In the basic work, a new Cartesian basis based on two idempotents was proposed, in which the motion of a particle is described by a single complex vector with internal and external components. This approach allows one to naturally relate the particle velocity to the geometric angle of the new basis and obtain the Lorentz factor without using traditional Lorentz transformations.

In this paper, we show that this same angle completely determines not only the velocity but also the momentum, rest energy, and total energy of the particle. As a result, energy and momentum acquire a simple geometric interpretation as projections of a single energy vector, and the fundamental relativistic relationship between them turns out to be a direct consequence of ordinary Euclidean trigonometry. This approach unifies velocity, momentum, and energy within a single geometric construct built on the new Cartesian basis.

A new Cartesian basis was previously introduced \[ \tag{1} \left\{ \ep,\, i\ep,\, \em,\, i\em \right\}, \] constructed from two mutually canceling idempotents \[ \tag{2} \ep^2=\ep, \qquad \em^2=\em, \qquad \ep\em=0, \qquad \ep+\em=1. \] In such a basis, the motion of a particle can be represented as a combination of external and internal complex components.

In this paper, we show that the angle \(\alpha\), which determines the external phase modulation of the motion vector, simultaneously determines the velocity, momentum, and total energy of the particle. In this case, the relativistic relationship between energy and momentum turns out to be a direct consequence of the usual trigonometric identity.

Consider the vector \[ \tag{3} V(t)= c\,e^{i\alpha} \left( \ep+ \em e^{i\omega t} \right), \] where \(c\) is the speed of light, \(\omega\) is the frequency of the internal cyclic motion, and \(\alpha\) is the angle of external phase modulation.

After expanding the parentheses, we obtain \[ \tag{4} V(t)= \ep c\,e^{i\alpha} + \em c\,e^{i(\omega t+\alpha)}. \] The first component refers to the external motion, while the second describes the internal cyclic process in the plane \[ \tag{5} \left\{ \em,\, i\em \right\}. \]

We will assume that the observed velocity is determined by the imaginary projection of the external complex component: \[ \tag{6} v=c\sin\alpha. \] Therefore, \[ \tag{7} \beta= \frac{v}{c} = \sin\alpha. \]

From the definition of the Lorentz factor \[ \tag{8} \gamma= \frac{1}{\sqrt{1-\beta^2}} \] and the relation \(\beta=\sin\alpha\), we obtain \[ \tag{9} \gamma= \frac{1}{\sqrt{1-\sin^2\alpha}} = \frac{1}{\cos\alpha}. \] Here, we consider the range \[ \tag{10} 0\leqslant\alpha<\frac{\pi}{2}, \] therefore \(\cos\alpha\) remains positive.

Thus, the inverse Lorentz factor is a real projection of the unit complex exponential: \[ \tag{11} \frac{1}{\gamma}=\cos\alpha. \] Moreover, the external component of the vector itself can be represented in the form \[ \tag{12} c\,e^{i\alpha} = c\cos\alpha + i c\sin\alpha = \frac{c}{\gamma} + iv. \]

Let \(m_0\) denote the invariant mass of the particle. The total relativistic energy is determined by the expression \[ \tag{13} E=\gamma m_0c^2. \] Taking into account formula (9), we obtain \[ \tag{14} E= \frac{m_0c^2}{\cos\alpha}. \] Consequently, \[ \tag{15} \cos\alpha= \frac{m_0c^2}{E}. \]

Formula (15) shows that the rest energy \(m_0c^2\) can be considered as the projection of the total energy \(E\) onto the direction corresponding to the real component of the complex exponential.

The relativistic momentum of a particle is \[ \tag{16} p=\gamma m_0v. \] Substituting \[ \tag{17} v=c\sin\alpha \] and \[ \tag{18} \gamma= \frac{1}{\cos\alpha}, \] we get \[ \tag{19} p= m_0c \frac{\sin\alpha}{\cos\alpha} = m_0c\tan\alpha. \]

After multiplying by \(c\) \[ \tag{20} pc= m_0c^2\tan\alpha. \] Using the expression \[ \tag{21} E= \frac{m_0c^2}{\cos\alpha}, \] we can rewrite formula (20) as \[ \tag{22} pc= E\sin\alpha. \]

Thus, the same angle \(\alpha\) defines two dimensionless energy projections: \[ \tag{24} \cos\alpha= \frac{m_0c^2}{E}, \qquad \sin\alpha= \frac{pc}{E}. \]

These relations allow a simple geometric interpretation. The total energy \(E\) isis the hypotenuse of a right triangle, the rest energy \(m_0c^2\) is the adjacent leg, and the quantity \(pc\) is the opposite leg.

Fig. 1. Total energy \(E\), rest energy \(m_0c^2\), and momentum component \(pc\).

For the triangle shown, the following relations hold: \[ \tag{25} \frac{pc}{E}=\sin\alpha, \qquad \frac{m_0c^2}{E}=\cos\alpha, \qquad \frac{pc}{m_0c^2}=\tan\alpha. \] The last expression directly relates the angle to the momentum: \[ \tag{26} \tan\alpha= \frac{p}{m_0c}. \] Therefore, \[ \tag{27} \alpha= \arctan \left( \frac{p}{m_0c} \right). \]

Thus, the angle \(\alpha\) can be determined not only through the velocity, but also directly through the particle's momentum.

We use the basic trigonometric identity \[ \tag{28} \sin^2\alpha+ \cos^2\alpha = 1. \] Substituting expressions (24), we obtain \[ \tag{29} \left( \frac{pc}{E} \right)^2 + \left( \frac{m_0c^2}{E} \right)^2 = 1. \] After multiplying by \(E^2\), \[ \tag{30} p^2c^2+ m_0^2c^4 = E^2. \] Thus, \[ \tag{31} E^2= p^2c^2+ m_0^2c^4. \]

The relativistic relationship between energy, momentum, and rest mass arises here as a direct consequence of the Euclidean trigonometric identity. Within the framework of this construction, its derivation does not require separately introducing the hyperbolic geometry of spacetime: the necessary hyperbolic dependencies are already contained in the relation \[ \tag{32} \gamma= \frac{1}{\cos\alpha}. \]

If we introduce the effective relativistic mass \[ \tag{33} m=\gamma m_0, \] then the total energy takes the familiar form \[ \tag{34} E=mc^2. \] In this case, the momentum can be written as \[ \tag{35} p=mv. \]

Since \[ \tag{36} v=c\sin\alpha, \] we get \[ \tag{37} pc= mc^2\sin\alpha. \] Since \[ \tag{38} mc^2=E, \] then \[ \tag{39} pc= E\sin\alpha. \]

At the same time, the rest energy is expressed in terms of the total energy: \[ \tag{40} m_0c^2= E\cos\alpha. \] Consequently, \(mc^2\) is the total energy modulus, and \(pc\) and \(m_0c^2\) are its mutually perpendicular projections: \[ \tag{41} mc^2=E, \qquad pc=E\sin\alpha, \qquad m_0c^2=E\cos\alpha. \]

This allows us to represent a single complex energy vector: \[ \tag{42} \mathcal{E}(\alpha) = m_0c^2+ i pc. \] Taking into account formulas (41), \[ \tag{43} \mathcal{E}(\alpha) = E\cos\alpha+ iE\sin\alpha = E e^{i\alpha}. \] Therefore, \[ \tag{44} \left| \mathcal{E}(\alpha) \right| = E = mc^2. \]

Thus, the complex exponential \(e^{i\alpha}\) can be used not only to represent the velocity vector, but also to combine the rest energy and momentum into a single energy quantity: \[ \tag{45} E e^{i\alpha} = m_0c^2+ ipc. \]

The total energy consists of the rest energy and the relativistic kinetic energy: \[ \tag{46} E= m_0c^2+ E_k. \] Therefore, \[ \tag{47} E_k= E-m_0c^2. \] Using expression (14), we obtain \[ \tag{48} E_k=m_0c^2 \left( \frac{1}{\cos\alpha}-1 \right). \]

At low velocities \[ \tag{49} \gamma \approx 1+ \frac{\beta^2}{2}. \] Therefore \[ \tag{50} E_k = m_0c^2(\gamma-1) \approx \frac{m_0c^2\beta^2}{2}. \] Since \(\beta=v/c\), we obtain the classical expression \[ \tag{51} E_k \approx \frac{m_0v^2}{2}. \]

In terms of the angle \(\alpha\), this formula is written as \[ \tag{52} E_k \approx \frac{m_0c^2}{2} \sin^2\alpha. \] Thus, classical kinetic energy is determined by the square of the spatial projection of the external complex component of the motion vector.

The resulting expressions can be combined into a system \[ \tag{53} \begin{aligned} v(\alpha) &= c\sin\alpha, \\[4pt] \gamma(\alpha) &= \frac{1}{\cos\alpha}, \\[4pt] p(\alpha) &= m_0c\tan\alpha, \\[4pt] E(\alpha) &= \frac{m_0c^2}{\cos\alpha}. \end{aligned} \]

Therefore, a single angle \(\alpha\) simultaneously determines the velocity, Lorentz factor, momentum, and total energy of a material point.

For \(\alpha=0\) \[ \tag{54} v=0, \qquad p=0, \qquad \gamma=1, \qquad E=m_0c^2. \] As \(\alpha\) approaches \(\pi/2\) \[ \tag{55} v\to c, \qquad \gamma\to\infty, \qquad p\to\infty, \qquad E\to\infty. \] Thus, the limiting speed of light arises directly from the geometric limitation of the angle.

This paper demonstrates that the motion vector constructed using the new Cartesian basis naturally introduces a single geometric parameter—angle \(\alpha\). This angle directly expresses the velocity, Lorentz factor, momentum, rest energy, and total energy of a particle. Furthermore, the rest energy and the quantity \(pc\) acquire a clear interpretation as two mutually perpendicular projections of the total energy, and the complex form of the energy vector unites them into a single algebraic construct.

The resulting representation allows one to derive the fundamental relativistic relationship between energy, momentum, and rest mass directly from the trigonometric properties of angle \(\alpha\), without resorting to Lorentz transformations or four-dimensional spacetime geometry. Thus, the new basis not only provides a compact notation for the fundamental relations of special relativity but also offers a unified geometric interpretation, in which velocity, momentum, and energy are different manifestations of the same angular parameter.

The vector \[ \tag{56} V(t)= c\,e^{i\alpha} \left( \ep+ \em e^{i\omega t} \right) \] contains the external phase parameter \(\alpha\), which can be considered a single geometric parameter of motion.

From the relationship \[ \tag{57} \sin\alpha= \frac{v}{c} \], expressions for the Lorentz factor, momentum, and total energy follow: \[ \tag{58} \gamma= \frac{1}{\cos\alpha}, \qquad p= m_0c\tan\alpha, \qquad E= \frac{m_0c^2}{\cos\alpha}. \]

In this case, the rest energy and the momentum component are two mutually perpendicular projections of the total energy: \[ \tag{59} m_0c^2= E\cos\alpha, \qquad pc= E\sin\alpha. \] Their combination leads to the complex energy form \[ \tag{60} E e^{i\alpha} = m_0c^2+ ipc, \] the modulus of which is \[ \tag{61} E=mc^2. \]

The fundamental relativistic relation \[ \tag{62} E^2= p^2c^2+ m_0^2c^4 \] turns out to be a consequence of the usual identity \[ \tag{63} \sin^2\alpha+ \cos^2\alpha = 1. \] Thus, velocity, momentum, rest energy, and total energy are linked into a single geometric construct determined by the angle \(\alpha\).