Part 2 - A Physical Approach


In this part, in addition to the pure mathematical methods, we will try to use some physical realities as well, to obtain the final form of the New transformation equations. In kinematics we are dealing with two basic physical quantities - distance and time. Quantity distance is very real, we can see it and we can touch it, which we can't say about the quantity of time. Time, as such, is a very mysterious quantity and in our article we are using many types of time. In the section $1.1$ we already define local time $\QTR{large}{t}$ and real time MATH. In the sections $1.3$ and $1.4$ we also define absolute time $\QTR{large}{\tau}$ and proper time MATH. Therefore the displacement rate of the radius vectors of the moving particle in respect to these different times generate different types of velocities, as shown three of them below, for arbitrary movement and for uniform movement separately.

 
MATH
(2-A)

On top of those velocities in $\left( 2-A\right) $, at the beginning of our article, we already used relative velocity MATH to derive New transformation equations and we need to identify that velocity as well. The type of relative velocity MATH is unknown for now, because we don't know in respect to which time we've made our calculation to find the displacement rate of the inertial systems. Therefore

 
MATH
(2-B)



2.1 - Movement of the Origin Inertial System $S^{\prime}$ Expressed by Local Time


In this section we will illustrate only, how the scientific community, from the birth of Special Relativity Theory until now, hasn't even imagined the existence of two types of velocities - local velocity and absolute velocity. Of course, in SRT there exists the idea of the Minkovsky velocity, which is not a real physical quantity, but just a handy instrument for mathematical calculation. Therefore, scientists don't distinguish these two velocities, have felled an "unknown trap". Existence of this entrapment, which brings us to a dead end, we like to show here.

Lets calculate the velocity of the origin of Inertial system $S^{\prime}$ with respect to the inertial system $S$ expressed by local time. If we put $\vec{r}\U{b4}=0$ in MATH and using velocity definitions $\left( 2-A\right) $, then for uniform local velocity of the origin of the inertial system $S^{\prime}$ we get:

 
MATH
(2.1-1)

From MATH we obtain a relation formula between local and relative velocities of the moving origin inertial system $S^{\prime}$.

 
MATH
(2.1-2)

In relative velocity formula MATH we still need to find coefficients $\beta_{1}$ and $\beta$. To do that we have two choices:

 
MATH
(2.1-3)



2.1.1 - First Choice - Relative Velocity $\vec{v}$ is the Same as Local Velocity MATH.


 
MATH
(2.1-4)

This first choice seems very intuitive and hard to deny, which is easily done by the founding fathers of Special Relativity Theory. But this is the trap which I am trying to describe here, and we need to avoid it. Before moving further let's expose how this trap works.

Inserting value $\vec{v}$ fromMATH into the relative velocity formula MATH we get the following relation between coefficients:

 
MATH
(2.1-5)

Solving MATH in respect to $\beta_{1}$ and remembering MATH we take only the positive root of $\beta_{1}$.

 
MATH
(2.1-6)

Also inserting value $1-\beta_{1}^{2}$ from MATH into MATH we get the expression for $\QTR{large}{q}$.

 
MATH
(2.1-7)

Because, according to MATH, $\beta_{1}$ is positive, then from MATH it follows that $\beta$ and $\QTR{large}{q}$ have the same sign which we already mentioned in the section $1.4$. Substituting value $\beta_{1}$ from MATH into MATH we obtain the following equation.

 
MATH
(2.1-8)

Solving MATH with respect to $\beta$ and substituting obtained $\beta$ into MATH we get the value $\beta_{1}$ as well.

 
MATH
(2.1-9)

After all of this preparation we can finally prove the contradiction of the first choice MATH. For that purpose, we need to recall the second limitation of MATH and compare coefficients $\beta_{1}$ from MATH and $\gamma$ from MATH. Here there is the possibility of only two assumptions.

 
MATH
(2.1-10)

  1. If we assume

     
    MATH
    (2.1-11)

    2. Solving equation MATH with respect to $\QTR{large}{q}$ we get the expression for $\QTR{large}{q}$ depending on the relative velocity MATH

     
    MATH
    (2.1-12)

    Conclusion (1)

    Expression MATH is contradictory to the fact of MATH, which manifests that coefficient $\QTR{large}{q}$ must be constant and doesn't depend on the relative velocity. Therefore, the first assumption is incorrect.


  2. If we assume

 
MATH
(2.1-13)

Then inserting value $1-\beta_{1}^{2}$ from MATH into transformation equations MATH and MATH we obtain

Even in this "entrapment choice" which appears as though we are following a logical path, the received New transformation equations MATH and MATH are very different from the Lorentz transformation equations.


Conclusion (2)

Transformation equations MATH and MATH are not looking bad. However, when we calculate successive transformations, it does not satisfy the linear transformation fundamental laws and therefore, according to the general guideline, we regard it as an incorrect transformation. Therefore, the first intuitive choice MATH has failed. In the section $2.2$ we prove that relative velocity MATH in fact is an absolute velocity



2.1.2 - Second Choice - Relative Velocity MATH is Not Equal to the Local Velocity MATH.


 
MATH
(2.1-16)

To analyze this second choice, we need first, to calculate the movement of the origin of the inertial system $S^{\prime}$ in respect to the inertial system $S$ using transformation equations MATH and MATH, which we will discuss in the next section.



2.2 - Movement of the Origin Inertial System $S^{\prime}$ Expressed by Real Time


Let us find now the velocity origin of the inertial system $S^{\prime}$ in respect to the inertial system $S$ expressed by real time and then we can make important conclusions from that. For that purpose, if we put $\vec{r}\U{b4}=0$ into MATH and also recalling velocity definitions $\left( 2-A\right) $, then for uniform real velocity of the origin of the inertial system $S^{\prime}$ we get:

 
MATH
(2.2-1)

And absolute time expression MATH when $\vec{r}\U{b4}=0$ becomes:

 
MATH
(2.2-2)

Now, in the real velocity expression MATH, dividing the left fraction numerator and denominator to the absolute time $\QTR{large}{\tau}$ and inserting also the value $\gamma$ from MATH into it, we get:

 
MATH
(2.2-3)

From the definition of velocities in $\left( 2-A\right) $ following, that absolute relative velocity $\vec{v}_{A}$ of the origin inertial system $S^{\prime}$ is:

 
MATH
(2.2-4)

From MATH we can calculate the expression for MATH and inserting the value MATH from MATH into it, we get:

 
MATH
(2.2-5)

Now substituting values MATH from MATH and MATH from MATH into MATH we obtain a vector equation:

 
MATH
(2.2-6)

Vector equation MATH will have only one solution, and that is:

 
MATH
(2.2-7)


Conclusion (3)

This is the second crucial result. From MATH we obtain the final proof that relative velocity MATH which we used to derive our New transformation equations is in fact not a local velocity but an absolute velocity.


Remark (4)

This is a very important point in the Armenian Theory of Relativity. Therefore, in the rest of this article we will omit index "A" denoting absolute velocity.

 

Using fact MATH, definition of velocities $\left( 2-A\right) $, also MATH and system of equations MATH we can define zero components of the absolute velocity of the moving particle in two inertial systems in the following way:

 
MATH
(2.2-8)

Also, the same way, we can define a zero component of the absolute relative velocity $\vec{v}$ as:

 
MATH
(2.2-9)

From MATH and MATH it follows that components of the absolute velocity of the moving particle or the absolute relative velocity satisfy the invariant relation.

 
MATH
(2.2-10)
Remark (5)

In the subsection $2.4.4$ we will prove that components of the absolute velocity of the moving particle form a four-vector.



2.3 - Calculation of Local Velocities and the Important Consequences


Lets calculate local velocities of the moving particle in two inertial systems $S$ and $S^{\prime}$ using the definition for velocities from $\left( 2-A\right) $.

 
MATH
(2.3-1)

To continue calculation MATH we need to find differentials MATH and $\dfrac{d\tau}{dt}$. For that purpose substituting notations MATH into MATH and then differentiating local times MATH and $\QTR{large}{t}$ by absolute time $\QTR{large}{\tau}$ and inserting also values MATH and MATH from MATH into it we get:

 
MATH
(2.3-2)

Substituting expressions MATH and $\dfrac{d\tau}{dt}$ from MATH into MATH we get the expression for local velocities.

 
MATH
(2.3-3)

Expressions in MATH are not addition formulas for the velocities of the moving particle but it is a local velocity conversion formula in the same inertial system. Therefore local velocity of the particle needs to be dependent only on the absolute velocity of the particle in the same inertial system and must be independent of the relative velocity MATH. But that only can happen when coefficients in MATH satisfy the following conditions.

 
MATH
(2.3-4)

Inserting values from MATH into the first equation of the limitation in MATH

 
MATH
(2.3-5)

Finally from MATH we obtain a relation between two constant coefficients $\alpha$ and $\beta$.

 
MATH
(2.3-6)
Conclusion (4)

MATH and MATH are the third crucial results. Inserting value $\beta_{1}$ and $\beta$ from MATH and MATH into MATH we find a value for constant coefficient $\QTR{large}{q}$ as well. Finally we obtain all necessary relations between the New transformation equation coefficients and they are:

 
MATH
(2.3-7)

Substituting coefficients MATH into MATH and MATH we obtain the final form of the New transformation equations, which we will call the Armenian transformation equations and they are:

 
MATH
(2.3-8)

Using MATH and inserting values of coefficients from MATH into real times expressions MATH we obtain

 
MATH
(2.3-9)
Remark (6)

Relation MATH manifest that real time is the local time. Therefore, from now on we are omitting all indexes "R" and "L" describing variables as real or local respectively. Particularly interval expressions MATH and MATH coincide to each other and we will call it Armenian interval of the event and denoted it by $\QTR{large}{s}$.

 
MATH
(2.3-10)

Substituting coefficients from MATH into MATH we obtain expressions for the differentials MATH and $\dfrac{dt}{d\tau}$.

 
MATH
(2.3-11)

Also substituting coefficients MATH into MATH we obtain a conversion formula for any arbitrary velocities.

 
MATH
(2.3-12)

From MATH it follows that for any arbitrary velocity we obtain a value for $\gamma$ expressed by absolute velocity or by local velocity.

 
MATH
(2.3-13)

Using notation for $\gamma$ from MATH we rewrite the velocity conversion formula MATH for any arbitrary velocity.

 
MATH
(2.3-14)

From local and absolute velocity conversion formula MATH follows their limits.

 
MATH
(2.3-15)



2.4 - Superposition and Transformation of the Absolute and Local Velocities


In this section we need to derive an addition formula for absolute and local velocities of the moving particle with respect to two inertial systems $S$ and $S^{\prime}$. Following our general guideline, we will obtain it in two different ways - by linear superposition and by differentiation, and they need to coincide each other. This is the important criteria, which asserts, that our new received transformation equations passes the mathematical test to be correct. To prove this, we need do the following:

  1. Use two successive transformations of the Armenian transformation equations MATH to obtain relations between absolute velocities of the moving particle in two inertial systems, using a linear superposition law by the scheme:

     
    MATH
    (2.4-1)

    2. Where operator $\widehat{A}$ is the Armenian transformation matrix


  2. Use a differentiation method to obtain relations between absolute velocities of the moving particle in two inertial systems, using Armenian transformation equations MATH.

Lets now consider three inertial systems $S$, $S^{\prime}$ and $S^{\prime \prime}$, where the absolute relative velocities are:

 
MATH
(2.4-2)



2.4.1 - Successive Transformations to Derive Superposition Formula for Absolute Velocities


We need to use Armenian transformation equations MATH, where coefficients MATH, as we've already mentioned, is constant and therefore doesn't depend on absolute relative velocities of the inertial systems. Lets make two successive transformations.

Straight and inverse Armenian transformation equations between three inertial systems as you can see below.

 
MATH
(2.4-3)

 
MATH
(2.4-4)

 
MATH
(2.4-5)

Implementing two successive straight transformations(it doesn't matter if we use straight or inverse transformations) MATH and MATH by scheme MATH. After eliminating $S^{\prime}$ we can get transformation MATH. We can calculate resultant velocity $\vec{u}$ in two ways - using local time transformation equations or radius vector transformation equations. In both ways we are getting the same result. Therefore it is reasonable to use here a more simple one, which is the local time transformation equations.

Substituting $t\U{b4}$ and $\vec{r}\U{b4}$ from MATH into the expression $t\U{b4}\U{b4}$ in MATH we obtain:

MATH

MATH

MATH

Now equating the above result to the time $t\U{b4}\U{b4}$ expression in MATH we get:

 
MATH
(2.4-6)

The equality MATH can only happen when the following relations exist for the resultant absolute velocity $\vec{u}$.

 
MATH
(2.4-7)

Two equations of MATH are consistent with each other and one can prove the correctness of the first equation using the second equation. From the addition formula MATH we can obtain a subtraction formula for absolute velocities, just by exchanging primes and absolute relative velocity $\vec{v}$ with $-\vec{v}$.



2.4.2 - Differentiation Method to Derive Superposition Formula for Absolute Velocities


For this purpose we need to use the inverse Armenian transformation equations MATH and differentiate local time $\QTR{large}{t}$ and radius vector MATH by the absolute time $\QTR{large}{\tau}$. Also inserting differentials MATH and $\dfrac{dt}{d\tau}$ from MATH into it, we obtain the superposition formula for absolute velocities.

 
MATH
(2.4-8)
Remark (7)

Resultant absolute velocity expressions are received in two different ways - by successive transformation MATH and by differentiation MATH coincide to each other. That means the Armenian transformation equations pass the mathematical test to be correct.



2.4.3 - Addition, Subtraction and Transformation of the Absolute Velocities


Using values for $\gamma$ fromMATH we can represent addition and subtraction formulas for absolute velocities MATH the following way:

One can prove that the addition formula for absolute velocities given by the second equations of MATH satisfy the associative law.

 
MATH
(2.4-11)

Multiplying the first equations of the systems MATH and MATH by $\QTR{large}{c}$ and using expressions MATHand MATH we can express addition and subtraction formulas by zero and spatial components of the absolute velocities only, which is the transformation equations for absolute velocities.

 
MATH
(2.4-12)

Using absolute velocity transformation equations MATH and making a straight calculation also recalling formulas MATH and MATH we obtain the following invariant relation:

 
MATH
(2.4-13)
Remark (8)

The addition and subtraction formulas MATH also invariant relation MATH clearly shows, that absolute velocity transforms as a four-vector.



2.4.4 - Addition and Subtraction of the Local Velocities


Dividing the second equations of the MATH and MATH into the first equations and using velocity conversion formulas MATH we obtain addition and subtraction formulas for local velocities.

 
MATH
(2.4-14)
Remark (9)

Local velocity or Newtonian velocity MATH does not transforms as part of a four-vector.

Using a straight forward calculation from MATH we can obtain the expressions for local velocities similar to the first equations of MATH and MATH.

 
MATH
(2.4-15)

The formulas MATH we can obtain also from the first equations of MATH using relations MATH and MATH.

Multiplying together the first and second equations of MATH we receive an interesting identity between local velocities.

 
MATH
(2.4-16)