As a continuation of the previous work, a new method for describing motion in the Cartesian basis \(\{\ep,i\ep,\em,i\em\}\), based on a general rotation of complex planes, is considered. It is shown that the inverse Lorentz factor arises naturally as a consequence of the conservation of the vector's modulus when the rotation angle changes, without introducing additional relativistic postulates. Analytical expressions for the velocity vector and the displacement vector are obtained, describing both the translational motion and the internal rotational structure.

Particular attention is given to the geometric interpretation of the obtained results. The paper demonstrates how a change in relative velocity leads to a continuous transition between the representations \((ct,y,z)\) and \((x,y,z)\), and also examines the limiting case of motion at the speed of light. For clarity, an interactive 3D visualization is provided, allowing one to change the velocity, select a coordinate system, and explore the trajectory geometry in the new basis.

Motion in the paradigm of our Cartesian basis \(\{\ep,i\ep,\em,i\em\}\) can be obtained in several ways. One of them is based on the phase difference between two complex planes: \[\tag{1} V=c\left(\ep e^{ig}+\em e^{ih}\right). \] Here \(c\) is the speed of light, and \(g\) and \(h\) are independent phase angles of two complex planes. This approach can be used to describe the relative positions of two independent complex spaces.

In this paper, we will consider another method, fundamentally different from the previous one. To do this, we take the velocity vector from formula (11) of the previous paper \[\tag{2} V'(t)= c\left(\ep+\em e^{i\omega t}\right), \] and perform the same rotation of both complex planes, modulating the vector by the factor \(e^{i\alpha}\): \[\tag{3} V(t)= c\,e^{i\alpha} \left( \ep+ \em e^{i\omega t} \right). \] Here, the angle \(\alpha\) defines the overall rotation of the Cartesian basis relative to the initial position.

Using the properties of idempotents and expanding complex exponentials using Euler's formula, we obtain a representation of the vector in a Cartesian basis. \[\tag{4} V(t)= \begin{pmatrix} c\cos(\alpha)\\ c\sin(\alpha)\\ c\cos(\omega t+\alpha)\\ c\sin(\omega t+\alpha) \end{pmatrix}. \] To convert the rotation angle into a familiar physical velocity, we use the physical interpretation obtained in this work, and introduce the relative velocity \[\tag{5} \beta={v\over c}, \] where \(v\) is the velocity, varying between zero and the speed of light \(c\). Then the rotation angle can be written as \[\tag{6} \alpha=\arcsin(\beta). \] The velocity vector is transformed to the form \[\tag{7} V(t)= \begin{pmatrix} c\bar\gamma\\ v\\ \cos(\omega t)\,c\bar\gamma - v\sin(\omega t)\\ \sin(\omega t)\,c\bar\gamma + v\cos(\omega t) \end{pmatrix}, \] where \(\bar\gamma=\sqrt{1-\beta^2}\) is the inverse Lorentz factor [1]. In this case, the absolute value of the first two coordinates is preserved: \[ (c\bar\gamma)^2+v^2=c^2, \] from which the expression for the inverse Lorentz factor follows directly. For \(\beta=0\), we have \(\alpha=0\), so expression (7) transforms into the original velocity vector (2).

If we establish the correspondence \[\tag{8} \begin{aligned} \ep &\longleftrightarrow ct,\\ i\ep &\longleftrightarrow x,\\ \em &\longleftrightarrow y,\\ i\em &\longleftrightarrow z, \end{aligned} \] then the coordinate \(i\ep\) coincides with the familiar relative velocity \(v\), while the coordinate \(\ep\), corresponding to time, automatically acquires a factor of \(c\bar\gamma\). Thus, the relativistic contraction of the time component arises directly from the geometry of the new Cartesian basis, without introducing additional postulates. Another method for obtaining the Lorentz factor is discussed in a separate paper.

Integrating the velocity vector (7) in the standard way, we obtain the displacement vector \[\tag{9} L(t)= \begin{pmatrix} c\bar\gamma t \\ vt \\ r \bar\gamma\, \sin(\omega t) + {v \over \omega} \cos(\omega t)\\ -r \bar\gamma\, \cos(\omega t)+ {v \over \omega} \sin(\omega t) \end{pmatrix}, \] where \[ r={c\over\omega}. \] Here, the first two coordinates describe translational motion, and the last two coordinates preserve the internal rotational structure associated with the complex plane \(\{\em,i\em\}\).

The special case of \(\beta=1\), that is, when the velocity \(v\) reachest the speed of light. In this case, \(\bar\gamma=0\), and the displacement vector takes the form \[\tag{10} L(t) \bigg|_{v=c}= \begin{pmatrix} 0 \\ ct \\ r \cos(\omega t)\\ r \sin(\omega t) \end{pmatrix}. \] Compared to the original case, the coordinate corresponding to time vanishes, while the coordinate \(i\ep\), associated with directed motion, becomes equal to \(ct\). In other words, at the maximum velocity, the contribution of the time component is completely transferred to the motion component. This can be viewed as a geometric expression of the law of conservation of the total magnitude of a vector in a given basis.

The figure below shows the spatial portion of the displacement vector of a point \(L(t)\) for \[c=1, \quad \omega=2\pi, \quad r=1/2\pi. \] The graph displays one turn of the spiral.

Use the mouse wheel to zoom, and drag the left mouse button to rotate the graph. The toggle allows you to choose between displaying in (x, y, z) or (ct, y, z) coordinates, since it is impossible to simultaneously represent all four spatial coordinates on a two-dimensional screen.

This paper shows that motion can be represented as the result of a complex modulation of a new Cartesian basis \(\{\ep,i\ep,\em,i\em\}\). In this case, the relative velocity is determined by the angle of rotation of the complex plane, and the inverse Lorentz factor arises naturally as the projection of the unit vector onto the corresponding coordinate axis. Analytical expressions for the velocity vector and displacement vector are obtained, and their geometric properties in coordinates \((x,y,z)\) and \((ct,y,z)\) are examined.

It should be noted that all transformations in this work are performed within the velocity vector. \[ V(t)=c\,e^{i\alpha}\left(\ep+\em e^{i\omega t}\right), \] the modulus of which remains constant and equal to the speed of light: \[ |V(t)|=c. \] This means that a change in relative velocity does not change the total norm of the vector, but only redistributes it between the translational and internal rotational components. Within the framework of the proposed geometric model, such invariance can be viewed as a mathematical expression of the law of conservation of energy: motion changes the structure of the distribution of the vector's components, but not its total modulus.