Part 1 - Pure Mathematical Approach


In this part we will use a purely mathematical approach to derive transformation equations for relative movement, without using physical realities. In part 2 we will exercise the received New transformation equations using a physical approach to find the relation between coefficients and eventually to obtain the final form of the Armenian transformation.



1.1 - Three Dimensional Species Transformation


We need to use a $3$ dimensional approach which will best describe our existence. Let us describe an event of the moving particle in the inertial system $S$ by MATH and in the uniform moving inertial system $S^{\prime}$ by MATH, where $t$ and $t\U{b4}$ are the local times and $\vec{r}$ and $\vec{r}\U{b4}$ are the radius vectors of the moving particle. Relations between these radius vectors with respect to two inertial systems in general can be represented in the following way.

 
(1.1-1)

Three vectors $\vec{v}$, $\vec{r}$ and MATH or $\vec{v}\U{b4}$, $\vec{r}\U{b4}$ and MATH are not coplanar, therefore the resolution of radius vectors $\vec{r}\U{b4}$ and $\vec{r}$ in those three directions are unique.

Transformations of the local time $t\U{b4}$ and $t$ in general can be constructed from infinite scalar terms as showing below:

 
MATH
(1.1-2)

Since the local time and space in the inertial systems are homogeneous and isotropic, therefore time-space transformation must be linear in respect to the local time and radius vector, and that can only happen when the first power of time and radius vectors are involved. Besides that, the other terms that contain the power of $\QTR{large}{v}^{2}$ vanish as well, because we assume that the inertial systems origin coincide ( MATH ) at $t\U{b4}=t=0$. Therefore in the local time transformation equations MATH we keep only the first two terms.

To calibrate the transformation equations coefficients, we use a universal constant - boundary velocity $\QTR{large}{c}$, which by the second principle of relativity has the same value for all the inertial systems.

Finally, as a $3$ dimensional species, we need to solve the following system of equations.

 
MATH
(1.1-3)

According to the first principle of relativity, all laws of physics, including transformation laws, must be the same in all inertial systems. Therefore all numerical coefficients in the transformations MATH must be equal to each other.

 
MATH
(1.1-4)

Coefficients $\alpha_{2}$ and MATH are not numerical quantities and the first principle of relativity doesn't apply for them and therefore they are not equal to each other. We need to find a transformation law for them as well. These time-like coefficients we will call real time and for the simplicity and physical-meaning purposes we denote them as:

 
MATH
(1.1-5)

Finally, according to the third principle of relativity, we can obtain the simplest transformation equations when:

 
MATH
(1.1-6)

Substituting primed values from MATH, MATH and MATH into the transformation equations MATH we get:

 
MATH
(1.1-7)
 
MATH
(1.1-8)

We need to be very careful, because in these transformation equations the real times MATH and MATH are not the physical local times of the inertial systems. In general, we need to assume, that they can be a function from both local times $\QTR{large}{t}$ and MATH and therefore we need to find those relations as well. All other coefficients can be constants or they need to be dependent on relative velocity MATH only.

To find the limitations of this coefficients, we need to approximate the New transformation equations MATH and MATH when MATH, and then they need to coincide with the Galilean transformation equations as shown below.

 
MATH
(1.1-9)

And

 
MATH
(1.1-10)

Transformation coefficients according to MATH and MATH need to satisfy the following limitations.

 
MATH
(1.1-11)

One more thing is clear, if we are looking for a new time-space transformation, different from the Galilean transformation, then on top of the coefficient limitations MATH, they must also satisfy the following condition:

 
MATH
(1.1-12)

To derive the New transformation equations, first we need to use only, as we have mentioned already, pure mathematics without limitations and then, if necessary, to use realms of the physical realities.



1.2 - Derivation of the Armenian Transformation Equations


  1. Inserting values MATH and MATH from MATH into the expression of $\QTR{large}{t}$ in MATH

    2. MATH

    MATH

    From the above calculation for the real time MATH we get:

     
    MATH
    (1.2-1)

  2. Inserting values $\QTR{large}{t}$ and MATH from MATH into the expression of MATH in MATH

    3. MATH

    MATH

    From the above calculation for the real timeMATH we get:

     
    MATH
    (1.2-2)

  3. Inserting value MATH from MATH into the expression of MATH in MATH

    4. MATH

    MATH

    MATH

    Therefore

    MATH

    From the above calculation we obtain a vector relation.

     
    MATH
    (1.2-3)

  4. Inserting value MATH from MATH into the expression of MATH in MATH

MATH

MATH

MATH

Therefore

MATH

From the above calculation we obtain a vector relation.

 
MATH
(1.2-4)

Equating MATH and MATH null vectors components to zero we obtain the following:

Equations MATH and MATH are expressing the relations between real times MATH and MATH into local times and radius vectors in the same inertial system, and vice versa relation as well.

 
MATH
(1.2-7)

Substituting values MATH from the first equation of MATH and MATH from the first equitation of MATH into the first system of equations MATH we obtain relations between real times MATH and MATH into local times $\QTR{large}{t}$ and MATH only.

 
MATH
(1.2-8)
Remark (1)

Not all received equations are independent. For example, equations MATH can be derived from the equations MATH using transformation equations MATH and MATH, also relation MATH.


For our convenience we need to denote

 
MATH
(1.2-9)
Remark (2)

When we have only one relative velocity, for simplicity purposes instead of using MATH we use just $\gamma$. When we have multiple inertial systems we then use MATH, MATH, MATH, and so on.


If we substitute $\alpha_{1}=\gamma$ and $\beta_{2}=\beta$ in the second system of equations MATH and then using the third system of MATH we obtain the following limitations between coefficients:

 
MATH
(1.2-10)



1.3 - Expressing the Armenian Transformation Equations by Real Time


Inertial systems with real times MATH, MATH which transformed according to MATH and radius vectors MATH, MATH which transformed by MATH and MATH, are the genuine components of the event. Also if we denote the real zero components of the event in a four-dimensional time-space as:

 
MATH
(1.3-1)

Then the New transformation equations expressed by the real time or by real zero components become:

Definition (1)

Any set of components MATH which transforms in a similar way to MATH and MATH, also the distance between two events which depend on time and on radius vector only and expressed by MATH, will be called a four-vector.


Remark (3)

From the above definition 1 it follows that the real interval $\QTR{Large}{s}_{R}$ in MATH can be dependent on time and radius vector of the event only, if the coefficient MATH is a constant. This is the first crucial result. Coefficient MATH is a truly universal time-space constant characterizing the medium of time-space or the moving particle.

 
MATH
(1.3-5)

Another important consequence following from the real interval expression MATH is that it is a monotonic increasing value, or more correctly, a strictly monotonic increasing value. Because, if there is no spatial movement at all, the real time is flowing nonstop and therefore the real interval $\QTR{large}{s}_{R}$ only increases. Therefore, we can implement and define the idea of absolute time or universal time for the event.


Definition (2)

Using a real interval expressionMATH we define the absolute time of the event, denoting it as $\QTR{large}{\tau}$, which is "flowing" by the boundary speed $\QTR{large}{c}$ and it is also invariant and strictly a monotonic increasing value. Absolute time $\QTR{large}{\tau}$ and it's differential is shown below.

 
MATH
(1.3-6)

Absolute time $\QTR{large}{\tau}$ doesn't have a singularity problem like the Lorentz transformation proper-time, but it has a real time characteristics. From MATH we can obtain the expression for real time and it's differential:

 
MATH
(1.3-7)



1.4 - Expressing the Armenian Transformation Equations by Local Time


To express the New transformation equations by local time MATH and $\QTR{large}{t}$, we need to insert real time values MATH and MATH from MATH into radius vector expressions MATH and MATH, also using notations MATH we get.

Transformation equations expressed by local time MATH and MATH in case of approximation of MATH need to coincide with the Galilean transformation equations, as we've already stated, and that can only happen, when in addition to the conditions MATH, MATH and MATH also exists the condition:

 
MATH
(1.4-3)

From the transformation equations MATH or MATH we get the invariant expression for the event, which we will call the local interval and denoted as $\QTR{large}{s}_{L}$

 
MATH
(1.4-4)

By four-vector definition in section $1.3$, the interval must depend only on time and radius vector of the event, and will never be the function of the relative velocity. Therefore, the coefficient in the formula of local interval MATH need to satisfy the following condition:

 
MATH
(1.4-5)

Therefore the invariant local interval expression MATH becomes:

 
MATH
(1.4-6)

From the condition MATH follows that the coefficient $\QTR{large}{q}$ in MATH has a sign of $\beta$ and in general,we assume, that it is a real number. Because at this moment we do not know the sign of the coefficient $\QTR{large}{q}$, therefore we cannot talk about absolute time derived from $\QTR{large}{s}_{L}$. Here we can talk only about proper-time, denoting it by MATH. This proper-time and it's differential is:

 
MATH
(1.4-7)