It is logical to assume that if we can decompose a scalar by a vector, the same way it can and roll. Most interesting is that you can collapse the global vector back to a scalar, but a part of him. It is possible to prove mathematically, but our goal is to use a minimum of formulas, and only the most necessary. To understand the principle of convolution from the pattern (1), which shows the cylinder C in three-dimensional space. His projection (convolution) on a two-dimensional space can consist of as a circle (Fig. 1a) and rectangle (Fig. 1b).

Fig.1. Different projections of a three-dimensional cylinder to a two dimensional space.

Similarly, we can project the three-dimensional cube on a two-dimensional space, of which might be square or, in General case, a parallelogram. Already now it is obvious that the realization of a particular projection will depend on the way of measuring it, and, of course, the characteristics of the space. With the first of them is the Lorenz-factor — we are already familiar, and the second will discuss later. Now we are interested in how you can collapse the global velocity vector.

Recall that the global velocity vector looks like: \[\mathbf{R} = \frac{1}{\gamma} \left\{\pm 1,\, \pm \beta,\, \pm \beta^2,\, \ldots,\, \pm \beta^n \right\}, \quad \gamma = 1/\sqrt{1 - \beta^2} \qquad (4.1)\] Let's choose the direction and this means that the sign in front of its coefficients. We do this for simplicity, but will work in a more General way. Then our vector will become: \[\mathbf{R} = \frac{1}{\gamma} \left\{1,\, \beta,\, \beta^2,\, \ldots,\, \beta^n \right\} \qquad (4.2)\] Let's leave only the first two coordinates, and the rest — turn (project) on the second: \[\mathbf{R} = \left\{\frac{1}{\gamma},\, \beta \right\} = \left\{\sqrt{1 - \beta^2},\, \beta\right\} \qquad (4.3)\] Check: \[\mathbf{R}\cdot \mathbf{R} = 1 \qquad (4.4)\] How did we do? Very simple. This takes part of the vector \(\mathbf{R}\) transform according to the following algorithm: \[\mathbf{R} = \left\{\frac{1}{\gamma},\, \frac{1}{\gamma} \sqrt{\beta^2 + \left(\beta^2 \right)^2 + \ldots + \left(\beta^n \right)^2} \right\} \qquad (4.5)\] I.e., for vector parts, you just need to use a reverse algorithm from the primary. Also, if you produce the action of convolution on power series, we get the following output \[\frac{1}{\gamma} \sqrt{\beta^2 + \left(\beta^2 \right)^2 + \ldots + \left(\beta^n \right)^2} = \beta \qquad (4.6)\] which leads us to the formula (4.3). It will reflect one of the possible projections of a global vector. If we domnain the resulting projection of the speed of light, get one of the reflections on our real world: \[\mathbf{V} = \mathbf{j_0} c\sqrt{1 - \beta^2} + \mathbf{j_1} v \qquad (4.7)\] where: \(\mathbf{j_0}, \mathbf{j_1}\) — the unit vector of the two coordinates: time and space, and \(v\) — familiar to us speed. Folded thus a vector is graphically depicted in figure (2a).

Fig.2. Two variants of the convolution of the global vector.

You can think of some other projection, like this: \[\mathbf{R} = \left\{\frac{1}{\gamma},\,\frac{\beta}{\gamma},\, \beta^2 \right\} \qquad (4.8)\] Then by multiplying by \(c\) the velocity vector will take the form: \[\mathbf{V} = \mathbf{j_0} {c \over \gamma} + \mathbf{j_1} {v \over \gamma} + \mathbf{j_2} {v^2 \over c} \qquad (4.9)\] But if we are in three-dimensional space and know that one of the spatial coordinates, the velocity is \(v / \gamma\), the two remaining coordinates will have to be split between the third term (the third vector) in (4.9): \(v^2 / c\). I.e. the probability that this vector is located on one of these coordinates is the same, as long as the length of this vector in any case was the same. Mathematically it can be formulated as \[\mathbf{V} = \mathbf{j_0} {c \over \gamma} + \mathbf{j_1} {v \over \gamma} + \mathbf{j_2} {v^2 \over c} \sin(\psi) + \mathbf{j_3} {v^2 \over c} \cos(\psi) \qquad (4.10)\] where: \(\psi\) is the probability which may vary in the range \(0\ldots 2\pi\). Thus, the vector located on the second and third spatial coordinates, represents the probability cone with base radius \(v^2 / c\), and can be located anywhere on the surface with equal probability (Fig. 2b). This figure reflected only the three spatial coordinates and appropriate them unit vectors: \(\mathbf{j_1}, \mathbf{j_2}, \mathbf{j_3}\), because to draw a four-dimensional space, which would display all global vector, it is not possible.

In this section we presented some variants of the convolution of the global vector. It is obvious that this kind of transformation, it lost part of the parameter vector, and this must be taken into account in further applications. For example, in operations with two (or more) vectors cannot use their rolled counterparts, the result of such an operation will be incorrect. However, convolution is necessary to understand the processes occurring in real finite-dimensional space.