_{}

**Late analysis results of the paper
“On the electrodynamics of moving bodies“ – kinematical part**

Libor Neumann

Prague, Czech Republic

**Abstract**

The paper presents analysis results of the original text of special relativity theory. The paper shows that the proof of the length contraction is incomplete. The velocity equations and the proof of the light velocity invariance were misinterpreted. The coordinate axes are not orthogonal and no origin translation exists in time-space. The validity space of the transformation is moving plane in classical three-dimensional space. The mathematically valid transformation does not support relativistic effects. There is no possibility to fulfil all necessary physical features of the relativity transformation together from the mathematical point of view. The serious physical use of the transformations is excluded.

PACS: 03.30.+p

The paper “On the Electrodynamics of Moving Bodies” (EMB) was published more 100 years ago [1]. The original EMB paper was not peer reviewed before publication.

Are the ideas of the special relativity theory valid or not? It is not clear yet.

The EMB paper is the background of the specific field of modern physics. Many papers are based on the ideas described in the EMB text. Many papers are dealing with the relativity and directly or indirectly use the ideas of the EMB text.

General relativity [2] is directly based on the EMB conclusions. Relativity theory is an ordinary part of physical courses and scientific research.

The papers and the discussions impeaching validity of the special relativity still exist [3, 4, 5] on the other site.

Two groups of experts exist. One majority group believes in correspondence of the EMB ideas with real world. The other group believes in the opposite.

Some measurable effects exist which can be interpreted as the indirect prove of EMB validity. The effect could be explained the different way as well. Existence and validity of a different explanation of the effects was not disproved. No direct generally accepted experimental proof of EMB validity exists. Many attempts were made; many teams were working in many scientific projects focused to the direct proof of relativity theory. None was successful yet. Remember the experiments with travelling atomic clocks [6] or the gravity waves detection experiments [7] as examples. Many experiments are interpreted as an experimental proof of special relativity theory by one group of experts and the other experts disagree with the interpretation.

Results of my gravity measurements [8] can be interpreted as disproving of general relativity theory. It was the reason I had made a decision to try to find the place where a mistake could be.

It was the reason for my analysis. I made the decision to start from the roots, from the original EMB paper. I tried to understand special relativity theory in detail. I started the analysis of the EMB.

The analysis results surprised me. They are very different from everything I read. It was the reason for publishing the analysis result.

The results of the analysis are described in form of questions, answers and explanations to enable understandable description of the main topics and respect the paper size. The analysis deals with the kinematical part of the EMB text.

The explanations refer many times to specific parts of the original EMB paper. The selected parts of original EMB paper are used in the explanations with unique identifiers. The identifiers EMBxx (where xx is unique number) are used.

**Question 1:** Does EMB text prove length contraction?

**Answer 1:** No. The proof in §4 is incomplete.

**Question 2:** Does the place in moving coordinate system
exist where the time is equal to the time in stationary coordinate system?

**Answer 2:** Yes. The moving plane exists where the
time is equal in both coordinate systems.

**Question 3:** Why the results of the sphere
transformations are different in case of light velocity limit proof in §3 and
length contraction proof in §4?

**Answer 3:** The transformation in §3 includes the time
transformation but the transformation in §4 does not.** **

**Question 4:** Does velocity equations in §5 describe
composition of velocities?

**Answer 4:** No. The equations describe transformation
of velocities?

**Question 5:** Does proof of light velocity limit in §3 correspond
with EMB light velocity invariance postulate described in §2?

**Answer 5:** No. The proof describes two different
physical effects. No proof of light velocity invariance is described in EMB.

**Question 6:** Is transformed coordinate system
translated relatively to original one.

**Answer 6:** Yes and no.

In case of time-space coordinate systems (4 dimensions) both coordinate systems have the same origin for any time.

In case of classical space coordinate systems (3 dimensions) the coordinate systems origins are translated. The distance is time dependent.

**Question 7:** Are coordinate axes orthogonal in transformed
coordinate system?

**Answer 7:** Yes and no.

In case of time-space coordinate systems (4 dimensions) time and one space axes are not orthogonal.

In case of classical space coordinate systems (3 dimensions) all coordinate axes are orthogonal.

**Question 8:** What is the validity space of the EMB
transformation with orthogonal axes and translated origins?

**Answer 8:** It is moving plane orthogonal to the
moving direction. The EMB (relativistic) transformation is equal to Galileo
transformation in the plane. No relativistic effect exists in the plane.

The EMB text in §4
(EMB12) includes
the statement that the equation (EMB11)
describes the ellipsoid of revolution without any proof. The statement must be
valid in the coordinate system, where (EMB10) is
equation of the sphere. In our case it must be valid only in the coordinate
system *k *(EMB2).
The equation (EMB11)
need not describe the ellipsoid in the coordinate system *K*. It must be
proved if the EMB statement is true or false.

It is not
generally true that the same geometrical configuration is described by the same
equation in any coordinate system. It is impossible to declare that the
equation of the ellipsoid in one coordinate system must be the equation of the
ellipsoid in second coordinate system. We can illustrate it in the following
example. The equation *x**=const.* can be the equation of a line in the
Cartesian coordinate system if *x*
is one of the coordinate axes. The same equation *x**=const.* can be the equation of a circle in the cylindrical coordinate system,
if *x* is distance from the origin.

Figure 1. The example of the different geometric shapes of the same equation in the different coordinate systems. The Cartesian coordinate system is on the left. The cylindrical coordinate system is on the right.

We can imagine the specific transformation mapping one Cartesian coordinate system with three coordinates measured in meters into the second Cartesian system where two axes are measured in meters and third axis is measured in feet. The similar transformation equation to (EMB9) describes the transformation in this case (without the time transformation – see Explanation 7). We can imagine using of the same procedure as described in EMB in §4 for our case. This procedure gives the same results, i.e. the sphere equation is transformed into “ellipsoid” equation. We know from experience that the transformation does not change the real sphere. The sphere is unchanged, but the equation describing the same sphere in the second coordinate system is different from the equation in the first coordinate system.

Figure 2. The example of the
circle transformation with the scale change in one axis. The scale of the
coorinare axis *x* is *γ* times smaller than the scale of the
coordinate axes *x, y, **h*. The original circle on the left
is transformed into the circle on the right. The equation on the left looks
like the equation of the circle and it describes the circle. The equation on
the right looks like the equation of the ellipse and describes the circle (the
solid line). Another equation on the right looks like the equation of the
circle and it describes the ellipse (the dotted line).

It is not possible
to tell what geometrical configuration is described by the equation in the
specific coordinate system without the knowledge of the coordinate system
features. **The isolated equation cannot be used for any statement dealing
with the geometrical configuration.**

** **

We suppose for the
analysis that the time scales in both coordinate systems are equal. We find the
answer from (EMB9) where we set _{} i.e. _{}.
We get

_{}

by modification.
The approximation *x = vt/2* of the equation

_{}

is useable for the
small values of *v/c*. The error is smaller than 7% for *v/c<0.5*.

This implies; the
situation *t**=t* exists in the plane perpendicular to the *x*
axis moving with the velocity approximately *v/2* in the stationary
coordinate system. The plane moves with the velocity approximately *–v/2*
in the moving coordinate system. It is approximately at the halfway between the
origins of both coordinate systems.

The time relations
in both coordinate systems are *t**>t* for all
points with the *x* (*x*) coordinate
smaller than the plane position and *t**<t* for all points with the *x *(*x*)* *coordinate greater than the plane one.
Therefore the event exists (at *t>0*) for any point with constant *x
>0* when the transformed time is equal to the original one. Before this
event the transformed time is smaller than the original one and after the event
it is up side down. The similar effect exists in the moving coordinate system
for *x**<0*.

The transformed
time at the origin of the moving coordinate system *k* is *t**<t*. It agrees with known equation (EMB13) for all positive times.

Transformed time
at the origin of the stationary coordinate system *K* is *t**>t*. It is the opposite situation. The equation (EMB13) is not valid there. It results from (EMB9)
for the origin of the stationary coordinate system where *t**=**b**t, *so

_{}

i.e. the opposite situation to the “time dilatation”.

We can see that *t=**t** *is valid at the origins of the coordinate systems only
at *t=**t**=0*. **The time transformation features of
EMB transformation are equivalent with the different time scales of the clocks
in different places. **

** **

The spherical wave and the result of transformation of the spherical wave in §3 are described

_{} (EMB5)

_{} (EMB6)

The sphere surface and the result of transformation of the sphere surface in §4 are described

_{} (EMB10)

_{} (EMB11)

The equations in §4 should be valid in any tome, the equations in §3 should describe spheres in any selected time.

We return to the
EMB text §4 describing the sphere and the ellipsoid. The time is defined in the
coordinate system *K* (EMB2) at the every point on the surface described by
equation (EMB11).
It is valid for *t=0* and *x≠0* that *t**≠0* according to (EMB9). The
points where *x≠0* exist on the surface described by (EMB10),
therefore *t**≠0* must be valid at those points in the coordinate
system *k*. The points where *x=0* exist on the surface, therefore
the points where *t**=0* exist too. It means the equation (EMB10) describes
the set of points existing at the different times *t*. **The equation (EMB10) can
not describe the solid sphere at the specific time in the coordinate system k.**
Every time difference causes distance difference between the stationary and
moving coordinate systems.

We can find in the
equation (EMB11)
deduction from (EMB10) that **the equation for the time transformation (EMB9) is not
used**. The definition *t=0 *in (EMB11) is
not transformed. The time transformation is omitted without any valid reason.

This is the basic
difference from the transformation of (EMB5) into (EMB6). The
time transformation is used in this case. **Unused time transformation is the
cause of the different results of those two sphere transformations. **

** **

In the following
text we suppose that relations (EMB14) describe velocity composition in one specific coordinate system (not
transformation between two coordinate systems). We will solve the composition
problem of two orthogonal velocities. We have two orthogonal velocities *v*
and *w* in one coordinate system.

We will analyse
two different compositions with the different sequence of the velocities. The
first composition is named I and it is based on the reference velocity v. It
can be described symbolically as *v+w*. The second composition is named II
and it is based on the reference velocity w. It can be symbolically described
as *w+v*.

The case I. We
orient the coordinate system in the parallel way with the reference velocity *v*
as usual in the EMB text. The velocity components of both velocities are in
this case *v*_{x}*=v, v*_{h}*=0, w*_{x}*=0, w*_{h}*=w*. We gain from (EMB14) and (EMB15)

_{}

The case II. We
orient the coordinate system in the parallel way with the reference velocity *w*.
The velocity components of both velocities are in this case *v*_{x}*=0, v*_{h}*=-v, w*_{x}*=w, w*_{h}*=0*. We gain from (EMB14) and (EMB15)

_{}

We can see from both cases that the composed velocity magnitude is equal in both cases. The magnitude is different than the velocity magnitude of the composed velocities based on Euclidean composition of distances divided by the time interval.

The specific
velocity component magnitudes are different in both cases. The composed
velocity component in the *v* velocity direction is greater in the case I
than in the case II. By contrast the composed velocity component in the *w*
velocity direction is smaller in the case I than in the case II. **The
direction of composed velocities must be different in both cases.**

It is interesting that the composition of orthogonal velocities changes the magnitude of orthogonal component of the composed velocity.

Figure 3. Two results of the
velocity composition. Vector I describes the *v+w* EMB velocity
composition result (case I). Vector II describes *w+v* EMB velocity
composition result (case II). The dotted vector E describes the velocity
composition result *v+w* and *w+v* based on Euclidean distance
composition and the velocity definition.

We see that the equations (EMB14) describe vector operation which is not commutative. The result of the operation is dependent on the vector sequence. This feature is not acceptable in the velocity composition in physics.

Let us test a
hypothesis that (EMB14) describe velocity transformation between
two coordinate systems *k* and *K*.

The point position transformation equation (EMB9) and the velocity transformation equation (EMB14) can not be independent, if they are used for the transformation in physics. The relations between the position transformation and the velocity transformation must be independent on the sequence of the transformation and the differentiation. The result must be the same if we count the velocity first by the differentiation and transform the velocity or if we transform the positions and time first and then count the velocity. It can be described by the equation

_{},

where *T _{p }*is
the position transformation,

Let us deduce the
equations for *T _{v }*from (EMB9)
and compare the result with the EMB velocity equation (EMB14).

We are working in the inertial system, in system with the constant velocities. We can use differences for simplicity. We deduce the velocity transformation per components (with respect of the transformation direction).

_{}

_{}

_{}

We gather the
velocity transformation directly deduced from (EMB9)
is equal to (EMB14) (the opposite transformation direction is
reflected by the sign change of the coordinate system velocity *v *as used
in the EMB text). The velocity equation (EMB14) corresponds to the position transformation (EMB9)
as usual in physics.

**The result
implies that interpretation in the EMB text describing equations (****EMB14****) and others in §5 as the velocity composition in one
coordinate system is wrong.**
The statements in (EMB16) and (EMB17) text can be valid only for transformation into the
moving coordinate system.

The text EMB in §3 describes the equation (EMB6) as result of the (EMB5) transformation. It is described as a proof of the EMB transformation correspondence with the light velocity invariance postulate (EMB1).

_{} (EMB5)

_{} (EMB6)

The equation (EMB5)
is interpreted as the equation describing the spherical wave propagated with
the velocity *c* in the coordinate system *K*. The equation can
describe the parametric sphere with the centre at the origin of the coordinate
system *K*. The equation can describe the wave propagated from *t=0*
with the velocity *c.*

The equation (EMB6) is interpreted similar way. It is interpreted as the
equation describing the spherical wave propagated with the velocity *c* in
the coordinate system *k*. The equation can describe the wave propagated
from *t**=0* with the velocity *c.* It is interpreted in EMB
as the proof of the light velocity invariance.

Let us analyse the proof in more detail.

Equation (EMB5)
describes the parametric set of the spheres with the radius *c.t *and with
the centre at the origin of the coordinate system *K* in the Cartesian
coordinate system *K* with the axes *x,y,z*. The equation can be
used for description of the spherical wave propagated with the velocity *c*.

It is not sure if
the equation (EMB6) describes parametric set of the spheres
in coordinate system *k*. The EMB text does not contain any proof of this
interpretation of (EMB6). It is not possible to deduce from the
same mathematical form that the described geometrical entity is the same. It is
well known fact (see Explanation 1).

Let us assume
existence of the proof. In this case the equation (EMB6) describes the parametric set of the spheres in the
coordinate system *k* with the axes *x**,**h**,**z* with the radius *c.**t* and with the centre at the origin of the
coordinate system *k*.

Let us determine
how is one sphere with the radius *c.t *in the coordinate system *K*
transformed into the coordinate system *k*. What is the radius of the
transformed sphere? If the spheres have the same radiuses *R=c.t=c.**t**=r *in both coordinate systems and the light velocity *c*
is the same in both coordinate systems, it must be valid *t=**t** *at every point of the sphere surface.

If we suppose the
spherical wave, the equation *t=**t* must be valid for all *t* and all *t* at any point of the sphere. The set of
those points is whole space. It is in the contradiction with the analysed
features of the time transformation (EMB9)
analysed in the Explanation 2. We know that the equation *t=**t* is valid only in the plane perpendicular
to the moving coordinate system velocity direction at approximately half of the
distance between the coordinate systems origins. It is valid *t≠**t* at all other positions. We have two
possibilities:

- the result of the spherical wave transformation from the stationary coordinate system is not the spherical wave in the moving coordinate system,
- the light
velocity
*c*is not equal in both coordinate systems. The light velocity in coordinate system*k*is dependent on the point position and*c.**t*

Let us look at the
sphere centres. Every sphere from the set of spheres included in propagated
spherical wave in the coordinate system *K* has the centre at the origin
of the coordinate system *K* according to (EMB5).
Every sphere from the set of spheres (if they are spheres) included in
propagated spherical wave in the coordinate system *k* has the centre at
the origin of the coordinate system *k* according to (EMB6). The origin of the coordinate system *K* is
equal to the origin of the coordinate system *k* only at the time *t=**t**=0 *as usual in EMB. The centre of the transformed sphere
is at the other place than the centre of the original one at any other time. **The
spheres can not be identical.** **The spheres must have the different
centres moving each other with the velocity v.**

Let us note the
equation describing the transformed spherical wave or the sphere with the
centre at the origin of the coordinate system *K* (EMB2)
can be described in the Cartesian coordinate system *k* by equation

_{}

or if we consider
the scale transformation in *x**,**t* axes (see Explanation 6 and Explanation 1) the equation could be

_{}.

Compare it with (EMB6).

We summarize.
Equations (EMB5) and (EMB6) **do not describe the same physical effect and they
do not prove the light velocity invariance** of the transformations described
in EMB. No other proof of the light velocity invariance is described in EMB.

Let us analyze the EMB transformations from mathematical point of view.

We will analyse the transformations described in the EMB text as a vector transformation.

The mathematic area dealing with vector transformation is part of linear algebra (vector algebra, tensor algebra). We can read in linear algebra textbooks, that the basis of the coordinate system and the origins are essential part of the transformation. We can read [9 – chapter 3.4, chapter 5.1], [10], [11], [12] that the coordinate transformation is unambiguously related with the coordinate systems bases and origin shift.

General vector space (geometry) transformation consists of origin shift (translation) and change of basis [13]. Translation can be described by translation matrix (1) [14].

_{} (1)

General transformation can be described by general transformation matrix (2) [15]

_{} (2)

Where **a** is
basis change matrix and *T* is translation vector in result coordinate
system.

We will use EMB
notation where coordinates in the original coordinate system *K* are *x,y,z*
and coordinates in the result coordinate system *k* are *x,h,z*.

We can formulate a 4D hypothesis. It is possible to do it. Some parts of EMB text can be understand this way (EMB3), (EMB4), (EMB9). The point in our 4D space is an event.

General transformation in 4D space can be described by general transformation matrix

_{}

We can write
directly from (EMB9) the transformation matrix **b** in general
transformation matrix form

_{} (3)

We can see that
the (EMB9) transformation includes zero translation vector *T*.
The transformation includes only bases change. **The (****EMB9****) transformation in 4D space does not shift the
coordinate system origins. **

The EMB text (EMB2)
describes clocks “be in all respects alike”. It can be understand from the
mathematical point of view as both 3D coordinate systems are parameterized by
the time attribute at any point. The note “this “*t*” always denotes a
time in stationary system” can be understand as the attribute of the stationary
coordinate system is named “*t*”. The value of the attributes *t* and
*t* at different points is referred to the EMB
§1 text.

We can write
directly from (EMB9) the transformation matrix **b** in
general transformation matrix form

_{} (4)

**We can see that
the (****EMB9****) transformation in 3D space includes nonzero translation.** It is not simple translation. The
transformation includes bases change as well (compare with (1) and (2)).

Let us find the
inverse mapping to (EMB9). We solve inverse matrix **c**=**b**^{-1}
(3). We need determinant det(**b**). We calculate det(**b**)=*b ^{2}*

Inverse matrix **c**
is

_{}

We can write from
the inverse matrix **c**

*x=**b**v**t+bx**=**b**(**x**+v**t**)* and *t=**bt**+**b**v**x**/c ^{2}=*

**The inverse
matrix c in 4D space gives the same results as the EMB text in (****EMB7****) and (****EMB8****).**

Let us verify if all bases vectors are orthogonal.

We can describe
unit vectors of the coordinate system *k* base in coordinate system *K*
from inverse matrix **c**

e_{t}=(b, bv,0,0), e_{x}=(bv/c^{2},b, 0,0), e_{h}=(0,0,1,0), e_{z}=(0,0,0,1)

The unit vector lengths are

_{}

We compute dot
products of every pair of unit vectors. We find out that all dot products but
one is zero. **Every base vectors but one pair are orthogonal.** **The
exception is e**_{t}** and e**_{x}**.**

*e*_{t }*.** e*_{x}*=**b*^{2}*v/c ^{2}+*

We can make general 4D space conclusions:

- The (EMB9) transformation is not translation. It is only bases change.
- Two unit vectors change theirs size. Two unit vectors have the same size.
- The coordinate axes are not multiply orthogonal. Coordinate axes t and x are not independent each other.
- The inverse transformation is equal with the EMB transformation (EMB7) and (EMB8).

We note the 4D
space we are using is not metric space. It is affine space. It is not possible
to measure the distances or the angles in this space. The value of the dot
product e_{x
. }e_{t} has no physical meaning in affine space
(the same is valid for the unit vector length). Zero value is important for
perpendicularity test. It is possible to test perpendicularity in our case.

We can change our
space into the metric space for better clearness. We can multiply both “time”
axes *t* and *t* by
any velocity *v _{0}*. We gain the metric space. We can measure the
distances and the angles in that space. We get in this case

*e*_{t }*.** e*_{x}*=**b*^{2}*vv _{0}/c^{2}+*

_{}

and we can measure
the angle between the *x* and *v _{0}*

_{}.

We can see the
angle is dependent on the ratio *v/v _{0}*

_{}

for *v=v _{0}=c*.
It means that the

Figure 4. The schematic drawing
of the 4D EMB transformation. Only the two dimensions are shown in the figure.
The unit vectors of the both coordinate systems *K* and *k*,
coordinate axes and the three points *P _{1}*,

We name the
distance of the coordinate systems *K* and *k* origins in the
coordinate system *K* as *D**K* and the same distance in the coordinate
system *k* as *D**k*.

The transformation
matrix **b** in general transformation matrix form is (4) where *-**b**vt=**D**k*.

Let us find the
inverse mapping to (EMB9). We solve inverse matrix **c**=**b**^{-1}.
We need determinant det(**b**). We calculate det(**b**)=b. We see that **the (****EMB9****) transformation in 3D space changes scale of volume** because det(**b**)≠1 (for *v**>0*) (compare with Explanation 1).

Inverse matrix **c**
is

_{}

We can write from
the inverse matrix **c**

x*=**x**/**b**+**D**K= **x**/**b**-**D**k/**b*,

where *D**K=-**D**k/**b**=vt* and from the
original matrix **b**

*x**=**b**x+**D**k**=b**x-**bD**K*.

Let us verify if all base vectors are orthogonal.

We can describe
unit vectors of the coordinate system *k* base in coordinate system *K*
from inverse matrix **c**

e_{x}=(1/b, 0,0), e_{h}=(0,1,0), e_{z}=(0,0,1).

The unit vector lengths are

_{}

We compute dot
products of every pair of unit vectors of the coordinate system *k* base.
We find out that all dot products are zero. Every base vectors pair of
coordinate system *k* is orthogonal. **Every base vector in 3D space is
orthogonal to every other.**

We can make the first 3D space general conclusions:

- The (EMB9) transformation can be only combination of translation and scale change. It is impossible to be simple translation only.
- All coordinate axes are mutually orthogonal in both coordinate systems.
- One base vector changes its length. The transformation changes scales in one direction.

We are in
classical 3D space (see Explanation 6 and Explanation 7). EMB transformation works with t and t attributes. Both can be supposed as attributes
of every point in the 3D space. The equation *t**=**b**(t-vx/c ^{2})* from (EMB9)
can be interpreted as relation between both attributes.

We analyse
dependency of the origins translation on the *t* attribute value.

We find from *D**K* and *D**k* definition and from translation definition
applied to the (EMB9) the relation *D**K=vt *(in coordinate system *K*).* *It implies directly that exists
only one value of the *t* attribute for every specific *D**K *or* **D**k. * It implies
directly, that every point in the space has to have the same value of its *t*
attribute (only one translation exists for every point in the space – see
translation definition [14]).

We can write direct and inverse transformation

*x=b**x-**bD**K=**b**x-**b**vt=**b**(x-vt)*

*x=**x**/**b**+**D**K= **x**/**b**+vt*

Let us note different scales in both coordinate systems.

We can see that our inverse transformation equation is different from the equations (EMB7) and (EMB8). Inverse mapping does not include t variable as in EMB text.

We have additional general conclusions:

- The equality of the
*t*attribute value at every point of space is directly caused by the translation existence and by translation definition. - The
transformation is valid at any point of the space if it uses only
*t*attribute (no*t*attribute is used).

Let us analyse
transformation features linked with the t attribute. We use the relation between both attributes
*t**=**b**(t-vx/c ^{2})* from (EMB9).
The relation

It implies; there
exists that *x* where the product *v _{k}*

If the product *v _{k}*

There exists only
one value of *x* coordinate where the*t* attribute has the specific value. The value of the*t* attribute has to be different at any other
point with the different *x* (*x*)
coordinate. Therefore the **inverse translation dependent on the ***t*** attribute is valid only in the plane
perpendicular to the x (**

Let us find the
position of the plane. We know from the distance between origins comparison in
different coordinate systems *D**k=**bD**K* (scale change must be applied to the distance
description in different coordinate systems), then

*v _{k}*

We gain the inverse translation validity condition

*t **=**b**tv/v _{k}*.

The inverse
translation has to fulfil two conditions dealing with the relation between *t* and *t* in its validity scope i.e.
relation

*t **=**b**(t-vx/c ^{2})*

and the condition

*t**=**b**tv/v _{k}*.

We solve the position where both equations are valid

*b**tv/v _{k }=*

The solution is

*x=tc ^{2}(1/v-1/v_{k})*.

We see that the plane
position is dependent on the *t *attribute value (i.e. on the distance
between the origins) and on the relation of both constants *v* and *v _{k}*.
We can choose the ratio

* _{}*.

We get general inverse transformation

*x = **x**/**b**+ vt = **x**/**b**+ v _{k} *

We gain *a**=**b*, *x=0 *and *t**=**b**t *for *v*=*v _{k}*. The
inverse transformation is

*x = **x**/**b**+ v _{k}*

We gain *a**=1, v _{k}=*

_{}

for *t**=t*. The inverse transformation is

* x = **x**/**b**+ v _{k}*

We have infinite
number of choices to select a (relation
between *v* and *v _{k}* i.e. relation between

All selections are equivalent from mathematical point of view.

We have additional
specific conclusions valid in the case of using the *t* attribute in inverse transformation:

**The inverse translation is valid only in the plane perpendicular to the***x*(*x***) axis. It can not be valid in every point in 3D space.**- The general
inverse transformation exists for every value of the
*a*ratio. - The position of
the plane can be chosen. The plane position is unambiguously linked with
the relation between the
*t*and*t*attributes. The relation of the*t*and*t*attribute values can be understood as the*t*attribute scale selection. **The inverse transformation is different from the equation described in EMB text (****EMB7****) and (****EMB8****)**.

We can see the
situation for *t**=t* i.e. *a**=1*on
the figure 5.

**Figure 5.** Schematic drawing of the validity scope of
3D EMB transformation. Only one coordinate system dimension and the t and t attribute value are shown on the figure. ** **Coordinate
system *k* at time *t _{1}* is drawn in the coordinate system

We can analyse
relativistic effects in the 3D space transformation validity scope (in the plane)
for *t**=**a**t*
where *a* is chosen constant:

- Length
contraction – the length doesn’t exist in the
*x*direction. The distance is conserved in the*y*and*z*directions. Length contraction doesn’t exist.

- Time dilatation
– It is valid
*t**=**a**t*at any point in the plane where*a*is chosen constant. Time dilatation doesn’t exist. Only time scale change can exist. - Velocity transformation – we can substitute the velocity of the plane

_{}

into velocity transformation equation (in the *y*
(*h*) direction) (EMB14)

_{}.

We gain

_{}.

We can see identity. No velocity limit exists in the plane.
The velocity transformation includes only time scale factor *a*.

- Limited light velocity proof – the spheres (EMB5) and (EMB6) are placed out of the validity scope independently on the plane position. The proof is irrelevant.

** **

**The EMB translation
transformation features in the validity scope are equivalent to the classical
Galileo transformation features **(with the scale change).** **

**No relativistic
effects exist **in this
case. **The transformation with validity scope limited to the plane cannot be
use in physics. **We know that the inverse transformation (5) in 3D space is
not identical with the EMB text (EMB7)
and (EMB8). The coordinate transformation symmetry is missing
in the EMB transformation validity scope.

The physical use
of EMB transformations is based on the idea of the orthogonal coordinate
systems, on idea that the *x *and*
**t* axes are the distance and time axes
respectively, all scales are the same, the transformation is valid at any point
of the space and physical effects are invariant with regard to the
transformation. The physical use is based on the idea that the origin of the
moving coordinate system is translated from the stationary one (the translation
is time dependent with constant velocity).

The interpretation
of the transformation where there are two sloping space-time *x* and*t* axes and where the position of the axes is dependent
on the “moving” coordinate system velocity does not give any sense in physics. Moreover
the transformation does not translate the coordinate system origins. The
“moving” coordinate system has its origin at the same place as stationary one. But
this variant does not change scales and inverse transformation is symmetric and
equal to EMB text.

The EMB transformation has orthogonal axes and origin translation only in classical 3D space. This interpretation of the EMB transformation is mathematically valid only in the moving plane. But the transformation changes scale in one axis and inverse transformation differs from EMB text in this case. No relativistic effect exists in the validity scope.

We can resume.**
No interpretation of the EMB transformation can be used in physics. There is no
possibility how to fulfil all required physical features of the transformation
in one variant of the interpretation. **

It is useless to try to experimentally prove the special relativity theory described in the EMB text. It can not be successful. It can be only a game with words.

It is better to focus scientific effort to humility measurement in very different experiments. It is better to try to find a different interpretation of the experimental results. It is possible to discover new undiscovered and maybe quite surprising effects and new physical laws describing the world more exactly.

It would be very interesting to explain every experiment interpreted as a proof of the special relativity theory with respect of the EMB text analysis result. It is possible, that some of them can open the way to undiscovered physical law.

The EMB text influenced thinking of several generations of physicists. It would burden their minds and their physical imagination for next several years. It is the time to change the status and open a new space for a new discovery.

The conclusion does not decrease other very important contribution of A. Einstein to modern physics.

(EMB1)

The following reflexions are based on the principle of relativity and on the principle of constancy of the velocity of light. These two principles we define as follows:-

…

2. Any ray of
light moves in the “stationary” system of co-ordinates with the determined
velocity *c*, whether the ray be emitted by a stationary or by a moving
body.

(EMB2)

Let us in “stationary” space take two systems of co-ordinates, i.e. two systems, each of three rigid material lines, perpendicular to one another, and issuing from a point. Let the axes of X of the two systems coincide, and their axes of Y and Z respectively be parallel. Let each system be provided with a rigid measuring-rod and a number of clocks, and let the two measuring-rods, and likewise all the clocks of the two systems, be in all respects alike.

Now to the origin
of one of the two systems (*k*) let a constant velocity *v* be
imparted in the direction of the increasing *x *of the other stationary
system (*K*), and let this velocity be communicated to the axes of the
co-ordinates, the relevant measuring-rod, and the clocks. To any time of the
stationary system *K* there then will correspond a definite position of
the axes of moving system, and from reasons of symmetry we are entitled to
assume that the motion of *k* may be such that the axes of moving systems
are at time *t* (this “*t*” always denotes a time of stationary
system) parallel to the axes of stationary system.

We now imagine
space to be measured from stationary system *K* by means of the stationary
measuring–rod, and also from the moving system *k* by means of
measuring-rod moving with it; and that we thus obtain the co-ordinates *x,y,z*
and *x**,**h**,**z*
respectively. Further, let us the time *t* of the stationary system be
determined for all points thereof at which there are clocks by means of light
signals in the manner indicated in §1; similarly let the time *t* of the moving system be determined for all
points of the moving system at which there are clocks at rest relatively to
that system by applying the method, given in §1, of light signals between the
points at which the latter clocks are located.

(EMB3)

To any system of
value *x,y,z,t,* which completely defines the place and time of an event
in stationary system, there belongs a system of values *x**,**h**,**z*,*t* determining that event relatively to the
system *k*, and our task is now to find system of equations connecting
these quantities.

(EMB4)

_{}

(EMB5)

_{}

(EMB6)

_{}

(EMB7)

_{}

(EMB8)

_{}

(EMB9)

_{}

where

_{}

(EMB10)

_{}

(EMB11)

…at the time t=0 is

_{}

(EMB12)

A rigid body which, measured in a state of rest, has the form of a sphere, therefore has in a state of motion – viewed from the stationary system – the form of an ellipsoid of revolution with the axes

_{}

(EMB13)

_{}

(EMB14)

_{}

(EMB15)

_{}

(EMB16)

It follows from
this equation that from a composition of two velocities which are less than *c*,
there always results a velocity less than *c*.

(EMB17)

It follows,
further, that the velocity of light *c* cannot be altered by composition
with a velocity less than that of light. For this case we obtain

_{}

** **

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