"The Action"

by Alexander V. Frolov

Scientific Expert of the Russian Physical Society
P.O.Box 37, 193024, St.-Petersburg, Russia
Tel: 7-812-2747877
Email: frolov@mail.dux.ru

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"Action is a curvature of the World"
Pavel D. Ouspensky, A New Model of the Universe, 911

"At any conditions the action is just the number" Herman Weyl, Gravitation and electricity, 1918


This epigraph discloses the technology for total time rate control, and I hope that this paper will allow you to use the mathematical tool I found and joined with a well-known conclusion about quantum of action. This new interpretation for the notion of Time lets us develop a new branch of physics: the physics of space-time engineering. Several examples demonstrated in this paper some natural law that is discovered by the author in 1990. It was found that natural mass-objects are described by whole number values of the curvature. This property allows us to consider the space-time as a secondary induced resonance effect of the mass process. The concept can be used as an applied tool for energy-mass and mass-energy transformations, gravitational propulsion systems and experimental research work on the chronal technologies.

Time as a Radius of 4-dimensional Resonator:

The motion of a point along a closed trajectory is the process that creates the dynamical one-dimensional space-time, i.e. the line. It is a process and there is some period of the process. If the line is closed there is a resonance phenomenon and parameter radius R. According known mathematical notion the curvature of this one-dimensional space is defined as 1/R:

F.1:   r1 = 1/R [ 1/m ]

where R is the radius and r1 is the linear curvature.

The process of motion for the line will create the dynamical surface, i.e. two-dimensional space-time (sphere), and the resonance parameter for its two-dimensional curvature (also according to any mathematical textbook):

F.2:   r2 = 2/R [ 1/m ]

F.3:   r3 = 3/R [ 1/m ]

It is the parameter for resonance processes of the dynamical structure of three-dimensional objects.

It is possible to use the radius as a description for periodical processes in one-dimensional closed space of circumference only. So, some value of radius that is measured in (N+1) space is a period of time for processes in dynamic N-space. The "time axis" as radius for circumference is direction that have place out of line. It is a new direction, a new dimension.

The dynamic structure of 4-dimensional objects is created as a change of 3-dimensional structure in a new direction. The parameter for it is a 4-curvature:

F.4:   r4 = 4/R [1/m ]

This understanding of the time nature allows us to suppose some methods to create a local rate of time. In any case, it is necessary to change the density of energy in space. For one-dimensional space (line) it is a linear density of energy, for example: the well-known density of an electric current. For its surface is the energy of electromagnetic wave.

Also it is possible to change 3-dimensional density of energy (electromagnetic energy or density of matter) in a volume of space to create a local time effect.

Electrodynamic Nature of Mass-Effect:

In 1923, L. de Broglie proposed that mass - particles must have wave properties. He used the formulation E=hf and E=pc, where p is impulse, h is the Planck constant, f is frequency, m is mass and c is light velocity. Then he joined both parts in the equation hf=pc. For a wave-length =c/f this formula results in =h/p [1].

There is other logical branch for this idea which leads to a broader concept of mass. Instead of E=pc, by de Broglie, let us take the formula for energy of rest mass:

F.5:   E = mc2

The energy of electromagnetic field energy (wave energy) is:

F.6:   E = hf

In strength of the wave-particle duality the equation is written:

F.7:   mc2 = hf

and mass can be represented as electromagnetic oscillations

F.8:   m = hf / c2

where (h/c2) is a new constant between the mass (m) and frequency of oscillation (f). Let us call it a chronal constant, because it demonstrates mass and time correlation:

F.9:   m = (hf / c2) (1 / T)

where T = 1/f is the period of oscillations.

In other words, the product of the mass and period is a constant value

F.10:   mT = (hf / c2) = constant

The chronal constant is a ratio of the elementary quantum of action (h) to the square of the velocity of light (c2) and is equal to 0.73725 x 10-50 [Js2/m2].

In other words, there is no time separately from mass, by F.10. Main masses of our time-system are masses of the Sun system.

Taking into consideration Heizenberg's formula:

F.11:   h = p x

the mass-time equation can be represented in a new view:

F.12:   mT = px / c2

Let's check that the measures are correct in this equation:

F.13:   [kg][s] = [kg][m][m][s2] / [s][m2] = [kg][s]

So, F.9 is a true equation that demonstrates the correlation between some mass and period of electromagnetic wave oscillations. The next conclusion is obtained: Mass of a particle is the result of electromagnetic energy oscillations. By such an approach, the frequency can be calculated by F. 8 for any known mass value. For example, a proton has a frequency value of about 8.1 x 1026 [Hz].

Over-Light Velocity:

Interesting conclusions for velocity of motion can be obtained from F.12:

F.14:   mT = px / c2 = m u x / c2

We can remove "m" from F.14 and obtain the next formulation:

F.15:   T = u x / c2

For a velocity u=c, the wave-length:

F.16:   = cT = c c x / c2 = x

It is a photon state: the wave-length value of the object determines the size of the local area (space) of the object. For a velocity 0 < u < c, the wave-length of an object is lesser than possibility of determination of the object at position x:

F.17:   = (u/c) x

It means that the object has some space of positions, object can move between different positions since all area of positions are determined. It is the nature of space itself for material objects are moving with u < c.

For a velocity ut > c, formula F.17 can be represented as:

F.18:   T = ut x / c2 = (c + u) x / c2


F.19:   Tc = c/f = = (c + u) x / c

The results of the formula for wave-length demonstrate that with the size of the object (wave-length) it is possibile to determine the objects position x:

F.20:   = (1+ u/c) x

Demonstrations of such objects in real space has some analogy with potential fields since the change of energy density of the object takes place in all space at the same moment.

Planet Curvature Value:

Before continuing, let us assume that the theory of similarity for microcosm and macrocosm is true, hence the planet can be considered as elementary particle in certain sense.

Substitute matter into the wave-length formula:

F.21:   = h/(mu) [ m ]

where h is Plank's constant, m is mass and u is velocity of our planet to calculate the value:

F.22:   = 3,725 x 10-63 [ m ]

In strength of the supposition that velocity of light for 4-space is only factor 4:

F.23:   E3= m3 c2 = 9 m3


F.24:   E4= m4 c2 = 16 m4

where m3 is mass of a 3-space and m4 is mass for a 4-space description.

Note this important supposition: The total energy of a system is the same independent of different dimensionality descriptions. It means that the same amount of energy (but in different forms of energy) must be considered in 3-space and 4-space description for one certain system. In other words, the total net energy of the Universe is the same independent of the dimensionality that is used by anobserver.

So, we must write:

F.25:   E1 = E2 = E3 = E4 =...

and in our case:

F.26:   9 m3 = 16 m4

According to F.21 mass is:

F.27:   m = h/(u)

Now we obtain the correlation:

F.28:   (16 h)/(4u) = (9h)/(3u)

where 3 is the wave-length in 3-space, and 4 is the wave-length is 4-space. Since the system is the same, as velocity is the same. So, there is simple ratio:

F.29:   4= (16/9) 3

Substitute the value for 3 from F.22 in F.29 and obtain the value:

F.30:   4 = 66,22 x 10-64 [ m ]

that conforms to the curvature:

F.31:   r4 = 1/4 = 151,00 x 1060 [ 1/m ]

Note: it is a whole number value.

On the other hand, the period of planet rotation around Sun is equal to 31557600 seconds, that conforms to the oscillation frequency value:

F.32:   f = 1/T = 3,168861 10 - 8 [ 1/s ]

The wave-length for electromagnetic oscillation in this case is:

F.33:   em= c/f = 9,46 x 10 16 [ m ]

and curvature for this length (radius of resonator) is equal to the whole number value also:

F.34:   rem = 1057,00 x 10-20 [ 1/m ]

The correlation for the two results obtained in F.31 and F.34 is:

F.35:   (rem / r4 ) = 7 x 10-80

The mathematical connection for results of the two different descriptions of he same natural object (planet) is the confirmation for the Whole Number Value Law. This Law is valid for any natural system element. Let us demonstrate this Law in other examples.

Bohr's Atom Space-Time:

In F.35: The curvature for Bohr's radius R = 0,52917 Angstroms is:

F.36:   r = 1/R = 3,0075 10 9 [ m ]

According to F.3, the linear curvature for Bohr's atom is equal to the unit:

F.37:   r1 = r3/3 = 1,0025 x 109 [ m ]

That is right since it is a simple atom, a unit matter engine. Some deviation k = 1,0025... is a demonstration of a non-ideal resonance state in a real system.

Space-Time of Proton:

To calculate the wave-length of proton let's use formula F.9:

F.38:   m = hf / c2 = h / c2 x c / = h / c

or = h / mc which is the version of de Brogle wave-length for velocity u = c.

So, for a proton m = 1.6726231...10-27 kg, the wave-length = 0.75676739... (mathematical degree is omitted) and the curvature is equal to:

F.39:   r = 1/ = 132141.000...

It is a whole number value up to the third sign that demonstrates the particle proton as some spatial resonance process.

DNA- molecule Space-Time:

Another example is the DNA helical molecule. The unfolded spire-length period of it is equal to 71.4417 Angstroms, that conforms to a curvature value:

F.40:   r DNA = 14,0000 x 107 [ 1/m ]

This is a very good resonator since its whole number value goes up to the fourth sign. This accuracy is the base for the supposition about resonance mechanics for energy transformation and information telecommunication on the molecular level in biosystems. The nature of the Time Effect is a motion only and resonance structure of matter elements superposition for a DNA molecule that allows us to use this motion as power source and information receiver for biosystems.

Note that information is not spreading in this case with some velocity but all 3-space is changing at the moment when some curvature of 3-space hs place. So, orientation for such sort receiver is not important. Information signal is change of energy density in space.

Now let's make a calculation for the displacement of spiral DNA branches. Two branches have a period of 34 Angstroms and a displacement of 23.8 Angstroms. In other words, the 'back wave' is displaced relative to the 'direct wave' on 0.7 of the wave period. It is equal to 50.0 Angstroms and corresponds to the curvature value:

F.41:   r2 = 2 x 108 [ 1/m ]

If you take into consideration the two-dimensional structure of the spire the curvature value is unit:

F.42:   r1 = 1 x 108 [ 1/m ]

So, branches of the DNA molecule have a unit value shift (by the curvature measurements method) from the zero-state. In this zero state theoretically both branches are joined together. Code structures of each branch are contra-directional to each other. There is a very interesting analogy here with the elementary particle world that can lead to the secret of the electron-positron pair creation process.


It would be premature to speak about the validity of this concept since the assumptions underlying it are open to question. It is necessary to make the experimental verification of this approach to the resonance nature of matter and vacuum. The Law of Whole Number Value for the curvature of natural elements of matter, demonstrated in this paper, is a mathematical tool for different technologies. In the case of its practicality the control on the rate of time and curvature of space, on the stability of matter (mass-energy transformations) and vacuum (energy-mass transformations) is not fantasy but technology.

Also, this Law shows that the absolute natural system for calculations is a decimal system and the absolute natural system for measurement is "meter/sec" system. There is the possibility to establish a new standard meter definition.

Since absolute systems are impossible in accidental phenomenon, the Law of Whole Numbers for curvature values of elements of matter show the material World is created. This is the main conclusion of this paper.