We will call our 4D world the familiar space with three Cartesian coordinates (X, Y, Z), plus the fourth coordinate - time (t). We will project the global vector onto it and observe unusual effects, which can only be obtained after the transition from the real - to our real space. For example, it is very easy to explain the so-called “expansion of the Universe” without using hypothetical dark energy.

We remind our readers that our hypothesis assumes the existence of a multidimensional real space, where many options are possible simultaneously, and where all the amazing effects that we can observe in our real space in the form of reflections or projections occur. You can go from the actual space to the real one using one of the properties of the global vector - convolution (projection).

Generally speaking, the global vector can be projected onto any other spaces, and if such spaces are ever discovered, then the tool for working with them already exists :) But since our space has the spatial and temporal characteristics described above, we will work with it further.

As before, here we will consider the movement of a mathematical point that does not have the dimensions we are accustomed to, but during its movement it can form geometric figures familiar to us. This resembles the representation of electric or magnetic fields, which are also mathematical abstractions. But these fields can take computable forms, with which physicists work in a completely real way.

If such a point moves rectilinearly with the velocity \(v\), then its global velocity vector (GVV) is described as follows: \[ \mathbf{V} = \frac{c}{\gamma} \sum \limits_{n=0}^{\infty} \mathbf{j_n} \beta^n \tag{1.1}\] Here \[ \beta = {v \over c}, \quad \gamma = {1 \over \sqrt{1 - \beta^2}}, \quad n=0,1,2,3,\ldots \tag{1.2}\] where: \(\gamma\) - Lorentz factor, \(c\) - the speed of light, \(\mathbf{j_n}\) - unit vectors of coordinates of multidimensional space.

But further it will be convenient for us to work with the specific GVV, in which the constant \(c\) is absent. Such a velocity vector looks like this: \[ \mathbf{R} = \bar\gamma \sum \limits_{n=0}^{\infty} \mathbf{j_n} \beta^n \tag{1.3}\] Here: \(\bar\gamma = \sqrt{1 - \beta^2}\) - the inverse Lorentz factor (introduced for convenience).

Let us also recall a very important property of the global velocity vector \[ \mathbf{R} \cdot \mathbf{R} = |\mathbf{R}| = 1 \tag{1.4}\] responsible for the law of conservation of energy, which is always observed here, unlike quantum mechanics or STR. This is why the space of the global vector (real space) is called "unit".

And since the speed \(v\) is constant, the length of the global length vector is found by simply multiplying GVV by time \(t\). Let us just note that \(t\) is time in the fixed coordinate system of our 4D world: \[ \mathbf{L} = \bar\gamma\, t \sum \limits_{n=0}^{\infty} \mathbf{j_n} \beta^n \tag{1.5}\] We will need the formulas given above for further work.

The global vector exists and works in a multidimensional space, and we cannot see it in its entirety. But we have full access to its projections onto our 4D world, which we, in general, see, feel, and measure.

But to understand the projection, we need to remember that the motion of a point in space is always one-dimensional. That is, it is always possible to find a coordinate system in which the motion of a point can be described by only one spatial coordinate and time. This is the key difference from STR, where an inertial reference frame is needed to describe the motion. If it is not there, the Lorentz transformations do not work. In STR, this applies, for example, to a photon. But in the global vector system, there are no such restrictions.

Since the motion is one-dimensional, we can project the GVV like this: \[ \mathbf{j_0} \to \mathbf{a} \\ \mathbf{j_1} \to \mathbf{x} \\ \mathbf{j_2}, \mathbf{j_3}, \ldots \to \mathbf{y}, \mathbf{z} \tag{1.6}\] That is, to the coordinates of time and space that are familiar to us. In the formula: \(\mathbf{a}, \mathbf{x}, \mathbf{y}, \mathbf{z}\) - unitvectors of our 4D space-time.

Then formula (1.3) will take the following form: \[ \mathit{\mathbf{R_{\perp}}} = \mathbf{a} \bar\gamma + \mathbf{x}\, \bar\gamma\, \beta + \mathbf{d} \tag{1.7}\] Here the first term of the sum is responsible for the time coordinate, the second - for the velocity coordinate along X. The third vector \(\mathbf{d}\) is the sum vector of projections of the remaining coordinates GVV with \(n \geqslant 2\), which we will find further. It should be responsible for preserving the unity of the real space according to formula (1.4). But it must be projected onto the remaining two coordinates Y and Z uniformly, since any point in the YZ plane is no different from any other. This can be done using random distribution as follows: \[ \mathbf{d} = \mathbf{y}\, \beta^2 \cos(r) + \mathbf{z}\, \beta^2 \sin(r), \quad r=rand(2\pi) \tag{1.8}\] Here the function \(rand(2\pi)\) produces random values in the range \(0..2\pi\) each time it is called. Now we can write down the general formula for the projection of GVV onto our 4D world \[ \mathit{\mathbf{R_{\perp}}} = \mathbf{a} \bar\gamma + \mathbf{x}\, \bar\gamma\, \beta + \mathbf{y}\, \beta^2 \cos(r) + \mathbf{z}\, \beta^2 \sin(r) \tag{1.9}\] and verify that \[ \mathit{\mathbf{R_{\perp}}} \cdot \mathit{\mathbf{R_{\perp}}} = 1. \] Now the value of the vector \(\mathbf{a}\) is responsible for time, and the values at the vectors \(\mathbf{x}..\mathbf{z}\) are responsible for the value of velocity in the space of our 4D world, decomposed into coordinates X, Y, Z respectively. And the entire vector \(\mathit{\mathbf{R_{\perp}}}\) reflects the motion of a point in a moving coordinate system.

Formula (1.5) already partially answers this question: the global length vector (GVL) is obtained by multiplying GVV by time \(t\), of course, provided that our point has a dimensional motion. It remains to find the projection of GVL onto our 4D world, which we will do by analogy with GVV (1.9): \[ \mathit{\mathbf{L_{\perp}}} = \big( \mathbf{a} \bar\gamma + \mathbf{x}\, \bar\gamma\, \beta + \mathbf{y}\, \beta^2 \cos(r) + \mathbf{z}\, \beta^2 \sin(r) \big) t \tag{1.10}\] By the way, if we again go from specific GVV to absolute - that is, multiply it by \(c\) again, and expand \(\beta\), we get the following picture: \[ \mathit{\mathbf{V_{\perp}}} = \mathbf{a}\, \bar\gamma c + \mathbf{x}\, \bar\gamma\, v + \mathbf{y}\, {v^2 \over c} \cos(r) + \mathbf{z}\, {v^2 \over c} \sin(r) \tag{1.11}\] And GVL in this case will become like this: \[ \mathit{\mathbf{L_{\perp}^{V}}} = \mathbf{a}\, c \bar\gamma t + \mathbf{x}\, \bar\gamma\, \ell + \mathbf{y}\, {v^2 t \over c} \cos(r) + \mathbf{z}\, {v^2 t \over c} \sin(r) \\ \ell = c \beta t = v t \tag{1.12}\] Here the time in the moving coordinate system \(\bar\gamma t\), and the length \(\bar\gamma \ell\), have classical changes depending on the speed. But the last two terms are very different from the classics. We will look at these differences further on the interactive graph.

Below we present the graph calculated by formula (1.10). The figure is interactive: you can rotate it and change its size using the mouse (or your 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 almost from zero to one. Recall that one in this case means that the point has reached the speed of light; in this case, the cone turns into a circle, and the movement of the point along the X axis turns into movement along the Y and Z axes.

For greater or lesser detail, you can change the number of points on the graph (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 points on the graph are the probable direction of the motion vector of our point at different times. This is a kind of analogue of the wave function from quantum mechanics. The time itself \(t\) is chosen to be 1 unit. With its further growth, the cone on the graph will simply increase proportionally in size.

The paper proposes an approach to describing the motion of a point in our 4D space-time through the projection of a global vector from a multidimensional real space. This method allows us to explain some observed phenomena, such as the expansion of the Universe, without introducing hypothetical dark energy.

The global vector retains its unit norm, which reflects the strict law of conservation of energy. The projection onto the coordinates of time and space demonstrates new effects of the distribution of motion in directions different from the main velocity vector, which is clearly shown using interactive graphs. The work creates a basis for a deeper analysis of physical processes through multidimensional models

Before moving on to complex motion - rotation around the center, let's discuss an interesting topic that physicists often ignore. How can we prove that the speed of light is the same in all reference systems? Why does this happen? I suggest you familiarize yourself with simple proofs of this property of the speed of light.