17. CONTRACTION OF SPACE
At first man studied the space around him to the limit of horizon where
the sky is joined with the earth. In the course of time, after many years
of evolution he widened that horizon to billions and more light years and
narrowed it down to a dimension of elementary particles. On that long journey
there were a great jumps ahead, and sideways as well, which slowed down
the rhythm of man's penetration to the unknown. The theory of relativity
has both possibilities, to be the great penetration to the unknown, and
to be the sideways which turns aside the course of research and in that
way slows it up.
The question of space and time is of fundamental importance, not only
in the theory of relativity but in physics in general. This is why no theory
can be accepted if it does not treat these two concepts correctly.
Until the appearance of the theory of relativity, space and time were
two separate entities and they were treated as absolute magnitudes. In
the theory of relativity these notions became relative and mutually dependent.
So, instead of Euclid's three dimensional space, Minkowski's four dimensional
space appears, where time is the fourth dimension. The characteristics
of space and time relative to the reference space - body, become dependent
on motion or more exactly, dependent on the speed of motion relative to
the reference space. Because of motion, the contraction of space appears
in the direction of motion, that is, the contraction of one space dimension
in the direction of motion, contraction of the length. With the contraction
of space the contraction of the body in the direction of its motion appears.
Lorentz deriving his famous transformation explained or more precisely,
he tried to explain the negative result of Michelson's experiment. However,
Einstein accepted his transformation and rejected the explanation.
In case of Michelson's experiment according to Lorentz, the contraction
of the body is in the moving system in which it is at rest, and is caused by the
effect of ether on atoms and molecules which means all together on the
whole body which moves within it.
Einstein does not acknowledge the ether or any other privileged coordinate
system, which could give a motive to introduce the idea about the ether.
According to him, the contraction of the body appears due to motion, so
there is no contraction in a system where the body is at rest, but in a
system in relation to which the body moves. According to that, Michelson's
equipment was at rest in the system where the measurement was made and
there could not be any contraction, so the effort that Lorentz made to
prove the contraction was useless. In regard to that, the question arises,
if the contraction given by equations of transformation really exists or
it is an illusion achieved by means of mathematics. We will consider this
question, in the way that Einstein did, as it is in science literature
and in a new way.
The procedure for determining the contraction of space, body or length,
which are all the same, will be accomplished in cases of four transformations:
the Lorentz transformation and the transformations No. 2, No. 4 and No. 5.
In the transformations No. 1 and No. 3,
the coordinate system 
and the light wave move in the opposite direction and there a dilatation
appears instead of contraction. Because of this the equations of these
transformations will not be examined in detail, nor will comparison be
made with other transformations. In order to come to a conclusion it is
enough to analyze four transformations.
17.1 Contraction of space according to the special
theory of relativity
Before we look into the method for determining contraction in the
scientific literature we will see how Einstein solved this problem by means of a rod [6].
Quotation: "I will place the rod on the
-axis of , so that
its beginning is at the point = 0,
and the end falls at the point = 1.
What is the length of the rod relatively to the system 
? In order
to find this out, we first have to ask ourselves, where the beginning and
the end of the rod lay relatively to in a certain determined
time 
in the system .
For both points it is found for time = 0
from the first equation of the Lorentz transformation
|
 |
(17.1) |
so, two points have the distance 
.
But relatively to ,
the rod moves at speed 
. The result
is that the length of the rigid rod, which moves at speed in
the direction of one's own longitudinal axis, is meters. This
means that the rod is shorter when it moves than when it is at rest. It
becomes shorter the faster it moves." End of quotation.
In citated text Einstein uses equation derived by Lorentz
transformation. However, he does not respect the condition on which that equation
is derived nor what it means.
In equations derived by Lorentz transformation
|
 |
(17.2) |
and , in these equations, are coordinates
of the position of the light wave propagating along and
-axis of the systems and
respectively. The times and 
are the times
of the coming of the light wave across and
coordinate respectively.
The Eqs. (17.2) are derived on condition that when one of ,
, and is equal to zero then
all others must be zero too. For example, if =0 then must be
=0, =0 and =0.
Accordingly, when =0 then can not be
"
" but 
. Consequently, Einstein's
proof of the contraction is incorrect and looks like joke.
17.2 Contraction of space according to the scientific literature
17.2.1 Contraction of space according to the Lorentz transformation
Three examples [10], [11] and [12] have been taken for the analysis
from the voluminous scientific literature. All three refer to the Lorentz
transformation because there were no others.
Let us see how it is treated in literature [10]:
Quotation: "Let 
be the length
of the rod in the system for which it is connected and where it is at rest relatively
to that system. Let us take two systems
and .
The latter moves at a speed relative to the former,
in such a way that its motion stays along the mutual -axes
and the axes 
and 
stay respectively parallel.
So, for the coordinate points in those two systems the Lorentz transformation could be applied
 |
Let the rod be connected to the system (Fig. 17.1) so that
it is in the plane 
parallel with ,
that is, with the -axis. In the system
let us mark the beginning of the rod with abscissa 
,
and the end of the rod with 
. In the
system let the abscissa of the beginning be 
and of the end 
.
 |
| Fig. 17.1 |
Then
|
 |
(17.3) |
is the length of the rod in the system which moves relatively to the
system . Of course, in the system
this is proper length or the length at rest.
The length of the same rod in the system ,
in relation to which the rod and the system
are moving at speed , will be
|
 |
(17.4) |
According to the Lorentz - Fitzgerald hypothesis 
should
be shorter than .
We note that the position of the two points in a moving system, that
is, two points of a body which moves relatively to an observer, have to
be determined simultaneously, because of the relativity of time. Simultaneity
refers to time in the system from which the observation is made. Simultaneity
of determination in the body's own system, that is, the one in which the
body does not move is not obligatory, because there one time is connected
to the body. But, according to Einstein's theory of relativity, what is
simultaneous in one system is not simultaneous in another system which
is in motion.
When the position of the beginning and the end of the rod are determined
from system then is the same,
but isn't.
Therefore we start from the Lorentz transformation of the coordinates
|
 |
(17.5) |
Both these times, 
and 
, are equal, so that
|
 |
(17.6) |
or
|
 |
(17.7) |
Thus
|
 |
(17.8) |
End of quotation.
Contraction is treated in a similar way and the same results are
obtained in [11].
Thus one arrives at the result that the contraction does not appear
in system in which the rod is at rest and it can be concluded
that nothing happens to the rod, but that the observer, from system ,
only sees the contraction due to motion even though it does not exist.
This contraction is in accordance with Einstein's understanding, but not
with Lorentz, who derived the transformation in order to prove that the
contraction happens in a system which moves and in which the body is at
rest. This was done in order to explain the negative results of Michelson's
experiment where the measurement was made in a system (the earth), which
moves relatively to the "absolute inertial system" - the ether.
However in the literature [12] the opposite results have been obtained.
There it begins with the same equations, but which have been solved for
coordinates of the system in the function of the coordinates
of the system which moves, so
|
 |
(17.9) |
here also it is claimed that 
, so, it is evident that
|
 |
(17.10) |
or
|
 |
(17.11) |
and
|
 |
(17.12) |
As can be seen, contraction of the length of the rod here is in the
system , however in the previous case it was
in the system .
Let's see what will happen in the following three transformations using
the same procedure for determining the contraction of space as in the first
quoted case of the Lorentz transformation.
17.2.2 Contraction of space according to transformation No. 2
In this case, according to Eqs. (12.22), the coordinates
in a system are
|
 |
(17.13) |
After substitution 
, and by subtraction we obtain
|
 |
(17.14) |
or
|
 |
(17.15) |
and
|
 |
(17.16) |
The contraction is in system
(or the dilatation in the system ),
but its magnitude differs from the magnitude of the contraction
in the first case, that is, contraction according to the Lorentz transformation.
17.2.3 Contraction of space according to transformation No. 4
Coordinates in the system
are given by Eqs. (12.24)
|
 |
(17.17) |
and from that at
|
 |
(17.18) |
and
|
 |
(17.19) |
In this case there is no contraction in any system.
17.2.4 Contraction of space according to transformation No. 5
According to Eqs. (12.25) the coordinates
in a system are
|
 |
(17.20) |
so it is at
|
 |
(17.21) |
or
|
 |
(17.22) |
and
|
 |
(17.23) |
Finally, we also obtain the opposite case. Namely, according to this
transformation the contraction of the rod appears in a the system where
the rod is at rest, that is, in system . Of course, in the system
relative to which the rod moves, the dilatation of the rod appears, and
that is contradictory to the theory of relativity.
What is to be concluded from this? We come to the conclusion that every
transformation gives a different value of contraction. In case of four
transformations three contradictory possible solutions are obtained: in
system relatively to which the rod moves, either contraction
occurs, or there is no change, or dilatation of the rod occurs. Such results
are certainly unacceptable. How can such contradictory results be arrived
at? An error has occurred somewhere. And certainly there is an error. The
error is in accepting that light wave comes to the ends of the rod at the same time,
that is . If the following was used
 |
which is defined by the fundamentals of the theory of relativity, the
calculation would be correct, but that result would not have been in accordance
with the theory of relativity, that is with Einstein's hypothesis on contraction.
Therefore was reached by "looking," as was the convenient result
that
 |
The incorrectness of the previous method of confirming the existence
of contraction and determining its magnitude can be proved in another way.
Namely, the basic principle of the special theory of relativity is that
the speed of light in both inertial systems
and
is the same and it is equal to the velocity of light in vacuum. If the
procedure in determining of the length interval and the time interval in
the systems and
is correct then by division of
the length interval with the corresponding time interval we should obtain the speed equal
to the light velocity in vacuum in both systems. This ascertainment will
be done later on, that is, after considering the dilatation of time in
the theory of relativity.
17.3 A new way of determining the contraction of space
Before we start to consider this method of determining the contraction
of space let us remind ourselves of the remarks made and emphasized earlier
on. First of all these are as follows: the coordinates
and are the coordinates of the light ray apex's
(or light wave front) position, which moves along
and -axes of the coordinate systems
and respectively. The axes
and are parallel; the motion of the origin of the
system is along the -axis,
and the motion of the light ray or the wave is followed only along
the and -axes.
Let us remember, Einstein himself gave 
in Eqs. (15.1), (15.2) and (15.3) that is 
and
and from there also 
.
This is a starting point in deriving the Lorentz transformation [Eqs. (10.1) and (10.2)].
In agreement with this we can also substitute 
and 
.
Coordinates and
are coordinates of the light ray apex on the -axis
of the system at times and
respectively, and nothing else. The same is valid for
, , 
and 
of the system .
On the basis of above presented we come to the conclusion that the
new way of determining the contraction of space is in the spirit of the
basic idea of the theory of relativity.
17.3.1 Contraction of space according to the Lorentz transformation
The coordinates in the observed systems
and are given in the form
|
 |
(17.24) |
After substitution and
and by subtraction we obtain
|
 |
(17.25) |
or
|
 |
(17.26) |
and
|
 |
(17.27) |
So, this means that the contraction occurs in the moving system
, but in that system Einstein's rod is at rest,
which is contrary to the theory of relativity. Besides, the coefficient of the space
contraction is not and from this it results that the Lorentz's
hypothesis about contraction is not correct even in a mathematical sense.
This was so when observation was made from the system
. Earlier on we saw that the opposite effect is obtained
if we make the observation from the system .
This has been presented in Eqs. (17.6) and (17.10).
Let us check if this will occur if we use the new way of determination
of the contraction. So, like in Eq. (17.9)
After substitution 
and 
and by subtraction we have
or
and
As can be seen we obtain the same result as in the previous case. This
proves the correctness of new method of determining of the contraction,
because if the contraction exists, even just in a mathematical sense, it
cannot depend on the place where it is observed from. Especially if one
insists that it occurs in the case of real bodies - rods.
17.3.2 Contraction of space according to transformation No. 2
According to Eqs. (12.22) the coordinates
in a system are
|
 |
(17.28) |
After substitution and
and subtraction we obtain
|
 |
(17.29) |
or
|
 |
(17.30) |
and
|
 |
(17.31) |
As in the previous case, the contraction is in the system
in which the body is at rest, but the magnitude of the contraction is different.
17.3.3 Contraction of space according to transformation No. 4
As mentioned earlier, this transformation and the next No. 5, have been
derived from the condition of invariability of the equation for the propagation
of the light plane wave or the sound plane wave.
According to Eqs. (12.24) the following may be written
|
 |
(17.32) |
As before, by substitution and
and by subtraction we obtain
|
 |
(17.33) |
or
|
 |
(17.34) |
and
|
 |
(17.35) |
As in the previous cases contraction appears in
system in which the body is at rest, but the magnitude
of contraction is different from the two previous cases.
17.3.4 Contraction of space according to transformation No. 5
According to Eqs. (12.25) it is
|
 |
(17.36) |
After substitution and
and by subtraction we have
|
 |
(17.37) |
or
|
 |
(17.38) |
and
|
 |
(17.39) |
As in the three previous cases the contraction
is in system in which the body is at rest.
Its magnitude also differs from the magnitudes in all three previous cases.
Naturally, instead of contraction in the system
we can say dilatation in the system , but it would not be correct,
because contraction arises in the coordinate system ,
but only in a mathematical sense.
So, according to the new way of determining the contraction of space,
body or length, in all four transformations the contraction occurs in the
coordinate system in which the body is at rest, while this
system moves with uniform translation relatively to the system .
This contraction - shortening does not depend on where the system is
being observed from and it has some logic. Because the coordinate system
, which moves after the light or sound wave, reduces the space
or length along the -axis, which the wave takes up in its motion.
This reduction, subtraction, increases with the speed of the system
. What is it, if it is not a contraction of length or space?
If the contraction were a physical reality, then the length
of the rod (from the origin of the system to the front of the wave)
would shorten almost to zero, if the speed of the system
got close to the velocity of light.
In Fig. 17.2 a contraction process is shown. It is assumed that
, in other words, that the coordinate system
moves after the light wave at a speed which is equal to one third of the velocity of light.
 |
| Fig. 17.2 |
After the first second the light wave passed along the
-axis a distance which is proportional to the length of three divisions
in system and reached the point 
.
During this time, the origin of the coordinate system
passed one third of that distance, that is the distance which corresponds to the length
of one division, and reached the point 
.
So, = 3 divisions, and = 2 divisions.
In the next second the wave will pass the next same length, and then will
be = 6 divisions, and = 4 divisions,
that is, the wave will reach the point 
, and origin of the
system the point 
.
So, 
= 3 divisions and 
= 2 divisions
and in that way 
where 
= 3 / 2.
For the different speeds
of the system the value of the contraction coefficient
are different too. With an increase
of the speed ,
and
are reduced, as well as their difference, because the system
is getting closer to the light wave. If the system had the speed
equal to the velocity of light, then and
would be equal to zero, their differences would be equal to zero too,
and the contraction coefficient would be infinitely great.
The old method of determining the contraction of space did not pass
the test. It was shown that at the same speed of the system
according to the old method can be: contraction, dilatation or
no change to the rod depending on the transformation which is being used, or depending
on where it is being observed from. In other words it seems that the rod
can change, that is, be shorter, remain the same, or be longer under the
same physical conditions. What happens to the rod does not depend only
on its motion, but also on the choice of the coordinate transformation
which is used. Simply said it depends only upon applied mathematics, which
is unacceptable.
The new way of determining the state of the contraction or dilatation
does not have this shortcoming. It confirms the same state for all the
coordinate transformations - contraction in the system in which
the rod is at rest, as Lorentz asserted. When we say the rod we think of
the length and not of the body. However, baring in mind that each coordinate
transformation gives a different value for contraction, the logical question
arises: "Can the contraction be accepted as a realistic physics process?"
The answer, of course, is negative. Simply said the contraction in question
is not a real physical process but a pure product of mathematics. A mathematician
would say: "It depends on the type of the variable substitution".
The realistic physical process of contraction occurs when some bodies
move through some environments which resist that motion. This contraction,
certainly depends on the speed of the body's motion, but also on the characteristics
of that body: neutral particles, electrified particles, solid bodies etc.
The resistance to motion and contraction also depend on the connection
of the body with that environment which surrounds it and the effects it
produces by moving. For example, an electrified body in motion generates
an electromagnetic field and establishes new relations with the surrounding
environment. It can interact in various ways with the environment, inductive,
capacitive, nuclear, gravitational etc. The environment can strongly resist
an increase of the body's speed - particle speed, above a certain value,
such as the speed at which the "electromagnetic barrier" breaks - the velocity
of light. However the contraction of a body without doubt varies according
to the characteristics of that body and its connection to the environment,
not according to Lorentz and Einstein's calculations. The contraction
results from physics and not from mathematics.
Finally we can conclude that Einstein's contraction of space is not
a physical reality but a pure illusion based on mathematics.
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