Another beautiful version of the convolution of the global vector is obtained by using quaternions [1] for this. In this case, it is no longer necessary for the point to move in a straight line and evenly, as it was necessary previously. Recall that a quaternion consists of a real and an imaginary part just like a complex number, but the imaginary part consists of three imaginary values: \[q = a + \mathbf{i} b + \mathbf{j} c + \mathbf{k} d \tag{2.1}\] where: \(a, b, c, d\) -- real numbers, \(\mathbf{i}, \mathbf{j}, \mathbf{k}\) -- imaginary units , which are given by the following rule \[\mathbf{i}^2 = \mathbf{j}^2 = \mathbf{k}^2 = \mathbf{i} \mathbf{j} \mathbf{k} = - 1\] \[\mathbf{i} \mathbf{j} = -\mathbf{j} \mathbf{i} = \mathbf{k} ,\quad \mathbf{j} \mathbf{ k} = -\mathbf{k} \mathbf{j} = \mathbf{i} ,\quad \mathbf{k} \mathbf{i} = -\mathbf{i} \mathbf {k} = \mathbf{j} \tag{2.2}\] Although here the imaginary parts are called vectors, but from the point of view of vector algebra, they are not exactly vectors. Therefore, we will call them imaginary vectors.

Then the global multidimensional velocity vector (GVV), in the general case, can be collapsed to four-dimensional space in this way: \[\mathbf{V} = \frac{c}{\gamma} \sum \limits_{n=0}^{\infty} \mathbf{j_n} \beta^n \to c\, e^{\mathbf{i}a} e^{\mathbf{j}b} e^{\mathbf{k}d} \tag{2.3}\] As in the case of a multidimensional vector, the modulus of the imaginary vector is also equal to the speed of light: \[ c\, | e^{\mathbf{i}a} e^{\mathbf{j}b} e^{\mathbf{k}d} | = c \tag{2.4}\] Let's take a look at special cases.

This case arises if we need to reflect (collapse) GVV onto a two-dimensional space, into two of its coordinates. Then GVV can be represented as follows: \[\mathbf{V} = c\,e^{\mathbf{i}a} \tag{2.5}\] where the angle \(a\) is found through the arcsine \(\beta\): \[a = \arcsin(\beta), \quad \beta= {v \over c} \tag{2.6}\] Moreover, in a more general case, all these quantities can depend and change on time \(t\) \[a(t) = \arcsin(\beta(t)), \quad \beta(t)= {v(t) \over c} \tag{2.7}\] but below we will still use a simpler version of formula (2.6), implying (2.7). Then, expanding the resulting expression, and remembering that \(\cos(\arcsin(\beta)) = \sqrt{1 - \beta^2}\), we get: \[\mathbf{V} = c \left( \sqrt{1 - \beta^2} + \mathbf{i}\beta \right) \tag{2.8}\] And now we ask our readers to pay attention to all the beauty of the moment in which we can rewrite the same formula, but not for imaginary vectors, but for real ones: \[\mathbf{V} = c \left( \mathbf{j_0} \sqrt{1 - \beta^2} + \mathbf{j_1} \beta \right) \tag {2.9}\] Here \(\mathbf{j_0}, \mathbf{j_1}\) are unit vectors. We can check the correctness of such a transition like this: \[ \left| \mathbf{j}_0 \sqrt{1 - \beta^2} + \mathbf{j}_1 \beta \right| = \left| \sqrt{1 - \beta^2} + \mathbf{i}\beta \right| = 1 \tag{2.10}\] The same rule for the transition from imaginary to vector coordinates can be extended to higher dimensions.

Here we can use the same principle and decompose formula (2.3) first into quaternions, and then go over to vectors. The 4D quaternion GVV looks like this: \[\mathbf{V} = c\, e^{\mathbf{i}a} e^{\mathbf{j}b} e^{\mathbf{k}d}, \quad a = \arcsin(\beta_1),\, b = \arcsin(\beta_2),\, d = \arcsin(\beta_3) \tag{2.11}\] Here \[ \beta_1= \beta_1(t) = {v_1(t) \over c},\, \beta_2= \beta_2(t) = {v_2(t) \over c}, \, \beta_3= \beta_3(t) = {v_3(t) \over c} \tag{2.12}\] Expanding formula (2.11) we get the following quaternion: \[\mathbf{V} = c\, (A + \mathbf{i} B + \mathbf{j} C + \mathbf{k} D) \tag{2.13}\] wherein \[A = \sqrt{1 - \beta_1^2} \sqrt{1 - \beta_2^2} \sqrt{1 - \beta_3^2} - \beta_1 \beta_2 \beta_3 \] \[B = \beta_1 \sqrt{1 - \beta_2^2} \sqrt{1 - \beta_3^2} + \beta_2 \beta_3 \sqrt{1 - \beta_1^2 } \] \[C = \beta_2 \sqrt{1 - \beta_1^2} \sqrt{1 - \beta_3^2} - \beta_1 \beta_3 \sqrt{1 - \beta_2^2 } \] \[D = \beta_3 \sqrt{1 - \beta_1^2} \sqrt{1 - \beta_2^2} + \beta_1 \beta_2 \sqrt{1 - \beta_3^2 } \tag{2.14}\] Moving on to vectors, we get: \[\mathbf{V} = c\, (\mathbf{j_0} A + \mathbf{j_1} B + \mathbf{j_2} C + \mathbf{j_3} D) \tag {2.15}\] Here \(\mathbf{j_0}, \mathbf{j_1}, \mathbf{j_2}, \mathbf{j_3}\) are unit vectors. Checking for a module from a function allows us to conclude that our calculations are correct: \[ |\mathbf{j_0} A + \mathbf{j_1} B + \mathbf{j_2} C + \mathbf{j_3} D| = |A + \mathbf{i} B + \mathbf{j} C + \mathbf{k} D| = 1 \tag{2.16}\]

For an example of convolving a multimerongo unit vector into a four-dimensional quaternion, we will take the rotation of a point along a circle around an axis, plus additional movement at a constant speed along that axis. Mathematically, this can be expressed as follows: \[ \beta_1 = {v \over c} \sin(\omega t),\, \beta_2 = {v \over c} \cos(\omega t),\ , \beta_3 = {v_3 \over c} \tag{2.17}\] where \(v\) is the speed of the point moving along the circle, \(v_3\) is the speed of the point moving along the axis, \(\omega\) is the rotational speed, radians per second, \(t\) -- time. The folded vector is calculated according to the formula (2.14-2.15), where we substitute (2.17).

The pictures below will show examples of the implementation of the resulting formula. Note that the X coordinate is the direction \(\beta_1\), Y is the direction \(\beta_2\), and Z is the direction \(\beta_3\) (movement along the axis). Time in the figures appears parametrically.

If we now integrate each direction from (2.14), which are the projections of the velocity onto the coordinates, then we will get a graph of the movement of a point in space. This is what the following graphs will show.

In Figures 9 and 10, we have obtained the direction of the point's velocity, and in Figures 11 and 12 - a graph of the movement of the point in space, in XYZ coordinates. It was also correct to display the time coordinate, which in formula (2.15) is represented by the real part \(A\), but this cannot be done on a plane.

The following figures will show the direction of velocities and the movement of a point (their integrals) at light speeds along one of the coordinates.

We integrate the global velocity vector.

Another option

We integrate the global velocity vector.

Another option in which all speeds are equal to the speed of light.

We integrate the global velocity vector.

According to the classical theory of general relativity, in the latter version, where the linear and circular velocities are equal to the speed of light, time in a moving frame of reference must completely stop. But if we write the element \(A\) according to the formula (2.14), which is responsible for the time coordinate in the moving coordinate system, then it turns out that this coordinate is non-zero, and depends on time in a fixed coordinate system as follows: \[ A = - \int \limits_0^t \sin(\omega t) \cos(\omega t)\, dt = - {\sin(\omega t)^2 \over 4\pi} \tag{2.18}\] The movement of a point in space in this case can be seen in Figures 19-20.