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

where **R** is radius and **r _{1}** is linear curvature.

Process of motion for the line will create the dynamical surface, i.e. two-dimensional space-time (sphere), and the resonance parameter for it is the two-dimensional curvature (it also disclosed in any mathematical textbook):

**
r _{2} = 2/R [ 1/m ]
F.2**

** Let's introduce the spatial curvature
**by similar arguments:

**
r _{3} = 3/R [ 1/m ]
F.3**

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

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

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

**
r _{4} = 4/R [1/m ]
F.4**

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

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

*There is other logical branch for this idea* *and this way
lead to more wide concept of mass:*

Instead of **E=pc**, by de Broighl, let us take the formula for energy
of mass

**
E =mc ^{2 }
F.5**

Energy of electromagnetic field energy (wave energy) is

**
E = hf
F.6**

In strength of the wave-particle duality wrote the equation:

**
mc ^{2} = hf
F.7**

and mass can be represented as electromagnetic oscillations

**
h**
**
m = -------- f
F.8**
**
c ^{2}**

where* (h/c^{2}) *is new constant

**
h 1**
**
m = ------- ------
F.9**
**
c ^{2} T**

where

In other words,

**
h**
**
mT = ----- = const
F.10**
**
c ^{2}**

The chronal constant is ratio of elementary quantum of
action ( h ) to the square of the velocity of light ( c^{2} ) and
it is equal to **0.73725 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 the Heizenberg's formula

**
h = Dp Dx
F.11**

mass-time equation can be represented in new view

**
Dp Dx**
**
m T = -----------
F.12**
**
c ^{2}**

Let's check the measures correctness in this equation

**
[kg][m][m][s2]**
**[kg][s] = ------------------- = [kg][s]
F.13**
**
[s][m2]**

So, F.9 is true equation that demonstrate the correlation between some mass and period of electromagnetic wave oscillations and next conclusion is obtained: Mass of particle is result of electromagnetic energy oscillations..

By such sort approach the frequency can be calculated by F. 8
for any known mass value, for example, proton has frequency value about
**8,1 10 ^{26} [Hz].**

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

** **
**
Dp Dx
m Du Dx**
**
mT = --------- = --------------
F.14**
**
c ^{2}
c^{2}**

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

**
DuDx**
**
T = ------------
F.15**
**
c ^{2}**

For velocity **u=c **
the wave-length

**
c c Dx**
**
L = cT = ---------- = Dx
F.16**
**
c ^{2}**

It is ** photon state**: the wave-length value of the object
determine the size of the local area (space) of the object.

For velocity

**
L = (u/c)Dx
F.17**
** **

It means that object has some space of positions, object can
be moving between different positions since all area of its positions is
determined. ** It is the nature of space itself **for some material
object that is moving with

For velocity **u
> c**, formula F.17 can be represented as

**
DuDx
(c + u)Dx**
**
T = ---------- = -----------------
F.18**
**
c ^{2}
c^{2}**

and wave-length

In result the formula for wave-length demonstrate that in this case
the *size of the object (wave-length) is more that possibility of
determination of the object position Dx:*

**
L = (1+ D/c) Dx
F.20**

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

Before the continuation 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 in matter wave-length formula

**
L = h/(mu) [ m ]
F.21**

where **h** is Plank's constant, **m** is mass and **u**
is velocity, parameters of our planet to calculate the value:
**
L = 3,725 10 ^{ - 63} [ m ]
F.22**

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

**
E _{3} = m_{3} c^{2} = 9 m_{3}
F.23**

and

where **m _{3}** is mass of 3-space and

Note important supposition: The total energy of system is the same independently of different dimensionality description. 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, total net energy of the Universe is the same independently of dimensionality of measurement system that is used by observer.

So, we have to write:

**
E _{1} = E_{2}= E_{3} = E_{4} =...
F.25**

and in our case**
9 m _{3} = 16 m_{4}
F.26**

According to F.21 mass is

**
m = h/(Lu)
F.27**

Now we obtain the correlation

**
(16 h )/( L _{4 } u)= (9h)/(L3_{
}u)
F.28**

where** L _{3}** is wave-length in 3-space, and

**L _{4} = (16/9) L_{3}
F.29**

Substitute value for **L3** from F.22 in F.29 and obtain the
value

**L _{4} = 66,22 10^{- 64} [ m
]
F.30**

that conform to curvature

**
r _{4} = 1/L_{4} = 151,00 10^{ 60}
[ 1/m ]
F.31**

Note: it is 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 = 1/T = 3,16886 10 ^{ - 8} [ 1/s ]
F.32**

It is own resonance frequency for our planet.

The wave-length for electromagnetic oscillation in this case is

**L _{ em} = c/f = 9,46 10 ^{16} [ m ]
F.33**

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

**
r _{em} = 1057,00 10^{- 20} [ 1/m ]
F.34**

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

**
(r _{em}/r^{4})=7 10^{-80}
F.35**

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

The curvature for Bohr's radius **R = 0,52917 Angstrom **is

**
r = 1/R = 3,0075 10 ^{ 9} [ m ]
F.36**

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

**
r _{1} = r_{3} /3 = 1,0025 10^{9} [
m ]
F.37**

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

**
h
h c
h**
** m = ------
f = ------ --- = --------
F.38**
**
c ^{2}
c^{2} L
c L**

**
h**

or ** L = -------**
that is version of de Broighl wave-length for velocity** u
= c.**

**
mc**

So, for proton **m = 1,6726231...10 ^{-27} kg**, the
wave-length

**
r = 1/L = 132141,000... ** **
F.39**

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

Another example is DNA helical molecule. The unfolded spire-length period
of it is about
**71,4417 Angstrom, **that conforms to a curvature value

**
r _{DNA} = 14,0000 10^{ 7} [ 1/m ] **

This is very good resonator since it is whole number value up to fourth sign. This accuracy is the base for supposition about resonance mechanics for energy transformation and information telecommunication on the molecular level in biosystems. The nature of Time Effect is motion only and resonance structure of matter elements superposition for DNA molecule allow 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 have place. So, orientation for such sort receiver is not important. Information signal is change of energy density in space.

Now let's make calculation for displacement of spiral DNA branches. Two branches have period 34 Angstrom and displacement 23,8 Angstrom. In other words, the 'back wave' is displaced relatively to the 'direct wave' on 0,7 of the wave period. It is equal to 50,0 Angstrom and correspond to curvature value

**
r _{2} = 2 10^{8} [ 1/m ] **

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

**
r _{1} = 1 10^{8} [ 1/m ] **

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

It would be premature to say about the validity of this
concept since the assumptions underlying it are opened to question. It
is necessary to make the experimental verification of this approach to
the resonance nature of matter and vacuum. Physical sense of the
action is the motion in some direction that is placed outside of the object.
The natural actions are whole numbers, it is discrete counter of the space-time.
The Law of Whole Number Value for curvature of natural element of matter,
demonstrated in this paper, is mathematical tool for different technologies.
In case of its practicable 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 show that any natural system, for example, the
planet, atom or DNA molecula can be used for calculation of the "meter/sec"
system units. There is possibility to establish new standard metre definition.

I believe now you have clear view to the **action as curvature and
the action as the number**.

**Alexander V. Frolov**

Scientific Expert of Russian Physical Society

P.O.Box 37, 193024, St.-Petersburg, Russia

This paper was published at first in Proceedings of International Scientific Conference "New Ideas in Natural Sciences", June 1996, St.Petersburg, p.123-135, "PiK" Co.,1996. Copyrigth 1996-1999 Alexander V. Frolov. All rights reserved.