Due to this property, further, it will be possible to apply simple math in the derivation of many well-known relativistic formulas and well-known experiments, which are not always understandable from the classical point of view.

From this work we can take the principle of obtaining the global vector length. This global vector speed need daminozide at time \(t\): \[\mathbf{L} = \frac{ct}{\gamma} \sum \limits_{n=0}^{\infty} \mathbf{j_n} \beta^n, \quad \beta = v/c \qquad (5.1)\] where: \(\gamma\) — the Lorentz factor, and \(\mathbf{j_0\ldots j_n}\) — the unit vectors of the orthonormal basis. Here, for simplicity, before the coefficients of the decomposition, we chose the pros. If you selected other values, the meaning will remain the same. Also, imagine the global vector length and even in this form: \[\mathbf{L} = \sum \limits_{n=0}^{\infty} \mathbf{L_n}, \quad \mathbf{L_n} = \mathbf{j_n} \frac{ct}{\gamma}\beta^n \qquad (5.2)\]

Fig.1. Global vector length and its reflection on the time axis.

While we talk about rectilinear motion with constant speed \(v\); it is a simple case, but very suitable to understand the meaning. Let's take the vector from (5.1) only the first coordinate is \[\mathbf{L_1} = \mathbf{t} = \frac{ct}{\gamma} \mathbf{j_0} \qquad (5.3)\] which, obviously, will represent the time coordinate (see Fig. 1). Thus, from the geometry of vectors is also clear that: \[ct' = \frac{ct}{\gamma} \qquad (5.4)\] From this we can immediately conclude that the time in the moving coordinate system is different from fixed to \(1 / \gamma\) or \(\cos(\alpha)\): \[t' = t / \gamma = t \cos(\alpha) = t \sqrt{1 - \sin^2(\alpha)}\qquad (5.5)\] \[\sin(\alpha) = \beta = v/c \qquad\] \[\cos(\alpha) = \sqrt{1 - \beta^2} = 1 / \gamma \qquad\] In expressions (5.5) and figure (1) reveals the geometric meaning of the Lorentz factor, and the reason for the reduction of time in the moving coordinate system. In this picture you can also clearly see the difference of time in mobile (\(ct'\)) and stationary (\(ct\)) coordinate system.

In addition, it can be concluded that the global vector of length \(\mathbf{L}\) is the time coordinate for a mobile system. And Vice versa. We can prove that from the point of view of the moving system stationary will look like this: \[\mathbf{J} = \sum \limits_{n=0}^{\infty} \mathbf{j_n}, \quad \mathbf{j_n} = \mathbf{L_n} \frac{ct}{\gamma}\beta^n \qquad (5.6)\] i.e. that formula (5.2) on the contrary. This evidence is presented later, it follows from the theory of matrices, in the meantime, to create a complete picture, given only its output.