Now go back a little and remember the paradox of the first paragraph of this work. Let's say you can. For this decompose the scalar function (1.3) is \[f(x) = \sqrt{(4\sin(x))^2 + 1} \qquad (3.1)\] for a vector, theorem of the previous paragraph: \[\mathbf{f}(x) = \pm \mathbf{i}\cdot 4\sin(x) \pm \mathbf{j}\cdot 1 \qquad (3.2)\] where: \(\mathbf{i}, \mathbf{j}\) are unit vectors of the orthonormal basis. Now we need to choose the sign before the coefficients and we get exactly formula (1.1): \[\mathbf{f}(x) = \mathbf{i}\cdot 4\sin(x) + \mathbf{j}\cdot 1 \qquad (3.3)\] it turns Out that no paradox there, and our two points can simultaneously exist in different spaces, and the dimension will depend on the specific properties of space and how we measure! On these properties, we'll talk further.

Finally we got to the topic stated at the beginning: how to convert a unit vector. Just say that to obtain such a conversion, we can choose any function, but it is better to take one that correctly describes our space. It brought H. A. Lorentz, and later, this function was named in his honor Lorentz factor [1]: \[\gamma = {1 \over \sqrt{1 - (v/c)^2}} \qquad (3.4)\] where: \(v\) is the velocity of point \(c\) is the speed of light. Oboznachen the ratio of velocity as \(v/c = \beta\) and introduce the Lorentz factor as a vector (example 2): \[\mathbf{\Gamma}(x) = \left\{\pm 1,\, \pm \beta,\, \pm \beta^2,\, \ldots,\, \pm \beta^n \right\} \qquad (3.5)\] And for to the square of this vector is always equal to one, we should be multiplied by \(1 / \gamma\). Finally, the layout of the unit vector in the Lorentz factor would look like this: \[\mathbf{R} = \frac{1}{\gamma} \left\{\pm 1,\, \pm \beta,\, \pm \beta^2,\, \ldots,\, \pm \beta^n \right\} \qquad (3.6)\] thus: \[\mathbf{R} \cdot \mathbf{R} = 1, \quad |\mathbf{R}| = 1 \qquad (3.7)\] i.e. the length of the vector \(\mathbf{R}\) is always equal to one. In the future we will use to represent a single space through speed. So what? — you will ask. Beautiful formula, but it is related to our world?

The first thing that catches my eye — the formula (3.7) expresses a global law of conservation of energy in a closed system: total energy is always the same, regardless of the processes taking place. The second property, at first glance less obvious, but let's examine it in more detail. For this, assume that the speed \(v\) we have much less than the speed of light. Then all of the vector (3.6) we can leave only the first two significant coordinates \[\mathbf{R} \approx \left\{\pm 1,\, \pm \beta \right\}, \quad \beta \ll 1, \quad \gamma \approx 1 \qquad (3.8)\] and choose the sign (direction) of motion: \[\mathbf{R} \approx \left\{1,\, \beta \right\} \qquad (3.9)\] now let's damnosum (3.9) the speed of light and write out the resulting vector more clearly: \[\mathbf{V} \approx \mathbf{i}\cdot c + \mathbf{j}\cdot v \qquad (3.10)\] If we assume that our system \(\mathbf{R}\) - axis \(\mathbf{i}\) travels at the speed of light, the vector (3.10) reflects full speed in a closed system. Hence, it is logical to assume that the axis \(\mathbf{i}\) is the coordinate time. Then the axis \(\mathbf{j}\) is the space in which the point moves with the usual speed \(v\). Thus, the point moves in space and time, which fully corresponds to our reality.

Let's go further. If the speed of the ball is constant, then domnain (3.10) at time \(t\) we need to obtain the values of some extensions: \[\mathbf{L} \approx \mathbf{i}\cdot ct + \mathbf{j}\cdot l \qquad (3.11)\] Here the first term is our current time, however expressed in units of length, about how it's done with light-second or a light year. The second term — our usual path length \(l\) (move point).

If you still need to take into account the Lorentz factor, which we equated to unity in equation (3.8), we get the classical reduction of time and length depending on the speed [2]: \[\mathbf{L} \approx \mathbf{i}\cdot ct \sqrt{1-\beta^2} + \mathbf{j}\cdot l \sqrt{1-\beta^2} \qquad (3.12)\] True, in this case, it is necessary to consider the coordinates of a higher order that will change the overall picture, but that's in one of the following chapters. I want to draw your attention to the fact that the reflection of the picture of our world may not need too complicated formulas and theories, and everything is much easier :)