In this work, we will tell you about one of the most remarkable properties of the global vector of clotting. The basics of it already covered previously, now we will introduce this property in more detail and with examples. It allows you to turn a linear function into a discrete, and even probable! Ie how will look this function in fact will determine the case, or some yet unidentified characteristics of a specific space.

When the dot starts to move, and here we assume that it moves rectilinearly and uniformly, then its global velocity vector (GVV) takes the following form: \[\mathbf{V} = \frac{c}{\gamma} \sum \limits_{n=0}^{\infty} \mathbf{j_n} \beta^n, \quad \beta = v/c \tag{1.1}\] where: \(\mathbf{j_0, j_1,\ldots ,j_n}\) — singular orthonormal basis vectors, \(\gamma = 1 / \sqrt{1 - \beta^2}\) — the Lorentz factor, and \(v\) is a constant speed in real space. Let's go straight to one of the options is the collapsed vector according to the formula (4.10) from the 4th properties GVV: \[\mathbf{V} = \mathbf{j_0} {c \over \gamma} + \mathbf{j_1} {v \over \gamma} + \mathbf{j_2} {v^2 \over c} \sin(\psi) + \mathbf{j_3} {v^2 \over c} \cos(\psi) \tag{1.2}\] Part of this vector is shown in figure (1).

Remember that display of infinite-dimensional vector on a two dimensional plane is very limited and we can show the strength of three-dimensional space. Therefore, in our images, coordinate with unit vector \(\mathbf{j_0}\) we omit, for more informative other coordinates.

Fig.1. Convolution GVV according to the formula (1.2)

Fig.2. Convolution GVV according to the formula (1.8)

For GVV, the most important is energy, which is very simple and which must always equal the speed of light squared: \[E = \mathbf{V}\cdot \mathbf{V} = c^2 \tag{1.3}\]

Recall that this property follows from the reduced (normalized) global velocity vector \(\mathbf{R}\), the modulus of which is always equal to one, and \(\mathbf{V} = c\mathbf{R}\).

Check formula (1.2) on this property will give such a result! Let's look at another version of the convolution GVV, but first let's denote our three-dimensional coordinates as \[\mathbf{x} = \mathbf{j_1} {v \over \gamma}a_m, \quad \mathbf{y} = \mathbf{j_m} c \beta^m \sin(\psi), \quad \mathbf{z} = \mathbf{j_{m+1}} c \beta^m \cos(\psi), \quad m \ge 2 \tag{1.4}\] Then our rolled GVV will take the form: \[\mathbf{V} = \mathbf{j_0} {c \over \gamma} + \mathbf{x} + \mathbf{y} + \mathbf{z} \tag{1.5}\] And note that now the second and the third coordinate is not a specific value of \(n\), and randomly selected — \(m\). As such the property is of the formula (1.3) in the General case will not be executed, then to execute it we will domnain the first coordinate by a factor \(a_m\). It remains to find, using the property (1.3): \[c^2 = {c^2 \over \gamma^2} + {v^2 \over \gamma^2} a_m^2 + c^2 \beta^{2m} \sin(\psi)^2 + c^2 \beta^{2m} \cos(\psi)^2 \tag{1.6}\] \[a_m = \pm \gamma \sqrt{1 - \beta^{2(m-1)}}, \quad m \ge 2 \qquad \tag{1.7}\] from the diagram (1) select the plus in front of this coefficient. Substituting all this into formula (1.4) we get the following coordinates: \[x = v \sqrt{1 - \beta^{2(m-1)}}, \quad y = c \beta^m \sin(\psi), \quad z = c \beta^m \cos(\psi), \quad m \ge 2 \tag{1.8}\] Here we have removed the unit vectors, because the direction of the coordinates is clear without them (Fig. 2). By the way, you can check that the formula (1.8), given the coordinates of time, fully preserves the property (1.3). But for clarity let's prenormine coordinates, i.e. divide them by the speed of light. This will give us the opportunity to display them on the drawings, without looking at absolute velocity values: \[\bar x = \beta \sqrt{1 - \beta^{2(m-1)}}, \quad \bar y = \beta^m \sin(\psi), \quad \bar z = \beta^m \cos(\psi), \quad m \ge 2 \tag{1.9}\] We come to the most interesting visual demonstration of this option global convolution of a vector with this formula. We only recall that \(m\) is an integer and can vary from two to infinity, and the angle \(\psi\) in the range (\(0..2\pi\)). Draw the illustration using the program MathCAD. To do this, select 100 points from the interval of the angle \(\psi\) for each \(m\) and changing the relative velocity, we get the following graphs (Fig. 3-6). Each rolled vector is from the center (\(x=0\)) and to the circle (\(x=\beta\)) formed by them. The largest circle will be located rolled vector with \(m=2\), and smaller when \(m=3\), etc.

Fig.3. GVV convolution with β=0.1

Fig.4. Convolution GVV when β=0.25

In these figures we see projections of a global vector in four-dimensional space (displayed only three coordinates). They show a probabilistic picture of all the options, but in reality, in each moment of time, may be shown only one of such folded vectors. Interestingly, the length of any of them, given a coordinate in \(\mathbf{j_0}\), is always equal to one. Do not forget that we have switched to normalized coordinates.

Fig.5. GVV convolution with β=0.5

Fig.6. GVV convolution with β=0.8

Below we present two more images of the coiled GVV according to the formula (1.9), where the gaps between veroyatnostei vectors shaded, that creates a more realistic picture, because in fact the probability vectors at each \(m\) not 100, but an infinite number (Fig. 7-8).

Fig.7. GVV convolution with β=0.9

Fig.8. GVV convolution with β=0.9 (different view)

So, first we had only a scalar value, which is laid on the global vector in the Lorentz factor, then this GVV turned and got a set of stochastic vectors with discrete values \(m\). How can you not remember the wave-particle duality: not out of the ordinary geometry if it happens? But this is an interesting aspect of the GVV we will talk another time.

Convolution proposed in this work, it is possible to disperse and global vector lengths are provided that point also moves uniformly. Then GVV, it is sufficient to multiply by time \(t\): \[\mathbf{L} = \frac{ct}{\gamma} \sum \limits_{n=0}^{\infty} \mathbf{j_n} \beta^n \tag{1.10}\] Also, we should remind our readers that there was considered only one of the variants of the convolution of the global vector. Other options, as well as cases with nonuniform traffic, we will consider in future works.