In this section, we will derive the Lorentz gamma factor from our hypothesis. Although this derivation demonstrates the physical formation of gamma from the mechanism at the smallest scale, it ignores time as an independent dimension; therefore, it should not be accepted as the full derivation.

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 confined in formations like knots or vortexes locally. Hence, there is a constant circulation (rotation) in the confinement volume (at 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.

For our derivation, let us accept that the length of one circulation in the confinement volume of a static “particle” (a vortex e.g. electron) as one. Certainly, according to our paper, light travels the same distance linearly without circulating.

Please note that, when we mention a static “particle” here, we are referring to the entire vortex itself being static. An assumed “point particle” cannot be static according to both quantum mechanics and our hypothesis.

Figure 6.19 Confinement volume of a static vortex

Furthermore, according to our hypothesis, an accelerated vortex of the same kind should travel the same distance that the static vortex travels. (Section 6.5 on “Simultaneity” should be well examined, in order to notice the nuances in this derivation.)

However, the path of the accelerated vortex is surely different from the path of the static vortex in space-time. The path of the accelerated vortex is no longer a circle, but it transforms into a helix, where the height of the helix describes the speed of that accelerated vortex as a ratio to the speed of light. Additionally, the radius of the helix is not the same with the radius of the static vortex, and it decreases as the vortex accelerates. This decrease in radius can easily be calculated geometrically by equalizing the arclength of the helix to the circumference of the circulation of the static vortex.

Figure 6.20 Confinement volume of an accelerated vortex

E 6.16 β is a dimensionless ratio less then one

If Nrotations is the number of circulations, then

E 6.17 Height of the helix is the function of the amount of the circulations and the speed of that accelerated vortex

We can equalize the arclength of the helix to the circumference of the circulation of the static vortex. Please note that according to our definition of simultaneity, both static and accelerated vortexes have the same amount of Nrotations in the same distance towards time dimension. (r_R is the radius of the helix of the accelerated vortex, and r_A is the radius of the circumference of the static vortex)

E 6.18 Equalizing the arclength of the helix to the circumference of the static vortex

If we simplify the equation above, we can see the relation between the radiuses of accelerated and static vortexes:

E 6.19 The relation between two radiuses is inverse gamma

As a result, the relation between two radiuses is the inverse gamma. As the vortex accelerates, the radius of the helix tightens as a function of the inverse gamma. We have discussed the accelerated nature of gamma in the previous section by the help of trigonometric relations.

Figure 6.21 Confinement volume contracts more with acceleration

Beyond 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”

Please note that in this derivation, we assumed that the direction of circulation is perpendicular to the direction of the motion of the accelerated vortex as this was a simpler way to demonstrate our derivation mathematically. However, the gamma factor can also be derived by assuming that the direction of motion is towards the radial direction of the intrinsic circulation. Actually, the critical question here is whether this circulation is completely inside three-dimensions or it bulges out to hyper-dimensions. We will discuss this question in the following chapters.

In the next section, the derivation of gamma is expanded by including the time dimension.