This section is not a proper derivation, and it does not describe the physical reasoning of the tightening of the confinement volume with velocity. In the following derivations, we will present how the tightness of the confinement volume contracts with velocity and that this contraction ratio exactly is equal to the Lorentz gamma factor.

However, this section is still important, because it demonstrates the basic geometric relation between (inverse) gamma and velocity, and makes it easier to understand the following derivations.

Actually, the relation between (inverse) gamma and velocity is exactly the same as the relation between the basic functions of trigonometry (sin θ and cos θ). Please note that we will 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 6.14 β is a dimensionless ratio less then one

Figure 6.17 Trigonometric relations on unit circle

Most simply, if sin θ represents β, than cos θ represents inverse gamma (1/_Y).

E 6.15 β is the dimensionless ratio of v/c, and _Y is gamma

As sin θ approaches to one, cos θ approaches to zero. Similarly, as β approaches to one and velocity approaches to speed of light, (inverse) gamma (_Y) approaches to zero too.

This trigonometric relation is also a good way to visualize the accelerating nature of gamma (_Y). At first, when β starts to deviate from zero, small changes do not cause large increases at gamma (_Y). However, when β approaches to one (1), gamma (_Y) rapidly grows to infinity.

Figure 6.18 The accelerating nature of gamma

As a result, gamma (_Y) is not an abstract mathematical function; it physically represents the relativistic transformations (of mass and metrics). Our following derivations (simple and full) exactly present how gamma is physically derived from the basic mechanism at the smallest scale.