Earlier we gave a simple proof of the relativistic addition of velocities. Based on it we can obtain a global vector length (GVL), provided that our two points are moving in relation to a fixed coordinate system \((\mathbf{i_0, i_1,\ldots ,i_n})\) uniformly. GVL was then obtained by simply multiplying the GVV at time \(t\): \[\mathbf{L} = \frac{ct}{\gamma} \sum \limits_{n=0}^{\infty} \mathbf{j_n} \beta^n \qquad (2.1)\] Let's simplify this vector turning it by analogy with (1.2): \[\mathbf{L} = \mathbf{j_0} \frac{ct}{\gamma} + \mathbf{j_1} l, \quad l = vt \qquad (2.2)\] where: \(l\) is the length equal to the product of speed at the time. The first term in this formula represents the time coordinate in the moving point the first: \[ct' = \frac{ct}{\gamma}, \quad t' = \frac{t}{\gamma} \qquad (2.3)\] where: \(t'\) — time in a moving reference system (the first point). It will represent a classic relativistic time dilation [1]. Let's write \(\gamma\) for two reference systems that move relative to each other, based on the formula (1.10) the previous part of this work: \[t' = \frac{t}{\gamma_1 \gamma_2 (1 - \beta_1 \beta_2)} \qquad (2.4)\] Unfold Lorentz factors and get the desired formula for time dilation in the two moving relative to each other, pixels: \[t' = t \frac{\sqrt{1 - \beta_1^2} \sqrt{1 - \beta_2^2}}{1 - \beta_1 \beta_2} \qquad (2.5)\]

Consider the special case, when the speed of the two points is equal to: \[\beta_1 = \beta_2 = \beta_0 \qquad (2.6)\] This can be represented in the form of moving relative to a fixed reference system (earth) platform, which is the same speed the car is moving. There are two options: when the car is moving in reverse from the platform side, and when the car is moving in the direction of the platform.

Option 1. The car moves in reverse from the platform side.
Then, substituting in the formula (2.5) the total rate of (2.6), we get: \[t' = t \frac{1 - \beta_0^2}{1 - \beta_0^2} = t \qquad (2.7)\] What was to be expected: in this case, the car must be stationary relative to the earth and no time dilation should not occur.

Option 2. Cars moving in the same direction as the platform.
Then, substituting in the formula (2.5) the total rate of (2.6), we get: \[t' = t \frac{1 - \beta_0^2}{1 + \beta_0^2}\qquad (2.8)\] But, if the first point was moving relative to the earth, just twice as fast, the formula would be: \[t' = t \sqrt{1 - 4\beta_0^2}\qquad (2.9)\] These formulas are not considered in the special theory of relativity, meanwhile, they should be of interest to some practical experiments.