In this section, we will derive the Lorentz gamma factor from our hypothesis.

According to our previous discussions, elementary “particles” with mass are local deformations (strain) packages in the expanding space-time geometry, where the continuous expansion is locally confined in formations like knots or vortexes. Hence, there is a constant circulation (rotation) in the confinement volume (at the constant speed of light). In other words, the same distance that light travels is also circulated in the confinement volumes, because according to our hypothesis, all physical existence moves at the constant speed of light.

We have also seen that the metrics of timeas clock-ticks, distancespatial, and quantity of mass of an elementary “particle” is directly dependent on the tightness of the confinement volume. The extra contraction in the radius of the confinement volume that occurs with acceleration can be calculated geometrically, and the rate of this contraction gives the Lorentz gamma factor.

Graph of definitions:

Figure 6.22 Graph of definitions

Estring of the inert “particle” with mass is in red, and

Estring of the same type of “particle”, which is accelerated relatively according to the inert “particle”, is in violet.

Notes on definitions:

- We express the speed of an accelerated body in terms of speed of light, where β is a dimensionless ratio that cannot be greater than one (1).

E 6.20 β is a dimensionless ratio less then one

- We express the four-dimensional wavelength of the E_string as wt. In other words, wt is the distance towards time dimension taken for a one full N_rotation. The ratio of the wt to the E_string also represents the strain in that package.

- Nrotations is the number of intrinsic circulations in the confinement volume. Nrotations is the multiplier, which determines the quantity of measure, considering the definition of simultaneity (in Section 6.5), which states that “same kind of elementary “particles” with mass always have equal amount of Nrotations in the same distance towards time dimension (Dtd) (in any relativistic cases)”.

- r represents the radius of the confinement volume of an elementary “particle” (e.g. electron). It is an expression of the Compton wavelength.

E 6.21

Derivation:

Figure 6.23

As it can be seen from the figure, the four-dimensional distance (D4d) that light travels can be calculated as follows according to Pythagoras, and then the radius of the confinement volume can be solved.

E 6.22

After this generalized equation of the radius (r) of the confinement volume, the only variable that differs between the radiuses (r) of the confinement volumes of two different bodies (based on same type of elementary “particle”) is the geometric increase in wt (wt₁).

The inert observer should find out the increase in the accelerated body’s wt (wt₁) in order to calculate the radius of the confinement volume (r₁) in the accelerated body.

It is possible for the inert observer to solve wt₁ according to Pythagoras, if the speed of the accelerated observer (β) is known.

Figure 6.24

E 6.23

If we insert wt₁ from Equation 6.23 above into the general equation of r (the radius of the confinement volume, Equation 6.22), we will find out the r₁ of the confinement volume in the accelerated body.

E 6.24

As it can be seen on equation above, there are no variables referencing the accelerated body in motion except β.

Figure 6.25

Additionally, both D4d and wt₀ can be eliminated from the equation above. If we solve the equation according to Pythagoras below for wt₀ (please check above),

E 6.25

moreover, insert Equation 6.25 above instead of wt₀ in the equation of r₁ (Equation 6.24), we will get

E 6.26

This is the inverse gamma from Lorentz Transformation Equations.

E 6.27

Please note that according to definitions in this chapter, mass and metrics of spatial distance and clock-ticks are the functions of the tightness of the confinement volume. Therefore, this derivation of the gamma from the tightening of the radius of the confinement volume (of the knots and vortexes) is the physical formation of the relativistic transformations, and it is the explanation of the relativity phenomenon at the smallest scale.

These derivations of gamma have a very important conclusion, which we will study more closely in Section 9.3, on “The Exact Meaning of Relativity, Connecting Gravity with Quantum Mechanics”