Earlier we obtained the projection of the global vector onto our 4D world from a uniformly moving point. Now let's complicate the task and see how the unit space works in the case of a point moving in a circle with a constant speed \(v\).

In this work we will show a method for obtaining a global vector from any form of point movement, and we will look at the result in the form of interactive graphs. This method becomes possible due to a very important principle: the movement of a point in space is always one-dimensional. Of course, a time coordinate is added to one spatial coordinate.

In this formulation of the problem, it will be more convenient to denote vectors as matrices. For example, the initial vector of a point rotating around a circle in our 4D space will be written as follows: \[ \mathbf{R\! 0} = \begin{vmatrix} 1 \\ \beta \cos(\omega t) \\ \beta \sin(\omega t) \\ 0 \\ \vdots \end{vmatrix} \tag{2.1}\] Here: \(\beta = v/c\) - relative velocity, \(t\) - time, \(\omega = 2\pi\). For greater clarity, we can use an analogy - which vector is responsible for which coordinate: \[ \mathbf{R\! 0} \sim \begin{vmatrix} T \\ X \\ Y \\ Z \\ \vdots \end{vmatrix} \tag{2.2}\] Where it becomes clear that our point rotates in two coordinates: X and Y, while the speed along the Z coordinate is obviously zero. Further, on the graphs, this will become more clear.

Using the principle of one-dimensional motion of a point, stated at the beginning of the work, we must select a coordinate system where this principle is observed. That is, in such a system there should be only one non-zero spatial coordinate, and the time coordinate.

In the case of rotation of a point along a circle, it is not difficult to select the necessary coordinate system. To do this, we use the rotation matrix [1], extended to a multidimensional space:

\[ \mathcal{P} = \begin{vmatrix} 1 & 0 & 0 & 0 & \\ 0 & \cos(\omega t) & \sin(\omega t) & 0 & \\ 0 & -\sin(\omega t) & \cos(\omega t) & 0 & \\ 0 & 0 & 0 & 1 & \vdots \\ & & & \ldots \end{vmatrix} \tag{2.3}\] Then, by multiplying the rotation matrix and the velocity matrix, we obtain a new coordinate space, where rotation along a circle becomes the motion of a point in a straight line with a speed of \(v\): \[ \mathbf{R}_{\mathcal{P}} = \mathcal{P} \cdot \mathbf{R\! 0} = \begin{vmatrix} 1 \\ \beta \\ 0 \\ 0 \\ \vdots \end{vmatrix} \tag{2.4}\] This important point requires special attention, since in the future we will obtain similar one-dimensional movements from more complex forms. At the first stage, using rotation matrices, we must obtain a matrix in the form (2.4), and then carry out further transformations with it.

From the unit space hypothesis, and the global vector following from it, we know that in the case of one-dimensional (in space) motion, the projection of the specific GVV onto our 4D space will be as follows: \[ \mathbf{R}_{\mathcal{P}} \to \mathbf{R_{\perp}} = \begin{vmatrix} \bar\gamma \\ \bar\gamma \beta \\ \beta^2 \sin(r) \\ \beta^2 \cos(r) \\ \vdots \end{vmatrix} \tag{2.5}\] Where: \(\bar\gamma = \sqrt{1 - \beta^2}\) is the inverse Lorentz factor, and the function \(r\) produces random values in the range \(0..2\pi\) each time it is called. This follows from the formula (1.9). We also remember that the global vector has infinite dimensions, but in this case we can limit ourselves to four of its coordinates.

To return to the original coordinate system, from where we looked at the rotating point (2.1), we need to do everything in reverse order: find the inverse rotation matrix \[ \mathcal{P}^{-1} = \begin{vmatrix} 1 & 0 & 0 & 0 & \\ 0 & \cos(\omega t) & -\sin(\omega t) & 0 & \\ 0 & \sin(\omega t) & \cos(\omega t) & 0 & \\ 0 & 0 & 0 & 1 & \vdots \\ & & & \ldots \end{vmatrix} \tag{2.6}\] and multiply it by the resulting GVV \[ \mathbf{R_{\perp}'} = \mathcal{P}^{-1} \cdot \mathbf{R_{\perp}} = \begin{vmatrix} \bar\gamma \\ \bar\gamma \beta \cos(\omega t) - \beta^2 \sin(\omega t) \cos(r) \\ \bar\gamma \beta \sin(\omega t) + \beta^2 \cos(\omega t) \cos(r) \\ \beta^2 \sin(r) \\ \vdots \end{vmatrix} \tag{2.7}\] Let us verify that in this case the main GVV energy principle: \[ \mathit{\mathbf{R_{\perp}'}} \cdot \mathit{\mathbf{R_{\perp}'}} = 1\] In essence, in this paragraph we have refined the initial vector \(\mathbf{R\! 0}\) taking into account the unit space. Now the resulting vector \(\mathbf{R_{\perp}'}\) contains all the necessary elements and we can begin to display it on graphs.

Below is a graph of the relative velocity calculated by formula (2.7). The figure is interactive: it can be rotated and resized using the mouse (or fingers - on mobile devices). It shows only three coordinates X,Y,Z, since it is impossible to display more of them on the plane.

The shape of the graph depends on the relative speed of the point \(\beta\), which can be changed from almost zero to one. Recall that one in this case means that the point has reached the speed of light; in this case, the ring turns into a sphere.

For greater or lesser detail, you can change the number of graph points (N). A large number of points can load your computer's processor, but it is better to detail the whole picture. You can also expand the graph to the full screen of your browser.

The following graph will show the relative length, which is obtained by integrating (2.7) over time \(t\): \[ \mathit{\mathbf{L_{\perp}}} = \begin{vmatrix} \bar\gamma t \\ \bar\gamma \beta {\sin(\omega t) \over \omega} + \beta^2 {\cos(\omega t) \over \omega} \cos(r) \\ - \bar\gamma \beta {\cos(\omega t) \over \omega} + \beta^2 {\sin(\omega t) \over \omega} \cos(r) \\ \beta^2 \sin(r) t \\ \vdots \end{vmatrix} \tag{2.8}\] We have stretched the graph by about 6 times along the X and Y axes for better perception. Time \(t\), by default, is chosen as one period of the point's rotation around its axis. It can be changed from 1 to 3 in the field called: «Periods».

The graph shows that when the point speed increases to the speed of light (\(\beta=1\)), its motion in the X-Y plane turns into a wave with a period of \(\omega t\), propagating along the Z axis. Moreover, we are talking about a wave of probabilities of a point being in a particular place in space. But if there are enough such points, then we will get a full-fledged wave! This is a key result that will be useful to us later.