Earlier we saw the scalar-to-vector theorem, from which a very interesting conclusion follows: any number is a projection of a vector (generally multidimensional) onto the number axis. By the way, we will call such a projection convolution of the vector.

One can imagine this conclusion in another way. Any movement of a point in space is one-dimensional at each moment of time. If a spatial axis of coordinates is placed on the motion curve of this point, then such motion will be spatially one-dimensional at any moment of time and on any of its segments. But after all, this is a certain numerical axis, where distances are located as numbers. And now - the most interesting! According to the scalar-to-vector transformation theorem, it follows that such a motion of a point can be decomposed into vectors. Those. the point moves, in the general case, in a multidimensional space, and our three spatial coordinates are just a property of our reality, for example, a limitation of our consciousness! Isn't this also a reason for limiting the maximum speed of movement?

One of the properties of our space describes Lorentz factor [1], whose expansion into a vector looks like this: \[\mathbf{R} = \frac{1}{\gamma} \left\{\pm 1,\, \pm \beta,\, \pm \beta^2,\, \ldots,\, \pm \beta^n \right\} \tag{1}\] Here: \[\beta = {v \over c} , \quad \gamma = {1 \over \sqrt{1 - \beta^2}} \tag{2}\] The speed of a point in the general case depends on time: \(v = v(t)\), which means that the dimensionless speed also depends on time: \(\beta = \beta(t)\). This generalization will be used in what follows.

Let's write the resulting vector in a different form: \[\mathbf{R} = \frac{1}{\gamma} \sum \limits_{n=0}^{\infty} \mathbf{j_n} \beta^n \tag{3}\] where: \(\mathbf{j_n}\) - unit vectors of coordinates of multidimensional space [2]. It should be noted that here we have chosen a positive sign in front of the dimensionless velocity, but in general it can be negative. This will depend on the initial conditions for the movement of the point.

The property of such a vector is the equality of its modulus to unity: \[|\mathbf{R}| = 1 \tag{4}\] Although such a vector itself is of mathematical interest, the physical meaning appears in it if it is multiplied by the speed of light \(c\): \[\mathbf{V} = \frac{c}{\gamma} \sum \limits_{n=0}^{\infty} \mathbf{j_n} \beta^n \tag{5}\] This is the Global Velocity Vector (GVV), which describes the movement of any point in multidimensional space. A property of GVV is the equality of its modulus of the speed of light \[|\mathbf{V}| = c \tag{6}\] which automatically implies the law of conservation of energy in the system. Examples of how this property can be used can be found in here.

More fully our four-dimensional space, and manipulations with it, we will present to you further, for now, let's select the first few coordinates from GVV and explain their meaning: \[\mathbf{V} = \frac{1}{\gamma} \big( \mathbf{j_0}\, c + \mathbf{j_1}\, v + \ldots \big) \tag{7}\] Here the coordinate \(\mathbf{j_0} c\) is responsible for the time, and in fact is the time coordinate. This will become clearer next. The coordinate \(\mathbf{j_1} v\) is the speed of the point in Cartesian coordinates.

For example, with a uniform point movement, we can get the global displacement vector by simply multiplying GVV by the time \(t\): \[\mathbf{L} = \frac{1}{\gamma} \big( \mathbf{j_0}\, ct + \mathbf{j_1}\, \ell + \ldots \big) \tag{8}\] Here the first coordinate is the time, the second one is the length of the point \(\ell\). On the physical graphs of Newtonian mechanics, it is the first two coordinates that are depicted. But the real space makes its own adjustments in the form of the Lorentz factor, therefore, at relatively high speeds, the contribution from other coordinates must also be taken into account (example). In the most general form, all this is taken into account in the global velocity vector (5).