2026-07-04
The origin of the Lorentz factor and energy invariant from the hyperbolic unit
In a previous work, we derived a formula for the fractional power of the hyperbolic unit, establishing a unique correspondence between the complex unit circle and the hyperbolic algebra. In the present paper, we show that this construction possesses a considerably deeper structure. Using the idempotent representation, the state can be decomposed into two mutually reciprocal components whose product remains invariant, making it possible to introduce an internal invariant of the system under consideration.
Based on this invariant, the parameters \(\beta\) and \(\gamma\) arise naturally, while the Lorentz factor [1] appears not as an initial postulate of special relativity but as the normalization factor of an idempotent state. As a result, compact representations of the fractional power of the hyperbolic unit are obtained in terms of the parameters \(\beta\) and \(\gamma\), together with its symmetric idempotent form, establishing a direct connection between complex and hyperbolic algebras and relativistic kinematics.
In our previous work, we demonstrated that the complex unit circle can be mapped uniquely onto the algebra of the hyperbolic unit \(\j\), for which \(\j^2=+1\). On this basis, a formula for the fractional power of the hyperbolic unit was obtained, together with an expression for its logarithm:
\[ \tag{1} \ln\j=i\pi(2k+1),\qquad k\in\mathbb Z. \] In the present work, we show that the same construction naturally leads to the Lorentz factor. Within this framework, the Lorentz factor emerges not as an external assumption, but as the normalization factor of an idempotent state preserving an internal invariant: \[ \large{\j^\alpha=\gamma(1+\j\beta)}. \]
Initial Formula for the Fractional Power
According to the results of the previous paper, the fractional power of the hyperbolic unit is given by
\[ \tag{2} \j^\alpha= \frac12 \left[ \left(1+e^{i\pi\alpha}\right) + \j\left(1-e^{i\pi\alpha}\right) \right]. \] A more general expression, taking into account the branches of the logarithm, is
\[ \tag{3} \j^\alpha= \frac12 \left[ \left(1+e^{i\theta}\right) + \j\left(1-e^{i\theta}\right) \right], \] where
\[ \tag{4} \theta=\pi\alpha(2k+1). \] Equation (3) simultaneously contains the complex phase \(e^{i\theta}\) and the hyperbolic unit \(\j\). Consequently, it naturally extends the conventional basis \(\{1,\j\}\) and motivates the consideration of an enlarged structure that also includes \(i\j\).
Idempotent Basis
This basis was introduced in our previous work. It is constructed from the two idempotents
\[ \tag{5} \e=\frac{1+\j}{2}, \qquad \eb=\frac{1-\j}{2}. \] These elements satisfy the properties
\[ \tag{6} \e^2=\e, \qquad \eb^2=\eb, \qquad \e\eb=0. \] Furthermore,
\[ \tag{7} 1=\e+\eb, \qquad \j=\e-\eb. \] Substituting (7) into equation (3) yields
\[ \tag{8} \j^\alpha= \frac12 \left[ \left(1+e^{i\theta}\right)(\e+\eb) + \left(1-e^{i\theta}\right)(\e-\eb) \right]. \] Collecting the coefficients of \(\e\) and \(\eb\), we obtain for \(\e\)
\[ \tag{9} \frac12 \left[ (1+e^{i\theta})+(1-e^{i\theta}) \right] =1. \] while for \(\eb\)
\[ \tag{10} \frac12 \left[ (1+e^{i\theta})-(1-e^{i\theta}) \right] =e^{i\theta}. \] Therefore, the fractional power of the hyperbolic unit takes the remarkably simple idempotent form
\[ \tag{11} \boxed{ \j^\alpha=\e+\eb e^{i\theta}. } \] This representation shows that the fractional power \(\j^\alpha\) modifies only one of the two idempotent components. However, the same expression can be rewritten in a more symmetric form by extracting the common phase factor:
\[ \tag{12} \j^\alpha= e^{i\theta/2} \left( \e e^{-i\theta/2} + \eb e^{i\theta/2} \right). \] Here, the common factor \(e^{i\theta/2}\) represents the overall complex phase, whereas the expression in parentheses describes the internal imbalance between the two idempotent components.
From Phase Imbalance to a Real Parameter
For further derivation of the Lorentz factor, we are interested not in the circular rotation itself, but in the ratio of the two internal components. Therefore, we introduce a real parameter \(Q\) describing the redistribution between idempotents:
\[ \tag{13} \j^\alpha=\e Q+\eb Q^{-1}. \] This notation preserves the main structural feature of an idempotent state: the components are mutually inverse. Therefore, their product is equal to one:
\[ \tag{14} Q\cdot Q^{-1}=1. \] It is this ratio that can be considered an internal invariant of the state. As \(Q\) changes, one component increases and the other decreases, but their product remains unchanged.
Sum and Difference of Components
Now let's consider the symmetric and antisymmetric parts of the state. We define:
\[ \tag{15} S=\frac{Q+Q^{-1}}{2}, \qquad D=\frac{Q-Q^{-1}}{2}. \] These quantities automatically satisfy the hyperbolic identity:
\[ \tag{16} S^2-D^2=1. \] Indeed:
\[ \tag{17} \left( \frac{Q+Q^{-1}}{2} \right)^2 - \left( \frac{Q-Q^{-1}}{2} \right)^2 = QQ^{-1}=1. \] Thus, from the sole condition of preserving the internal invariant \(QQ^{-1}=1\), a hyperbolic structure arises.
The Emergence of the β Parameter
Now we introduce a dimensionless quantity characterizing the relative imbalance of the two components. It is natural to define it as the ratio of the antisymmetric part to the symmetric part:
\[ \tag{18} \beta=\frac{D}{S}. \] That is,
\[ \tag{19} \beta= \frac{Q-Q^{-1}}{Q+Q^{-1}}. \] At this stage, \(\beta\) is not yet a velocity. It is simply a measure of the internal asymmetry of the idempotent state. If \(Q=1\), then the components are equal, \(D=0\), and therefore
\[ \tag{20} \beta=0. \] If \(Q\neq1\), a nonzero imbalance arises:
\[ \tag{21} \beta\neq0. \] Since \(D=\beta S\), identity (16) can be rewritten as follows:
\[ \tag{22} S^2-\beta^2 S^2=1. \] Hence
\[ \tag{23} S^2(1-\beta^2)=1. \] Therefore,
\[ \tag{24} S=\frac{1}{\sqrt{1-\beta^2}}. \] Now we denote the symmetric normalization part by \(\gamma\):
\[ \tag{25} \gamma=S. \] Then from (24) we immediately obtain:
\[ \tag{26} \gamma= \frac{1}{\sqrt{1-\beta^2}}. \] This is the Lorentz factor. In this construction, it appears as a normalization coefficient for the idempotent state while maintaining the internal invariant \(QQ^{-1}=1\).
Physical Interpretation of the Parameter β
The resulting quantity \(\beta\) was originally introduced as the ratio of the difference between the components to their sum. However, its form is completely identical to the dimensionless velocity in special relativity. Therefore, a physical identification can be performed further:
\[ \tag{27} \beta=\frac{v}{c}. \] Then formula (26) takes on the standard relativistic form:
\[ \tag{28} \gamma= \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}. \] It is important to emphasize that in this logic, \(\beta\) is not introduced in advance as a velocity. It first appears as an internal measure of the imbalance of idempotent components, and only then receives a physical interpretation as the ratio of the speed of motion to the speed of light.
Expressing Q in Terms of β
From Definition (19):
\[ \tag{29} \beta= \frac{Q-Q^{-1}}{Q+Q^{-1}}. \] Multiply the numerator and denominator by \(Q):
\[ \tag{30} \beta= \frac{Q^2-1}{Q^2+1}. \] Hence:
\[ \tag{31} \beta(Q^2+1)=Q^2-1. \] Expanding the parentheses, we get:
\[ \tag{32} \beta Q^2+\beta=Q^2-1. \] Moving the terms from \(Q^2\) to one side:
\[ \tag{33} Q^2(1-\beta)=1+\beta. \] Therefore,
\[ \tag{34} Q^2=\frac{1+\beta}{1-\beta}. \] And finally:
\[ \tag{35} \boxed{ Q= \sqrt{\frac{1+\beta}{1-\beta}}. } \] This same quantity can be written using a hyperbolic parameter:
\[ \tag{36} Q=e^\eta, \qquad \eta=\operatorname{artanh}\beta. \] However, in this derivation, the parameter \(\eta\) is not the original one. It appears only as a convenient way to write the already found relation \(Q\).
Compact Form of Fractional Powers
Let's return to the idempotent representation:
\[ \tag{37} \j^\alpha=\e Q+\eb Q^{-1}. \] Using definitions (15), we rewrite it in terms of sum and difference:
\[ \tag{38} \j^\alpha= \frac{Q+Q^{-1}}{2} + \j \frac{Q-Q^{-1}}{2}. \] But by definition:
\[ \tag{39} \frac{Q+Q^{-1}}{2}=\gamma, \qquad \frac{Q-Q^{-1}}{2}=\gamma\beta. \] Therefore:
\[ \tag{40} \j^\alpha=\gamma+\j\gamma\beta. \]
Or in even more compact form:
\[ \tag{41} \large{\j^\alpha=\gamma(1+\j\beta)}. \]
This is one of the central formulas of this paper. It shows that the fractional power of the hyperbolic unit can be written directly in terms of the parameters \(\beta\) and \(\gamma\).
Form via Idempotents
Using expression (35), we obtain the final idempotent form:
\[ \tag{42} \j^\alpha = \e \sqrt{\frac{1+\beta}{1-\beta}} + \eb \sqrt{\frac{1-\beta}{1+\beta}}. \] Or, via \(Q\): \[ \tag{43} \j^\alpha=\e Q+\eb Q^{-1}, \qquad Q= \sqrt{\frac{1+\beta}{1-\beta}}. \]
Formulas (41) and (43) are two equivalent representations of the same state. The first shows the connection with the Lorentz factor, the second the internal idempotent structure.
Relationship with Preservation of an Internal Invariant
Consider the product of two idempotent coefficients:
\[ \tag{44} Q\cdot Q^{-1}=1. \] This means that when the state changes, the internal normalization is preserved. If one component increases by a factor of \(Q\), the other decreases by a factor of \(Q\). Therefore, the change in state is not arbitrary: it occurs while preserving the invariant.
In this case, the observable symmetric part of the state is equal to:
\[ \tag{45} \gamma= \frac{Q+Q^{-1}}{2}. \] And the antisymmetric part is equal to:
\[ \tag{46} \gamma\beta= \frac{Q-Q^{-1}}{2}. \] Hence it follows:
\[ \tag{47} \gamma^2-(\gamma\beta)^2=1. \] Or:
\[ \tag{48} \gamma^2(1-\beta^2)=1. \] Thus, the Lorentz factor arises as a coefficient necessary to preserve the internal invariant in the presence of an imbalance between idempotent components.
Conservation of the Norm of the Energy State
The resulting compact form of the fractional power of the hyperbolic unit allows us to transition from the dimensionless state to the energy state of a material point. To do this, we multiply \(\j^\alpha\) by the rest energy \(mc^2\):
\[ \tag{49} \mathcal E = mc^2\j^\alpha. \] Using the compact form found earlier
\[ \tag{50} \j^\alpha=\gamma(1+\j\beta), \] we obtain:
\[ \tag{51} \mathcal E = mc^2\gamma(1+\j\beta). \] Since \(\beta=v/c\), the second term can be rewritten in terms of the velocity \(v\):
\[ \tag{52} \mathcal E = \gamma mc^2 + \j\gamma mvc. \] Let's introduce standard relativistic notations for energy and momentum:
\[ \tag{53} E=\gamma mc^2, \qquad p=\gamma mv. \] Then the energy state takes a particularly simple form:
\[ \tag{54} \boxed{ \mathcal E = E+\j pc. } \] In this notation, the energy \(E\) and the quantity \(pc\) have the same dimensions, so they act as two components of a single hyperbolic object. The real part corresponds to the energy component, and the hyperbolic part corresponds to the momentum component multiplied by the speed of light.
Now let's consider the conjugate energy state:
\[ \tag{55} \overline{\mathcal E} = mc^2\j^{-\alpha}. \] Since conjugation changes the sign of the hyperbolic part, we have:
\[ \tag{56} \j^{-\alpha}=\gamma(1-\j\beta), \] and therefore
\[ \tag{57} \overline{\mathcal E} = E-\j pc. \] The product of a state and its adjoint gives the square of its norm:
\[ \tag{58} \mathcal E\overline{\mathcal E} = (E+\j pc)(E-\j pc). \] Since \(\j^2=1\), we get:
\[ \tag{59} (E+\j pc)(E-\j pc) = E^2-p^2c^2. \] On the other hand, from the definition of the energy state it follows:
\[ \tag{60} \mathcal E\overline{\mathcal E} = m^2c^4\j^\alpha\j^{-\alpha}. \] But the fractional powers \(\j^\alpha\) and \(\j^{-\alpha}\) are mutually inverse:
\[ \tag{61} \j^\alpha\j^{-\alpha}=1. \] Therefore, the norm of the energy state is preserved:
\[ \tag{62} \mathcal E\overline{\mathcal E} = m^2c^4. \] Comparing two expressions for the same norm, we obtain the relativistic energy invariant:
\[ \tag{63} E^2-p^2c^2=m^2c^4. \] Thus, inIn this model, the law of conservation of energy-momentum is expressed as conservation of the norm of the energy state \(mc^2\j^\alpha\). When the parameter \(\alpha\) changes, the energy \(E\) and momentum \(p\) are redistributed between the components of a single hyperbolic object, but its norm remains unchanged.
This is why the Lorentz factor can be interpreted as a coefficient that ensures conservation of the energy invariant in the presence of a nonzero imbalance \(\beta\). In this formulation, \(\gamma\) arises not only from the normalization of the idempotent state, but also from the requirement to conserve the norm of the corresponding energy state.
Interpretation of the Result
The resulting conclusion shows that the fractional power of the hyperbolic unit connects three levels of description: the internal idempotent state, the Lorentz normalization, and the energy invariant. First, the coefficient \(gamma\) arises from the mutually inverse components \(Q\) and \(Q^{-1}\); then, after multiplying by \(mc^2\), this same coefficient becomes the energy normalization.
In the usual notation of special relativity, the quantity \(gamma\) arises from the requirement of invariance of the space-time interval. In the construction under consideration, it arises from the conservation of the internal idempotent invariant.
\[ \tag{64} QQ^{-1}=1, \] and then leads to the conservation of the energy norm.
\[ \tag{65} \mathcal E\overline{\mathcal E}=m^2c^4. \] After physical identification, \(\beta=v/c\), the imbalance parameter becomes the relative velocity, and the normalization factor \(\gamma\) coincides with the Lorentz factor. Therefore, within the framework of this model, the Lorentz factor can be viewed as a consequence of the conservation of the energy state norm.
Final Derivation Chain
The result can be represented as a sequential chain of transformations. Each subsequent step follows directly from the previous one and shows how the Lorentz factor, idempotent normalization, and energy invariant arise from the fractional power of the hyperbolic unit.
\[ \tag{66} \ln\j=i\pi(2k+1) \] The starting point is the logarithm of the hyperbolic unit. It defines the fractional power \(\j^\alpha\) and connects the complex phase with hyperbolic algebra.
\[ \tag{67} \j^\alpha= \frac12 \left[ (1+e^{i\theta}) + \j(1-e^{i\theta}) \right] \] Transition to an idempotent basis allows us to represent the state through two mutually inverse components:
\[ \tag{68} \j^\alpha=\e Q+\eb Q^{-1} \] Their product remains unchanged, which defines an internal invariant of the state:
\[ \tag{69} QQ^{-1}=1 \] The ratio of the antisymmetric part to the symmetric part determines the dimensionless parameter imbalance:
\[ \tag{70} \beta= \frac{Q-Q^{-1}}{Q+Q^{-1}} \] The half-sum of the reciprocal components defines the normalization coefficient:
\[ \tag{71} \gamma= \frac{Q+Q^{-1}}{2} \] The Lorentz factor follows directly from these two definitions:
\[ \tag{72} \gamma= \frac{1}{\sqrt{1-\beta^2}} \] Therefore, the fractional power of the hyperbolic unit takes a compact form:
\[ \tag{73} \j^\alpha=\gamma(1+\j\beta) \] The equivalent idempotent form shows how the same result is expressed through the imbalance parameter: \[ \tag{74} \j^\alpha = \e \sqrt{\frac{1+\beta}{1-\beta}} + \eb \sqrt{\frac{1-\beta}{1+\beta}} \]
After multiplying by the rest energy \(mc^2\), this same state acquires an energetic interpretation:
\[ \tag{75} mc^2\j^\alpha=E+\j pc \] Preserving the norm of this hyperbolic object yields a relativistic energy invariant:
\[ \tag{76} E^2-p^2c^2=m^2c^4. \] Thus, the entire derivation sequence begins with the logarithm of the hyperbolic unit, passes through an idempotent expansion, and ends with the preservation of the norm of the energy state. The Lorentz factor appears as a necessary normalization coefficient, ensuring the preservation of this invariant.
Conclusions
The fractional power of the hyperbolic unit, obtained from the map of the complex unit circle, has a natural idempotent representation. This representation separates the state into two mutually inverse components \(\e Q\) and \(\eb Q^{-1}\), the product of whose coefficients is preserved.
The preservation of the internal invariant \(QQ^{-1}=1\) gives rise to a hyperbolic identity. The relative imbalance of the components determines the parameter \(\beta\), and normalizing the symmetric part leads to the coefficient
\[ \tag{77} \gamma= \frac{1}{\sqrt{1-\beta^2}}. \] After identifying \(\beta=v/c\), this coefficient coincides with the Lorentz factor of special relativity. Moreover, \(\gamma\) appears not as an external postulate, but as a consequence of the normalization of the idempotent state.
Additional multiplication by the rest energy \(mc^2\) converts this state into energy form.
\[ \tag{78} mc^2\j^\alpha=E+\j pc. \] Since \(\j^\alpha\j^{-\alpha}=1\), the energy state norm is preserved:
\[ \tag{79} (E+\j pc)(E-\j pc)=m^2c^4. \] From this, the relativistic energy invariant follows directly.
\[ \tag{80} E^2-p^2c^2=m^2c^4. \] Therefore, within the framework of the proposed model, the Lorentz factor can be interpreted as a coefficient that ensures the conservation of the energy state norm. In other words, the law of conservation of energy-momentum manifests itself here as the conservation of the modulus of a hyperbolic object \(mc^2\j^\alpha\), while energy and momentum are its two interconnected projections.
Materials used
- Wikipedia. Lorentz factor.