Applicability of the special theory of relativity

Applicability of the special theory of relativity

Cochetkov Victor Nikolaevich
Chief specialist, FGUP "The operation
center of the facilities of the ground
space infrastructure" (FGUP "TSENKI")
vnkochetkov@google.com


This article seeks to determine whether the Lorentz transformation of the
special theory of the relativity is the only possible relationship between the
coordinates and time in inertial reference systems, as well as whether its findings
are requirements imposed by the conditions of the symmetry of space and time.

I. The introduction
Currently, the internet and various magazines contain numerous articles
devoted to the criticism of the special theory of relativity. It seems that only the
lazy do not criticize it. At the same time, we fail to see any article that supports the
special theory of the relativity. Perhaps its defenders consider it below their dignity
to engage in the polemics with the critics, or they forget what it is.

You can also note that the criticism of the special theory of the relativity
mainly consists of a description of the logical inconsistencies of its findings with
respect to the real presentation of space and time. But the special theory is an
idealized mathematical model, built under the certain conditions, and therefore the
results may not be made available outside the conditions set for it.
In my opinion, if we criticize the special theory, the criticism would have to start
with its mathematical model. Maybe it would be useful to re-examine this
mathematical model and test its conclusions through the conditions that lay in its
creation.

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I.1. The brief history of the creation of the special theory of relativity
At the turn of the XIX/XX century, the efforts of the greatest physicists of that
time established the special theory of the relativity. At the end of XIX century, two
of the most important sections of physics - mechanics and the electrodynamics
displayed serious contradictions. Mechanics contained the Galilean principle of the
relativity - full equality of reference systems, moving relative to one another.
In electrodynamics, the fundamental place was held by the idea of an ether, which
filled all of space and in which all physical processes took place, including
electromagnetic fluctuations. This required that the movement of particles and
fields should be described by coordinates tightly linked to the ether, which served
as an absolute reference system.
In years 1881, 1886/1887 the Michelson-Morley experiments were unable to
register an "ether wind". As a result, the ether theory of light, seemingly confirmed
by experiments, was shown to be inconsistent with classical mechanics. In 1889,
Irish physicist D. Fitzgerald proposed that the longitudinal l´ of a body moving at
the speed V through the ether, is reduced by:
l´ = l · [1 – (V2 / c2)]1/2 ( 1 )
where: c is the speed of the light,
l - the fixed length of the body.
In 1892 the Dutch physicist, H. Lorenz, added to the D. Fitzgerald hypothesis,
the idea of a "local" time t´ associated with the "true" universal time t in the
transformation:
t´ = t – [(x · v) / c2] ( 2 )
where: v is the speed of the movement of the body while passing the point of
the space with the coordinate x .
Also H. Lorentz modified the Galilean transformation for high speeds:
x1 = β · ( x2 – V · t2 ) ( 3 )
y1 = y2 ( 4 )
z1 = z2 ( 5 )
t1 = β · { t2 – [(x · V) / c2]} ( 6 )

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by introducing a "relativistic" factor β :
β = 1 / {[1 – (V2 / c2)]1/2} ( 7 )
Formulas (3) and (6) representing relations between inertial reference systems
were named the Lorentz transformations.
As early as in 1881 the English physicist D. Thompson suggested that a mass
М moving with speed v, will be greater than mass Мо in a state of rest, with the
value М being:
М = Мо / {[1 – (v2 / c2)]1/2} ( 8 )

I.2. The special theory of the relativity
In 1905, A. Einstein considered the basic the fundamental principles of the two
classic physical theories: from mechanics - the principle of the equality of all
inertial reference systems (the principle of the relativity), and electrodynamics - the
principle of the constancy of light speed.
The principle of the relativity states: in the all inertial reference systems, all
physical phenomena in the same state, operate the same way. That is, the
physical laws are independent (invariant) of the choice of the inertial reference
system. Therefore, the equations expressing these laws have the same form in the
all inertial reference systems.
The principle of the invariance of the speed of light: the speed of the light in
the vacuum is independent of the movement of the light source. That is, the
speed of light is the same in the all directions and in all inertial reference systems.
Using the principle of relativity and the principle of the constancy of the
speed of the light, Einstein restated the Lorentz transformations, giving them a
physical sense:
x1 = [x2 + (V · t2)] / [1 – (V2 / c2)]1/2 ( 9 )
x2 = [x1 – (V · t1)] / [1 – (V2 / c2)]1/2 ( 10 )
y1 = y2 ( 11 )
z1 = z2 ( 12 )
where: x1, y1, z1 – are the coordinates of the point А at time t1 in the fixed
inertial reference system O1x1y1z1;

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x2, y2, z2 – the coordinates of point А at time t2 in the moving inertial
reference system O2x2y2z2, as shown in Fig. 1.
t1 = {t2 + [( V · x2) / c2]} / [(1 – V2/ c2)1/2] ( 13 )
t2 = {t1 – [( V · x1) / c2]}/ [(1 – V2 / c2)1/2] ( 14 )
On the basis of formulas (9)-(14), the relationship between the projection vx2,
vy2 and vz2 of the speed at point А in the moving reference system O2x2y2z2 on the
axis of the Cartesian coordinates, and the similar projection vx1, vy1 and vz1 of the
speed of the same point А in the fixed inertial reference system O1x1y1z1 was
defined as:
vx1 = (vx2 + V) / {1 + [(V · vx2)/ c2)]} ( 15 )
vx2 = (vx1 – V) / {1 – [(V · vx1)/ c2)]} ( 16 )
vy1 = {vy2 · [1 – (V2 / c2)]1/2} / {1 + [(V · vx2)/ c2)]} ( 17 )
vy2 = {vy1 · [1 – (V2 / c2)]1/2} / {1 – [(V · vx1)/ c2)]} ( 18 )
vz1 = {vz2 · [1 – (V2 / c2)]1/2} / {1 + [(V · vx2)/ c2)]} ( 19 )
vz2 = {vz1 · [1 – (V2 / c2)]1/2} / {1 – [(V · vx1)/ c2)]} ( 20 )
According to the special theory of relativity the mass М(V), of the momentum
Р(V), and of the kinetic energy Ек(V), of the material point, moving at the speed V,
was expressed by the formulas:
М(V) = Мо / [1 – (V2 / c2)]1/2 ( 21 )
Р(V) = ( Мо · V ) / [1 – (V2 / c2)]1/2 ( 22 )
Ек(V) = Мо · c2 · {{1 / [1 + (V2 / c2)]1/2} – 1} ( 23 )
where: Мо is the mass of the material point at rest.
Finally, it may be noted that the special theory of relativity was established
primarily to explain the results of experiments (A. Michelson and others), leading
to the question of the constancy of the speed of the light (or more precisely to the
explanation of the constancy of the speed of the light).

II. The kinematics

II.1. "The special theory of relativity in general terms"
Here we relax the specific requirements of relativity to facilitate analysis.

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Suppose that space is homogeneous and isotropic and time is homogeneous
(that is, there is symmetry in space and the time).
We consider whether to use the principle of relativity: "in all inertial
reference systems, all physical phenomena in the same state, operate in the same
way."
In the absence of the need not to apply the principle of the invariance of the
speed of ight (that is, less stringent conditions apply),
suppose that there are two inertial reference system: fixed O1x1y1z1 and
moving O2x2y2z2, shown in Fig. 1 and that:
- the axes of the Cartesian coordinates of the systems O1x1y1z1 and O2x2y2z2
are parallel and equally directed;
- system O2x2y2z2 is moving in system O1x1y1z1 with constant speed V2 on
the Ox1 axis;
- a starting time (t1=0 and t2=0) in both systems is selected when coordinate
centres O1 and O2 of the systems match.
Based on the symmetry of space and time, the relationship between the time
and the coordinates of the same events in the two inertial reference systems: fixed
O1x1y1z1 and moving O2x2y2z2 can be written as follows:
x1 = β1 · ( x2 + V1 · t2 ) ( 24 )
x2 = β2 · ( x1 + V2 · t1 ) ( 25 )
y1 = β3 · y2 ( 26 )
y2 = β4 · y1 ( 27 )
z1 = β5 · z2 ( 28 )
z2 = β6 · z1 ( 29 )
where: x1, y1, z1 and x2, y2, z2 - the coordinates of the point А in the reference
systems O1x1y1z1 and O2x2y2z2, respectively;
t1 and t2 - the time value in the reference systems O1x1y1z1 and O2x2y2z2,
respectively;
β1, β2, β3, β4, β5 and β6 - the transition coefficients;
V1 - speed of system O1x1y1z1 relative to the system O2x2y2z2.

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y1
y
V
1
t1
V
t2

y
y
2
2
t

·
1 = t2 = 0
·
А
А


О1 ≡ О2
x2
x1
О1
О2
x2
x1

Fig. 1

Using the principle of relativity and the symmetry of the space and the time
provides:
V1 = – V2 = V ( 30 )
β1 = β2 = β ( 31 )
β3 = β4 = 1 ( 32 )
β5 = β6 = 1 ( 33 )
This system of equations (24 )-( 29) are simplified and will take the form:
x1 = β · ( x2 + V · t2 ) ( 34 )
x2 = β · ( x1 – V · t1 ) ( 35 )
y1 = y2 ( 36 )
z1 = z2 ( 37 )
And the transition coefficient β is not dependent on the values of the
coordinates x1, y1, z1, x2, y2, z2 and the times t1 and t2, and presumably could be a
function of the speed V of the reference systems O1x1y1z1 and O2x2y2z2 about
each other.
Of formulas (34) and (35) can be recorded the values of the times t1 and t2:
t1 = {[(β2 – 1) · x2] / (β · V)} + (β · t2) ( 38 )
t2 = {[(1 – β2 ) · x1] / (β · V)} + (β · t1) ( 39 )
We can state the following for transition coefficient β in formulas (34) and
(35):
- based on the principle of the relativity and the symmetry of space and the
time, the transition coefficient β can only be a real value;

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- the transition coefficient β will equal 1 with V = 0 (the boundary
condition);
- the transition coefficient β will equal 1 if it is not dependent on the speed V;
- if the direction of the axis of the Cartesian coordinate of systems O1x1y1z1
and O2x2y2z2 is taken, the transition coefficient β will be more than 0. As well,
negative values of the transition coefficient β transition will apply with a different
direction of the axes O1x1 and O2x2;
- while the meaning of the transition coefficient β > 1, the linear dimension
of the body moving on the inertial reference system, decreases in the direction of
motion and time, moving on the same inertial reference system, slows;
- while the meaning of the transition coefficient 0 < β < 1 the linear
dimension of the body, moving on the inertial reference system, increases in the
direction of the movement and time, moving on the inertial reference system,
accelerates;
- the principle of relativity and the symmetry of space and time determines
that in the case of the application of the transition coefficient β on the values of
the speed V, the transition coefficient β value unequivocally depends on the value
of the speed V (that is, one specific value of the speed V applies to only one
specific value of the transition coefficient β).
Formulas (24)-(29) unequivocally define the accord between the coordinates
x1, y1 and z1 of point А and time t1 in the fixed system O1x1y1z1 and the
coordinates x2, y2 and z2 of the same points А and the time t2 in the moving
system O2x2y2z2.
Using formulas (24)-(39), there may be obtained an unequivocal accord
between the projection vx2, vy2 and vz2 of the speed of point А in the moving
system O2x2y2z2 on the axis of the Cartesian coordinates and the similar projection
vx1, vy1 and vz1 of the speed of this point А in the fixed system O1x1y1z1:
vx1 = (vx2 + V) / {{[(β2 – 1) · vx2] / (β2 · V)} + 1} ( 40 )
vx2 = (vx1 – V) / {{[(1 – β2) · vx1] / (β2 · V)} + 1} ( 41 )
vy1 = vy2 / {{[(β2 – 1) · vx2] / (β · V)} + β} ( 42 )
vy2 = vy1 / {{[(1 – β2) · vx1] / (β · V)} + β} ( 43 )

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vz1 = vz2 / {{[(β2 – 1) · vx2] / (β · V)} + β} ( 44 )
vz2 = vz1 / {{[(1 – β2) · vx1] / (β · V)} + β} ( 45 )
Considering formula (40) for the case in which the transition coefficient β > 1
with the real values V, vx1, vx2, it may be noted that:
-
with the positive values vx2:
vx1 ≤ (vx2 + V) ( 46 )
-
with the negative meanings vx2:
vx1 ≥ (vx2 + V) ( 47 )
Equities (46) and (47) do not exclude that in β > 1, the possible existence of
the real value of the speed vx1 of the movement of the point in the fixed inertial
reference system O1x1y1z1, would be equal to the value of the speed vx2 of the
same point in the moving inertial reference system O2x2y2z2.
And studying formula (40) for the case in which the transition coefficient
0 < β < 1 with real values V, vx1, vx2, it may be noted that:
- with positive values vx2:
vx1 ≥ (vx2 + V) ( 48 )
or when V ≠ 0:
vx1 > vx2 ( 49 )
- with negative values vx2:
vx1 ≤ (vx2 + V) ( 50 )
The equities (48)-(50), show that the transition coefficient 0 < β < 1 may not
provide the real value of the speed vx1 of the movement of the point in the fixed
inertial reference system O1x1y1z1, which would be equal to the value of the speed
vx2 of the same point in the moving inertial reference system O2x2y2z2.
Of formulas (38)-(45) can be obtained the unequivocal accord between the
projection ax2, ay2 and az2 of the acceleration of point A in the moving system
O2x2y2z2 on the axis of the Cartesian coordinates and the similar projection ax1, ay1
and az1 of the acceleration of this point in the fixed system O1x1y1z1 :
ax1 = (ax2 · β-3) / {{[(β2 – 1) · vx2] / (β2 · V)} + 1}3 ( 51 )
ax2 = (ax1 · β-3) / {{[(1 – β2) · vx1] / (β2 · V)} + 1}3 ( 52 )
(ay2 · {{[(β2 – 1) · vx2] / (β · V)} + β}) – {[(β2 – 1) · vy2 · ax2] / (β · V)}

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ay1 = ———————————————————————————– ( 53 )
{{[(β2 – 1) · vx2] / (β · V)} + β}3
(ay1 · {{[(1 – β2) · vx1] / (β · V)} + β}) – {[(1 – β2) · vy1 · ax1] / (β · V)}
ay2 = ———————————————————————————– ( 54 )
{{[(1 – β2) · vx1] / (β · V)} + β}3
(az2 · {{[(β2 – 1) · vx2] / (β · V)} + β}) – {[(β2 – 1) · vz2 · ax2] / (β · V)}
az1 = ———————————————————————————– ( 55 )
{{[(β2 – 1) · vx2] / (β · V)} + β}3
(az1 · {{[(1 – β2) · vx1] / (β · V)} + β}) – {[(1 – β2) · vz1 · ax1] / (β · V)}
az2 = ———————————————————————————– ( 56 )
{{[(1 – β2) · vx1] / (β · V)} + β}3

II.2. The definition of the special speed
Assume that there is the value Vxкр of the projection vx1 of the speed of point
А in the fixed inertial reference system O1x1y1z1, which would be consistent with
the value of the projection vx2 of the speed of point А in the moving inertial
reference system O2x2y2z2 equal Vxкр. That is when:
vx1 = vx2 = Vxкр ( 57 )
Substituting value (57) in formulas (40) or (41), we get:
V 2
xкр = (β2 · V2) / ( β2 – 1 ) ( 58 )
Of formula (58), there should be an accord Vxкр with the speed V and the
transition coefficient β for any possible values of speed V:
Vxкр = ± (β · V) / ( β2 – 1 )1/2 ( 59 )
In the event that the transition coefficient β is the value β ≥ 1, we get that Vxкр
will be a real value (which is in accordance with conditions (46) and (47)), which is
written for further consideration as:
Vxкр = vxкр1 = ± (β · V) / ( β2 – 1 )1/2 ( 60 )
where: vxкр1 - the real value of having the speed dimension.
And if the transition coefficient β is the value 0 < β ≤ 1, we get that Vxкр will
be an imaginary value (which is in accordance with conditions (48)-(50), because
the speed of the point in the fixed reference system is always above the speed of the
same point in the moving inertial reference system with 0 < β ≤ 1), which is written
for further consideration as:

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Vxкр = ί · vxкр2 = ± (ί · β · V) / (1 – β2 )1/2 ( 61 )
where: vxкр2 - the real value of having the speed dimension, but ί is equal:
ί = ( – 1 )1/2 ( 62 )
From formula (58) there can be an accord for the transition coefficient β on
the value of the speed V for any possible values of the speed V:
β2
2
= 1 / [1 – (V2 / Vxкр )] ( 63 )
Then from formula (63), taking into account formula (60) for the transition
coefficient β, and having values β ≥ 1 which is denoted as β>, you can write:
β 2
2
> = 1 / [1 – (V2 / vxкр1 )] ( 64 )
And from formula (63) taking into account formula (61) for the transition
coefficient β, having values 0<β≤ 1 and which is denoted as β<, you can write:
β 2
2
< = 1 / [1 + (V2 / vxкр2 )] ( 65 )

II.3. The equation of accord for the transition coefficients
Consider three inertial reference systems: fixed O1x1y1z1 and moving
O2x2y2z2 and O3x3y3z3, shown in Fig. 2 and from which:
- the axes of the Cartesian coordinate of the systems O1x1y1z1, O2x2y2z2 and
O3x3y3z3 are parallel and equally directed;
- system O2x2y2z2 moving in system O1x1y1z1 with constant speed V2 on the
axis Ox1;
- system O3x3y3z3 moving in the system O1x1y1z1 with constant speed V3 on
the axis Ox1;
- the starting time (t1=0 , t2=0 and t3=0) in three systems is selected as when
their coordinate centres O1 , O2 and O3 match.
Based on formula (41), you can determine the value of the speed V23 of the
motion of point O3 on point O2:
V
2
2
23 = (V3 – V2) / {{[(1 – β2 ) · V3] / (β2 · V2)} + 1} ( 66 )
and the value of speed V32 of the motion of point O2 on point O3:

V
2
2
32 = (V2 – V3) / {{[(1 – β3 ) · V2] / (β3 · V3)} + 1} ( 67 )
where: β2 and β3 - the transition coefficients for the inertial reference
systems, moving relative to the fixed reference system at speeds V2 and V3,

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