In the previous sections on convolution multidimensional global velocity vector on two-three and four-dimensional spaces, we did not connect it with mass. In this part, we will not only relate the global velocity vector (GVV) to the mass, but also find its change in the direction perpendicular to the point's tangent. In this paper, we will assume that the mass of the body is concentrated at a point. To do this, we introduce GVV \[\mathbf{V} = \frac{c}{\gamma} \sum \limits_{n=0}^{\infty} \mathbf{j_n} \beta^n, \quad \beta = \frac{v}{c} \tag{3.1}\] in this form: \[\mathbf{V} = \frac{c}{\gamma} \left( \sum \limits_{n=0}^{\infty} \mathbf{j_{2n}} a^{n} d_{n} + \sum \limits_{n=1}^{\infty} \mathbf{j_{2n-1}} b^{n} d_{n} \right) \tag{3.2}\] where \[\gamma = {1 \over \sqrt{1 - a^2 - b^2}}, \quad d_n = \sqrt{(a^2 + b^2)^n \over a^{2n} + b^{2n}}, \quad a^2+b^2 \leqslant 1 \tag{3.3}\] In fact, formula (3.2) splits GVV from (3.1) into two subspaces (even and odd) perpendicular to each other. This makes it possible to set your own point motion function in each of them. For example, we can specify the rotation of a point like this: \[a = \beta \cos(\omega t), \quad b = \beta \sin(\omega t), \quad \beta = \frac{v}{c} \tag{3.4}\] Recall that here: \(v\) is the speed of the point (on a circle), \(c\) is the speed of light, and \(t\) is the time. The circular frequency is found as follows: \(\omega = 2\pi f\), where \(f\) is the rotation frequency of the point.

Checking GVV by formula (3.2), in which its modulus is taken, will give a positive result: \[|\mathbf{V}| = c \tag{3.5}\] Now we can deal directly with the convolution of the global vector. Since three coordinates are known to us, we can derive the fourth, and thus collapse GVV to four-dimensional space. But first, let's clarify that: \[a^2 + b^2 = \beta^2 \tag{3.6}\] Then, folding the global velocity vector, we get: \[\mathbf{V} = c \left( \mathbf{j_0} {1 \over \gamma} + \mathbf{j_1} {\beta \sin(\omega t) \over \gamma} + \mathbf{j_2} {\beta \cos(\omega t) \over \gamma} + \mathbf{j_3} \beta^2 \right) \tag{3.7}\] This vector (its three coordinates) is more clearly presented in the following graphs, in the construction of which the time \(t\) acts as a parameter. The first coordinate, as before, is responsible for the time in the moving coordinate system, and is not represented on the graphs. The correspondence with the graphs is as follows: \(\mathbf{j_1} \to X,\, \mathbf{j_2} \to Y,\, \mathbf{j_3} \to Z\). The speed of light on them, conditionally, is equal to 1.

Graph 22 clearly shows that in the direction of the Z coordinate, which is perpendicular to the tangent of the point's motion, an additional motion occurs proportional to the square \(\beta\). If we translate this into the language of impulses, then the following expression will be obtained for this coordinate: \[ p_3 = m c \beta^2 \tag{3.8}\] Here: \(p\) is the momentum of the point, \(m\) is its mass. When moving along a circle, its speed is found through the circular frequency \(\omega\) and the radius \(r\): \[ \beta = {\omega r \over c} \tag{3.9}\] Then the additional momentum along the third coordinate will be equal to \[ p_3 = {m (\omega r)^2 \over c} \tag{3.10}\] Since this impulse is applied to the mass on each revolution, then the additional force applied to the point, perpendicular to its movement, will be as follows: \[ F_3 = {m f (\omega r)^2 \over c} = {m f^3 (2\pi r)^2 \over c} \tag{3.11}\] Recall that \(f\) is the rotation frequency of the point. Hence, the change in body mass, directed perpendicular to the plane of its rotation, will be equal to: \[ \Delta m = m {f^3 \ell^2 \over c\, g} \tag{3.12}\] where \(g\) is the free fall acceleration, \(\ell = 2\pi r\) is the length of the circle along which the mass moves.

Thus, when a physical body moves along a circle, there should be a change in its weight in the direction perpendicular to the plane of rotation. It is interesting that the result obtained in formula (3.12) corresponds to the experimental one in [1], at maximum gyroscope rotation speeds. In fact, we are talking about antigravity. Of course, in accurate calculations, it is necessary to take into account the rotation of the planet, its movement in the solar system, the movement of the solar system itself, and so on. factors.