Through an original, purely mathematical and very elementary, study, I will demonstrate:

That Lorentz' formulas are interpretable in the classical space-time (euclidian
space-time),

That they have very fundamental natural mechanical representations,

That these formulas do not exclude Galilean transformations ,

That speeds greater than ** c** ,

transformations.

That the Einstein's principle of relativity is not the consequence of these transformations.

__Beginning of the study__

As a simplification the formulas are studied here only in a one dimensional
space *.*

Let ** S** an axis

This means that the motion of

By an adequate choice of the origin

Therefore we will take the equation

If ** E **is an event of coordinates

by définition of Lorentz' transformations :

Given the two systems** S **and

In the next, one puts:
** b **is a constant dépendant only of

page 1

__Another expression of__* t _{1}*

From ** (1)** we have :

If we replace

Otherwise

this last formula brings out :

1. The slowing of clocks
of ** S_{1}** , slowing proportional to the multiplicative
factor

2. The phasing-out of clocks of

term proportional to

And more over, the formula

In the following, I will represent systems

These never seen figures will allow us to catch the full meaning of Lorentz' transformations thoroughly.

__Representation of S and S_{1}
in the particular case where__ :

With these values one has

that give for ** x_{1}** and

Otherwise :

The unit of length in ** S** is considered to be equal to
2 centimeters .

These various values enable convenient vizualizing of

The different drawings of the figure 1 are obtained in the following manner.

For drawing 1, one has ** t=0** , and gives

*
x _{1} = 2x - 6t ; t_{1 }= 2t - 1/2 x .*

With

For example, when ** x** =

This mean that the point of

(at the date

indicates

Page 2

Similarly for

The point of

(in

Similarly for all the other values of ** x**, always with

To obtain the second drawing, one takes ** t =1** , that gives

vary as previously, which gives the system

One proceeds similarly to obtain drawing 3 by taking

In the drawings, the only times which are indicated are given by clocks placed on integer abscissas of systems.

The drawings of figure1 lead to the following remaks:

1. Units of length of ** S_{1}** (in the direction
of the speed) are constant, and smaller than the units of length of

In the drawings of figure1, the unit of

2. In ** S_{1}** , on each point, there is a clock,
clock that is used to date events that happen in this point of

When in

the clock of

An event taking place in Paris is dated by the Paris time zone, and not by that of Moscow.

**E _{1}** arise in

the date

This same event arise in ** S_{1}** at the point of
abscissa

the clock of

3. Simultaneous events in one system are no longer simultaneous in the other.

Let me first define what are two simultaneous events in a system.

Two events (two explosions, for example) are
simultaneous for a system, if, and only if the dates given by

the two clocks of the system that are found
in the places (at the core) of these events, indicate the same

numbers (the same time, the same date).

__Example__: The event **E _{1} **of abscissa

(an event indicated on the first drawing of figure 1), and the event

of date

are simultaneous relatively to

One notices that the coordinates of **E _{1}** in

Similarly, two simultaneous events in ** S **, not having the
same abscissa, are not simultaneous in

On the drawing no. 2 of figure1, the event

The event

The coordinates of

The coordinates of

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More simply by referring to the time zones of the earth, an explosion that occurs in Paris at 12 noon,

the time in the Parisian time zone, and an other explosion that happens in New-York at 12 noon,

the time in the New-York time zone, constitute, two simultaneous events, relatively to the

terrestrial time zone system. Relatively to an other time measurement system, these two events are no

longer simultaneous.

4. The measure of the speed of a moving body, relatively to a moving
system of reference,

must refer to a more precise definition than
commonly admitted, a definition that must

exclusively refer to clocks and abscissas
(or coordinates) of the moving referential system.

Let me define therefore the speed of a moving body, relatively to a
system of reference.

The definition is given in *S _{1}*

Let **M** be a point in translation on the ** S_{1 }**axis.

(date given by the clock of

Then **M** passes on the point of abscissa ** x'_{1}**
of

( date given by the clock of

The average speed of **M **, relatively to ** S_{1}**
,between points

The limit of this average speed when ** x'_{1}** tends
toward

at the point

__Example__ : Let us resume our drawings (see figure 2). Let **M**
be a point in translation along axis ** S_{1}**,

therefore along axis

is assumed to be constant.

In the first drawing of figure 2, **M **is found on ** x_{1}**
=

(date given by the clock of

In the second drawing of figure 2 , **M **is found on ** x'_{1}**
=

(date given by the clock of

The measured speed of **M,** relatively to ** S_{1}**
is :

(

The measured speed of **M, ** relatively to ** S **is
:

Remark : on the third drawing of figure 2, the coordinates of **M**
, relatively to ** S_{1} **are

We observe that (

This allows us to observe empirically that the speed of

If **M** is moving, relatively to ** S** , at the speed

that the speed of

in the previous case. I leave to the reader the care of this expériment.

I advise remaking drawings of the figures, taking ** c** =

Drawings, as well as the representation of

5. The measure of the length of a mobile segment, relatively to a system
of reference (in movement)

also has to refer to a far more rigorous definition
than that given by intuition.

This definition must refer exclusively to the abscissas(or coordinates) and the clocks of this system of reference.

The definition is: let **AB** be a straight segment
parallel to ** S_{1}** , a mobile or immobile segment
relatively to

The origin

of

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If the extremity

shows the same date

This number represents the difference of the abscissas, of the points
of ** S_{1}** , from which one sees simultaneously,

relatively to

This definition is valid in all referential systems.

__Example__: let the segment **AB ,** motionless relatively to**
S** .

**A** of abscissa ** 0** in

On the first drawing of the figure 1, the origin

and the clock of

On the second drawing of figure 1, the extremity **B** of **AB**
is found on the point of abscissa ** x'_{1} **=

of

Therefore, in ** S_{1}** , at " the instant "

and the extremity

Therefore, the lenght of segment **AB **is ** 2**
in

Any segment of length

We can see immediately on the drawings of the figure 1 , that a segment
of length ** 4 **, taken on

(and measured in

Page 6

__Mechanical systems that verify Lorentz' transformations.__

1. System built (arbitrary appearance) .

Let a motionless ** Ox** axis in a classical space time. This
axis is noted

Let an

relatively to

Axiomatically we take the unit of length ** S_{1}**
to be half the unit of

we place a clock , clocks that progress at the same speed twice as slow as those of

and that are phased out, each one compared to the other, phasing out being

of abscissa of

And moreover, the clock that is placed on

More rapidly, on ** S_{1}** we measure, abscissas and
times as indicated on the drawings of figure 1.

The manner to measure times in

The events taking place in Paris are dated by Parisian clocks, and those
taking place in New York are dated

by the clocks of that town. When one says that an explosion happened
at noon in New York ,

it is always implied that the hour is given by the clocks of that town.
Similary in all other cities.

Let us return to our systems. Let **E** an event (an explosion, for
example) that happens on the point of abscissa ** x** of

at the date

The abscissa ** x_{1} **measured in

in the center of this event

are given by :

With :

One passes from ** S** to

on the drawings of the figure 1, or the figure 2 .

One notices that knowing ** x_{1}** and

therefore by the formulas :

Page 8

We pass from ** S_{1}** to

Therefore we have two materially constructed systems, such that we pass
from one to the

other by the Lorentz' transformations of same constant ** c**
.

The symmetry of these transformations does not in the least imply that
** S **and

are analogous from a material viewpoint. In our drawings, systems are totally dissimilar,

and by seeing them, no one would think that they are equivalent,

and nevertheless, one passes from one system to the other by the same transformations.

We will see later that these two systems are not at all equivalent in
the description of some

phenomena, especially moving bodies in translation in ** S **at
speeds greater than

(pseudo) paradoxes in

its time zones. A very fast plane that leaving Paris at noon (Parisian time) can arrive at eleven

o'clock in New York ( New York time) , and nobody would be shocked as one would be in relativity.

2° Natural system in classical mechanics.

One can wonder whether, in classical mechanics, there are systems that
naturally obey

Lorentz' transformations, ** c** being a constant attached
to systems.

The reply is yes,they are, and it is even astonishing to see that they are very common.

Let a perfet elastic cord ** Ox** , brought to an orthogonal
referential

Any tranversal wave (of direction ** Oy **) on the cord is
defined by two (any) functions

At any date ** t** , the point of abscissa

** c** being a positive constant attached to the mechanical
characteristics of the cord,

** y **=

The sum of these two waves gives the state of the cord at any date

The cord is oriented left to right.

Let ** t'** and

As previously ,

Page 9

Let us calculate

We find that :

Let us put :

*k _{1}*

**
ct' - x' **=

In the transversal perturbation ** y **=

**
y **=

Which gives a new perturbation (which is entirely possible) of the same vibrant cord.

This new perturbation is the old one, moving at ** v** speed
along the

It is what is shown in the program (relauk.exe) accompagnying this text.

One can observe that if **M** is a motionless point of the starting
perturbation , this same point is in translation

at speed ** v** along the

Example : the transversal perturbation of the cord, defined by :
** y **=

admits the point of abscissa

In the perturbation put at speed ** v** :

the point of abscissa

"Permanent" points of the starting perturbation , are converted to "permanent"
points at speed ** v** in the

perturbation put at speed

We now wonder :

What our clocks
are on the cord ?

And what
they become when the perturbation is put at speed ** v **? .

The same questions are asked about the units.

Here are the answers.

On each point of the cord , in a motionless system , times are simply
given by a stationary periodical

referential wave (or perturbation), the unit of time being proportional
to the vibratory period.

The unit of length is defined by the same referential wave, the chosen
unit being proportional to the distance

of two consecutive "permanent" points .

In any system put at speed ** v** along

along

Let us specify this with an example: it is the example displayed when
one starts the attached program.

Page 10

For a motionless system, we take as the stationary referential wave :

(One can take ** c **=

The points of abscissa ** x **=

The unit of measure is the distance divided by

On each point of the motionless system is a clock , a clock that advances
one unit each time the wave

crosses under its feet the ** Ox** axis.

(The vibratory period of this wave is

When we put the system in motion at ** v** speed along the

motion are given by the referential wave put at

(Constants ** k_{1}** and

Each clock of the system in motion, always advances one unit each time
the referential wave at speed ** v** crosses

the

"permanent" points divided by

The "permanent" points of the referential wave put at speed

Now , "permanent" points are points of abscissas ** x** =

One sees that "permanent" points have drawn nearer according to multiplicative factor

A close observation of the animation of the attached program allows us to perfectly understand the above explanations.

This natural way of measure time and distance with the help of a referential
wave, completely explains

Lorentz' transformations without exiting the framework of classical
mechanics.

Complement : if **Eo** is the total energy , between two "permanent"
points of the motionless "stationary wave" ,

the energy that takes into account the potential energy of the forces
exerted by the wave on the extremities of the system

(forces exerted on the two permanent points), we find that the total
energy **E v** of the system at speed

The démonstration is outside the limits of this elementary study.

The system behaves, relatively to the acceleration, as a mass of

The results obtained on a vibrating cord are generalized to three dimensional
elastic mediums.

__Experimental study of moving bodies travelling in S at
speeds greater than c .__

The aim of this short study is to underline some paradoxes that prove
that ** S **and

to coherently describe the motion of a body moving at speeds greater than

body becomes paradoxical (or incoherent) in

Page 11

Let us consider the drawings of figure 3.

Still with :

Let **M** be a moving body in translation at speed ** v **=

At ** t** =

and the clock of

At the date ** t'** =

and the clock of

The speed of **M** measured in the referential ** S_{1}
**is : (

The measure of the speed of **M** , relatively to ** S_{1}**
, gives a value greater than that obtained in

beeing greater than

a smaller value than that obtained in

This paradox is simply due to the way of measuring times and distances
in ** S_{1}** .

Now let **N** be another moving body in translation at speed ** 4**
relatively to

( the time equation of

In

of

From this point of view ,

Now, let the body **P** moving at speed ** v** =

(the time equation of

At the date

At

As previously, with the help of figure 3 , we find that the coordinates of

In ** S_{1}** , we have therefore : the starting date
of

the arrival date of

In

Let us present this otherwise: in

This bullet arrives on ** x'_{1}** =

arrives

This paradox is analogous to those created by terrestrial time zone gaps.

Once again , ** S_{1}** is inadequate for a coherent
description of some phenomena.

This fact therefore allows to determine which of ** S** or

The motionless system is that in which no paradox exists, it is that in which the effect never precedes the cause.

It is always possible to add to ** S_{1}** a Galilean
system

( clocks and units of

In

The simultaneous existence of different types of referential systems
is not in the least contradictory.

A system of Lorentz (relativistic system) , can always be doubled by
a Galilean system in which it is easier to reason.

Page 13

The thoeretical study of speeds greater than

The relativity by the waving strings.

But what has been explained is enough to understand what happens.

__The paradoxe of twins__ : this "paradox" is famous, I'll quickly
mention it.

Let's take a system ** S_{2}** that coincides with

(at the end of the experiment , the origin

We notice then that the clocks of ** S_{2}** are late
compared with those of

accelerations of

In the attachedprogram, the T key allows this experiment.

We can also draw different pictures as those of figures 1,2,3 with three systems to visualize this paradox.

This paradox is explained by Lorentz' formulas, without the help of any other theory.

Cabala Serge . 1975 , 1976, 1977 , 1980 , .... ,1999 , january 2000

P.S.

My mains works on this subject are:

- This study.

- Relativity by Waving Strings. (1975-76) classified in Archives Originales
du CNRS (France) from 1977 (183 pages)

- Relativity by Elastic Mediums (1981, 185 pages)

- The Changes of variables that convert all functions of d'Alembertian
equal to zero into functions of same property ,

with extension to Klein-Gordon equations (1988, 129 pages)

All these works have been accepted , and an ex-director of the French
CNRS had underlined its quality.

They are neither taught not published on a large scale in order to
preserve the traditional relativity.

It is to be regretted to see that science has been transformed into
religion , by maintaining some unnecessary

and constraining dogmas , such as the principle of relativity and its
limit speed.

The formulas of relativity being only classical mechanics formulas
of elastic mediums, it is

unnecessary to wish to establish them from new principles, that arbitrarily
limit their applications,

and introduce a degree of strange thought .

When one speaks about relativity, one always quotes the Michelson and
Morley experiments,

done in 1881 and 1887 , on the detection of the wind of ether relative
to the earth.

Let us reflect a bit, after having read what precedes and use the program
on waves.

If electromagnetic fields are only particular "waves" of the ether
considered as an elastic medium,

the only manner to put these "waves", or electromagnetic fields, in
translation, is to use Lorentz' formulas,

and this in the framework of classical mechanics.

Atoms of matter are held together by "stationary" electromagnetics
fields .

The setting in uniform lengthwise motion of a ruler of any material,
puts the "stationary waves"

in motion , these "waves" are contracted in the direction of the motion
and are phased out (see the program) ,

the ruler's atoms are compacted and the ruler is shortened.

The wind of ether becomes thus undetectable by this experiment.

The explanation of the negative result of this experiment, by the contraction
of bodies in motion,

had already been given by Fitzgeral (1893), but the actual atomic constitution
of the matter, not being known

(the atomistic constitution of the matter being even hardly denied
by famous scientists) ,

this mechanical explanation seemed very difficult to admit.

One could not conceive, that materials of very different hardness at
same velocity, can be contracted

in the same way.

The Michelson and Morley experiment , finely analyzed is in favor of
the ether.

Page 14

This is near the analysis of the phenomenon of tides made in 17 th century, by Newton's detractors.

For these detractors: if the moon exerts an attraction on the earth
, a sea swell must exist

in front of the moon and a hollow at the antipodes side.

But there are always two sea swells, one in front of the moon and another
to lunar antipodes,

there is therefore no attraction.

The phenomenon of tides roughly explained by the universal attraction
is opposite to reality,

but the universal attraction finely used with acceleration (what I
do not do here) explains

perfectly why there are two swells.

It is similary with the Michelson and Morley experiment : roughly analized,
it is in disfavor of the ether,

finely analized, it becomes one of the proofs.

I hope I have communicated the pleasure of at last a true Cartesian relativity.

Page 15