an electronics engineer and programmer in the hope of realizing a high-frequency experiment on the direct determination of motion discreteness.  The experiment scheme is illustrated in figure 4, below.

The linear scale LS is mounted on a metal plate during the experiment, which slides on the two metal rollers on an inclined plane. The angle of inclination should be minimal to ensure minimal acceleration of the scale. This will get the maximum possible period of the quantum of motion (see the formulas of my hypothesis here [1, section 3]), commensurate with the pulse width from the linear scale sensor that increases the likelihood of a positive outcome experiment. The sensor of the linear scale is mounted on a bracket above the LS at a minimum distance. The contacting surface of the plate and rollers must be of high quality to reduce the coefficient of friction. If the results of the counting pulses from the generator, account for neighboring pulses from the sensor of the linear scale, are equal, the motion of the linear scale is discrete (atomically, in principle).

Alas, the experiment did not come to fruition at that time, due to the disengagement with said engineer, and unforeseen and unrelated circumstances. However, I was able to conduct an indirect experiment, proving the conclusion from the hypothesis and, therefore, the hypothesis itself.

In 2011, I registered my hypothesis on a-priority.ru and published it for peer-review on several Russian scientific sites. Recently, I have offered some interesting conclusions from the hypothesis, and perhaps the story of my hypothesis has only just begun?

My hypothesis on discrete motion

The below section comes from my two scientific publications on the matter (see attribution [1] and [2]). Here, you will find the full technical version of the hypothesis and the analysis of its validity.

I want to note that by the motion of physical bodies, I mean movement with acceleration in magnitude or in direction since the state of rest can be equated with rectilinear and uniform motion - and this does not exist in nature. So, a force F acts on a ball of mass m (see figure 5 below), the ball rolls on a smooth surface (the force of friction can be ignored) with acceleration a = F / m.

Let’s discuss some fundamental questions, and how my hypothesis will answer them:

Q: Does the ball's speed increase continuously over time?

A: No. The speed of the ball does not grow continuously in time, but discretely (in quanta), since a continuous increase in the speed of a ball moving under an action of a force is not permissible according to Newton’s law of motion: The instantaneous transition of the ball from the previous state to the next would contradict the inertness of the ball itself.

Q: What is the reason for the inertness of the ball m?

A: The inertness of the ball m is in its own gravitational field (OGF) since the inertial mass of the ball is proportional to its gravitational mass. Until the OGF of the ball m reacts to the action of the force F, the speed of the ball will not change.

Q: Why can't the ball move instantly to a new state of motion (to a new speed)?

A: Quantum of motion is the process of the ball's OGF response to the action of a force. This process is periodical and is accompanied by an elementary deformation of the ball m (the response of the OGF) and subsequent redeformation towards the motion (after the response of the OGF).

Q: What is this mechanism of the motion of the ball m?

A: The mechanism of the motion of a ball under the action of a force is a combination of very small elastic deformations and redeformations of the ball ~10^-23m, which is similar to the motion of a caterpillar. The ~10^-23m is a road elementary of the quantum of the ball's motion. This value is calculated according to the formulas displayed in my publication (see attribution [2]  – it equates the road elementary of the quantum of motion of the body m to the wavelength of its de Broglie (see formula 9 below). Each quantum of motion is characterized by elementary displacement, period, and elementary speed (see the enlarged graph of the ball speed - Figure 6 below).

Let us reiterate the hypothesis by describing the figures below in a simpler and, in my opinion, more understandable fashion:

Figure 5 shows the proposed mechanism of the motion of the ball m in the two moments. The ball's own gravitational field (OGF) is depicted by a faceless man.

• Moment 1. Force F acts on the ball - the ball is deformed by ~10^-23 meters because its OGF has not yet reacted to the action of the force and slows down the motion of the ball. In other words, the OGF of the ball is the reason for its inertia. At the same time, the deformation is elastic, and the speed of the ball m has not yet changed (see the graph of the speed - Figure 6). 

• Moment 2. The force no longer acts on the ball, since the ball's OGF reacted to this action and does not slow down the motion of the body. But the plane by the body - creator of the force F remains, and the body is redeformed in the direction of the motion. In this case, there is the quantum of the motion with elementary speed and with the period of the quantum of motion.

• Moments 1,2 - quantum of motion. Thus, the supposed mechanism of the body's motion is periodical.

From the above, it is clear that according to my hypothesis, the physical motion of the bodies occurs discretely (stepwise), that is, in separate quanta. The mechanism of a body’s motion under the action of the force is a moving oscillatory process - a wave process. Whence follows the explanation of the duality of the properties of material bodies (wave, body), which confirms the reality of my hypothesis. In 1924 Louis de Broglie put forward the hypothesis that dualism is not a feature of optical phenomena only, but has a universal character. Elementary particles of matter also have wave properties.

It should be noted that the duality of the properties of elementary particles (wave, particle) refers to quantum mechanics. Therefore, perhaps my hypothesis is a special case of quantum theory, just as classical mechanics is a special case of quantum mechanics.

The below figure 7 is illustrating the mechanism of movement:

The main elements of the hypothesis are the parameters of the quantum of motion: are elementary speed, time period, and elementary motion. The question then arises on how they relate?

T is the reaction period of the inert mass m, or the period of the quantum of the motion, can be expressed by the formula, which is the formula of my hypothesis:                                 

 𝑻 = m/K (formula of the Hypothesis 1)

K is the reaction speed of the inert mass [kg/s];
F is the force acting on the body m [N];
L is the change in the reaction rate of the inert mass when the force changes by 1 N— a value that claims to be a constant value and requires experimental determination [s / m].

Then the elementary change in the speed of the body (elementary speed of the quantum of the motion) takes the form:

aT = 1 / L, (formula 2)

a is the acceleration of the body m.

The graph of the relationship between the elementary speed, period, and elementary displacement of the quantum of motion of the body m, as seen in Figure 6, above.

The Quantum of motion by direction (for example) is a transition body, moving under the influence of centripetal force F, from the previous direction of the velocity (vector V1) in a subsequent (vector V2). This is change elementary of the direction of the velocity of the body - vector dV, the module is equal to 1/L (see figure 8).

Consider the uniformly accelerated motion of the body m under the action of a constant force F. In this case, the path of the body m until reaching the speed v = n × 1 / L, where n is the number of the quanta of the motion, is equal to: 

0 + 1/(a×L2) + 2/(a×L2) + … + n/(a×L2) = 1/2×(0 + n/(a×L2)×n = n2/2×a×L2 = v2/2×a (formula 3)

That corresponds to the kinematics of the uniformly accelerated motion and this validates and proves the Hypothesis.

The principle of optimal motion

This section is an explanation of the full version of the hypothesis [1] - with formulas repetition.

According to my hypothesis, the motion of physical bodies is discrete and has a quantum character. How do I depict this graphically? Which function is best for this purpose?

This is an exponential, it is a gradual function and corresponds to our inert world (see figure 9 below).

The main parameter of the motion is acceleration, so let's build a graph of motion quanta using the acceleration function.

The OGF response of the body is an exponent to the nominal acceleration value (OAB element), the deformation of the body. The body's redeformation or road elementary is accompanied by a drop in acceleration to 0, since the force does not act on the body (there is no opposition OGF, BAC element). The BAC element in time makes up 24.28% of the period T of the quantum of the motion, making it possible to call the motion more "discrete" than "continuous". Since the period of the quantum of the motion is very small, we perceive the nominal acceleration values ​​as "continuity."

To derive the formula for the exponential change in the acceleration of the quantum of the motion of a physical body, we apply two conditions:

1. The rate of change of the force acting on the body is equal to the reaction rate OGF of the body. In this case, the most optimal (effective) quantum of motion is obtained.

2. As a basic system, we consider the surface of the Earth and the bodies falling on it. This is the most optimal natural system for analyzing changes in the acceleration of a quantum of motion. In this case, the Earth's gravitational field pulsates (see figure 5, moment 2).

We derive the formula for the optimal change of the acceleration of the quantum of the motion of the body m (1): 

a = (1/m)×(exp(t×L×g) – 1) (formula 4)

a is the function of the optimal change of the acceleration of the quantum of the motion of the body m

g - is the intensity of the gravitational field of the Earth; t – time.

It is clear that the formula for changing the intensity of the Earth's gravitational field for the quantum of motion (falling) of bodies near the Earth's surface looks like this, because this formula does not depend on the mass of the falling body:

a = exp(t×L×g) – 1 (formula 5)

Formula (4) turns into formula (5) at m = 1 kg. And under this condition, we will build a graph of these formulas (see Fig.10). S1 × L = 0.7572 is a constant dimensionless quantity regardless of the value of L. How many times does L increase; S1 decreases by the same factor. But to calculate the optimal acceleration of various masses, we use L = 1 s/m. The optimal acceleration is calculated graphically by using the program Graph. These are the values ​​of the function according to formula (4), limited by the area S1.

Thus, a practical conclusion from the hypothesis of discrete motion of physical bodies and the basis of the Principle of optimal motion: for each mass of various moving objects (people, vehicles, industrial equipment when switched on, alternating current - as the mass of electrons, and so on, its optimal (effective) acceleration, which makes it possible to increase the efficiency of the motion. For example, for physical bodies with a mass of 1000 kg and more optimal (effective) acceleration aopt. = 7.4 m/s2, which is presumably applicable to optimize the motion of the missiles and modern aircraft.

Experimental confirmation of the hypothesis

I applied the principle of the optimal motion to experimentally prove the hypothesis (formula 1). The experiment consisted of measuring the efficiency of a direct current electric motor with a certain mass on its shaft at various accelerations at the moment of switching on. The maximum value of the efficiency is fixed at the optimal acceleration of the electric motor at the moment of its switching on, which confirms the hypothesis (see figure 11, below).

In addition to this experiment, a similar experiment was carried out, which also confirms the hypothesis (see video below):

In addition to my experiments confirming the hypothesis, in my opinion, was the discovery of argument fluctuations. In 1968-1969, the brothers Danil and Yakov Douboсhinski discovered the Macroscopic Quantum Effect based on an argument pendulum (figure 12 below). It consists of a low-friction pendulum suspension that moves in a vertical plane, at the free end of which a small permanent magnet is attached. An electromagnet is installed under the balance point of the pendulum. Electric current is supplied to the electromagnet with a frequency of 20 to 3000 Hz. The pendulum "selects" certain stationary quantum amplitudes of oscillations, at which it fully compensates for the energy consumption for the friction.

The fundamental significance of the Doubochinski pendulum lies in its ability to "jump" from one oscillatory regime to another (from one amplitude to another) like "quantum jumps" in atomic physics. In other words, Professor D.B. Doubochinski practically proved that mechanical motion (in this case, oscillatory) has quantum properties, that is, discontinuously (discretely). Here I should mention that Professor Doubochinski welcomed my hypothesis and suggested a theoretical proof of the hypothesis ( see formula 3).

AN Analysis of the VALIDITY aND reliability of the hypothesis

In 1924 Louis de Broglie put forward the hypothesis that dualism is not a feature of optical phenomena only, but has a universal character. Particles of matter also have wave properties. According to the hypothesis of Louis de Broglie, any body is a wave (the dual nature of physical bodies). According to my hypothesis, the motion of the physical bodies is a periodical oscillatory process, that is, a wave process. So my hypothesis is very close to that of Louis de Broglie.

In 2012, the group of Anton Zeilinger (University of Vienna) conducted a series of experiments to study the wave properties of C70 fullerene molecules in order to confirm Louis de Broglie's hypothesis. It was found that the wavelength of a molecule is equal to the vibration amplitude of its sphere. By analogy, I equated the road elementary of the quantum of the motion of a body m = 1 kg, falling in the gravitational field of the Earth (according to my hypothesis), to its wavelength according to Louis de Broglie, in order to approximately theoretically determine the value of the parameters of the quantum of the motion: period, road elementary and elementary speed.

The formulas obtained, are:

 L = 3√m/g×h = 3√1/9.8×6.6×10-34 = 5.37×1010[s/m] (formula 6)

Where h is Planck's constant.

T = 1/g×L = 1.9×10-12[s] (formula 7)

1/L = 0.186×10-10[m/s] (formula 8)

X = 1/g×L2 = 3.5×10-23[m] (formula 9)

In the attributions [2], a theoretical study of the quantum of motion of the planets of the Solar System was also carried out to analyze the areas of the exponents of the optimal change of the centripetal accelerations for the quanta of the motion of the planets on the basis of the formula 4. It is concluded that the areas of the exponents of the optimal change of the centripetal accelerations of the quanta of the motion of the planets of the solar system or the modules of the elementary change of the vectors of the velocities of the planets by the direction are constant for all planets S1 = 1/L ~ 1.86 * 10-11 [m/s] and S2 => 0.

It means our solar system can be an indicator of the consistency of the hypothesis of the discreteness of the motion of the physical bodies.

I present to you some complementing aspects to consider, all derived from my hypothesis:

• The electromagnetic nature of gravitation is, in connection with the discrete frequency of the motion of physical bodies, ~500 GHz (see formula 7). According to the hypothesis, Earth emits a pulsating gravitational field with the same frequency and enters into resonance with physical bodies, causing them to move (fall). The Sun emits a pulsating gravitational field with a frequency of ~500 GHz and enters into resonance with the frequency of the discrete rotation of the planets (with the frequency of the change of the modules of the elementary change of the vectors of the velocities of the planets by the direction), which causes the rotation of the planets around the Sun. That is, presumably, electromagnetic radiation in the terahertz range of ~500 GHz, which has a similar penetrating ability 

• The explanation of the "mysterious thrust" of the EmDrive engine by British engineer Roger Shawer. The magnetron of this motor emits electromagnetic waves at a frequency of 2.5 GHz. According to the hypothesis, this is only a few orders of magnitude less than the discreteness frequency of motion of the entire mass of the engine, which causes an insignificant thrust comparable to the measurement error. If the frequency of the magnetron or other emitter is increased to ~500 GHz, then the thrust of the EmDrive engine will probably become noticeable. Based on the above, I am assuming that the Emdrive is the prototype for the future "wave motor"

EPILOGUE

My hypothesis, and its mathematical and experimental foundation, prove that the motion of physical bodies has mechanical properties of “discreteness” and “continuity” with a predominance of “discreteness”; it shows us how movement is logically possible while still honoring Zeno’s aporias.

Yours truly, Vladimir Reznikov - you are welcome to contact me by email for further questions.

Attribution:

[1] V. A. Reznikov, “The principle of the optimal motion (full version)” / URL:http://cdn.scipeople.ru/materials/70765/The%20principle%20of%20the%20optimal%20motion.pdf

[2] V. A. Reznikov, “Theoretical analysis of reliability of the "Hypothesis of the atomic (quantum) motion" / URL: http://cdn.scipeople.ru/materials/70765/Theoretical%20....pdf   

Photos via Google & Illustrations © (2021) by V. A. Reznikov