2026-07-04
Geometric origin of the Lorentz factor and the energy invariant
How does a factor that increases indefinitely as a particle's velocity approaches the speed of light arise from a single rotation? In special relativity, the Lorentz factor [1] is introduced by the expression \(\gamma=1/\sqrt{1-\beta^2}\). In this paper, we show that in split geometry, this quantity has a simple meaning: it is the inverse real projection of a single external state of a particle.
The complete operator \(J(a,b)=\j^a(-\j)^b\) separates the internal periodicity and external motion between two orthogonal idempotent planes. The internal exponent \(a\) is responsible for the particle's eigenstate, and the external exponent \(b\) is related to its velocity. Therefore, the origin of the Lorentz factor can be considered independently of the origin of mass.
The main result of the article is that the standard relativistic invariant
\[ \tag{1} E^2-p^2c^2=E_0^2=m^2c^4 \] turns out to be a physical notation of the usual trigonometric identity for two mutually perpendicular projections of a single external state.
1. Two idempotent planes
Split geometry is based on two mutually complementary idempotents:
\[ \tag{2} \ep^2=\ep, \qquad \em^2=\em, \qquad \ep\em=0, \qquad \ep+\em=1. \] The hyperbolic unit is expressed through them as
\[ \tag{3} \j=\ep-\em, \qquad \j^2=1. \] In the extended i-basis, two independent complex planes arise:
\[ \tag{4} \mathcal S_i = \operatorname{span}_{\mathbb R} \left\{ \ep,\, i\ep,\, \em,\, i\em \right\}. \] The plane \((\ep,i\ep)\) will henceforth correspond to the external motion, and the plane \((\em,i\em)\) will correspond to the internal state of the particle. Their orthogonality is expressed by the equality \(\ep\em=0\).
2. Powers of Hyperbolic Units
To correctly determine powers, it is necessary to keep in mind that the logarithm of a hyperbolic unit can also be expanded in terms of idempotents. On the principal branches
\[ \tag{5} \ln\j=i\pi\em, \qquad \ln(-\j)=i\pi\ep. \] This yields two independent phase operators:
\[ \tag{6} \j^a = e^{a\ln\j} = \ep+\em e^{i\pi a}, \] \[ \tag{7} (-\j)^b = e^{b\ln(-\j)} = \ep e^{i\pi b}+\em. \] The first operator rotates only the \(\em\) component, and the second only the \(\ep\) component. Their product forms the complete state operator:
\[ \tag{8} \boxed{ J(a,b) = \j^a(-\j)^b = \ep e^{i\pi b} + \em e^{i\pi a} }. \] Under complex phase conjugation, we obtain
\[ \tag{9} J\overline J = \ep+\em =1. \] Thus, the external motion and internal periodicity can change independently, but the complete state retains the unit norm.
3. Internal and external parameters of state
The internal exponent is determined by the particle's intrinsic periodicity:
\[ \tag{10} a=\varpi t, \qquad \pi\varpi=\omega_{\mathrm{int}}. \] The external exponent is determined by the velocity of the center of mass:
\[ \tag{11} b = \frac{\arcsin\beta}{\pi}, \qquad \beta=\frac{v}{c}. \] In general, \(v=v(t)\), therefore \(b=b(t)\). From definition (11) it immediately follows
\[ \tag{12} \sin(\pi b)=\beta. \] Since the external phase belongs to the unit circle, its second projection is equal to
\[ \tag{13} \cos(\pi b) = \sqrt{1-\beta^2}. \] For motion in the chosen direction, the principal region \(0\leqslant b\leqslant 1/2\) is assumed, in which both quantities \(\sin(\pi b)\) and \(\cos(\pi b)\) are non-negative.
4. Geometric origin of the Lorentz factor
In standard relativistic mechanics, the Lorentz factor is defined by the expression
\[ \tag{14} \gamma = \frac{1}{\sqrt{1-\beta^2}}. \] Comparing formulas (13) and (14), we obtain its representation in terms of the external exponent:
\[ \tag{15} \boxed{ \frac{1}{\gamma} = \cos(\pi b) }, \] \[ \tag{16} \boxed{ \gamma = \frac{1}{\cos(\pi b)} = \frac{1}{\sqrt{1-\beta^2}} }. \] Consequently, \(1/\gamma\) is the real projection of a single external state, and \(\gamma\) is the inverse of this projection. When \(b=0\) the particle is at rest, the projection is equal to unity and \(\gamma=1\). When \(b\to1/2\) the real projection tends to zero, therefore \(\gamma\to\infty\).
It is important that the dimensionless operator \(J\) itself remains unity, not the particle energy. Unlimited growth of \(\gamma\) arises in the transition from the normalized geometric state to physical quantities measured relative to the decreasing projection \(\cos(\pi b)\).
5. From the External Phase to Energy and Momentum
To transition from dimensionless geometry to physics, we associate two normalized quantities with two orthogonal projections of the external phase:
\[ \tag{17} \boxed{ \frac{E_0}{E} = \cos(\pi b), \qquad \frac{pc}{E} = \sin(\pi b) }. \] Here \(E\) is the total energy, \(p\) is the momentum, and \(E_0\) is the rest energy. Formula (17) can be written as a single complex vector in the energy plane:
\[ \tag{18} \boxed{ E_0+i\,pc = E e^{i\pi b} }. \] This equality does not mean that energy or momentum become complex observable quantities. The complex notation merely combines two actual, mutually perpendicular projections of a single energy state.
The first formula (17), together with (15), yields
\[ \tag{19} E_0 = E\cos(\pi b) = \frac{E}{\gamma}, \] from where
\[ \tag{20} \boxed{ E=\gamma E_0 }. \] The second projection, taking into account \(\sin(\pi b)=\beta\), takes the form
\[ \tag{21} pc=E\beta. \] Substituting formula (20) here, we obtain
\[ \tag{22} \boxed{ pc = \gamma\beta E_0 }. \] If \(E_0=mc^2\), then formulas (20) and (22) become standard relativistic expressions.
\[ \tag{23} \boxed{ E=\gamma mc^2, \qquad p=\gamma mv }. \] 6. Energy Invariant as the Pythagorean Theorem
We square both parts of formula (17) and add:
\[ \tag{24} \left(\frac{E_0}{E}\right)^2 + \left(\frac{pc}{E}\right)^2 = \cos^2(\pi b) + \sin^2(\pi b) =1. \] After multiplying by \(E^2\), we obtain
\[ \tag{25} E_0^2+p^2c^2=E^2. \] Therefore,
\[ \tag{26} \boxed{ E^2-p^2c^2 = E_0^2 = m^2c^4 }. \] Thus, the energy invariant represents the Pythagorean theorem for two projections of the total energy: rest \(E_0\) and momentum \(pc\). The unusual minus sign appears only after moving the momentum component to the left side of the equation.
In normalized form, the relationship is especially clear:
\[ \tag{27} \boxed{ \left(\frac{mc^2}{E}\right)^2 + \left(\frac{pc}{E}\right)^2 =1 }. \] Essentially, formula (27) is a physical notation of the identity \(\cos^2(\pi b)+\sin^2(\pi b)=1\).
7. What Exactly Is Obtained from Geometry
A distinction should be made between the mathematical part of the construction and its physical interpretation. The definition of the external exponent \(b\) and the unity of phase rotation directly imply the following relations
\[ \tag{28} \sin(\pi b)=\beta, \qquad \cos(\pi b)=\sqrt{1-\beta^2} = \frac{1}{\gamma}. \] A new assumption of the model is the mapping of these geometric projections onto physical relations
\[ \tag{29} \boxed{ \sin(\pi b) \longleftrightarrow \frac{pc}{E}, \qquad \cos(\pi b) \longleftrightarrow \frac{E_0}{E} }. \] After this identification, the standard formulas for energy, momentum, and their invariant follow from the geometry of unit rotation. Therefore, we are not talking about an independent proof of special relativity, but rather about its geometric representation using the split operator.
8. Connection with the Hyperbolic Boost
In special relativity, the transformation of motion is conveniently described by the rapidity \(\eta\):
\[ \tag{30} \eta = \operatorname{artanh}\beta. \] For a hyperbolic unit \(\j^2=1\), the corresponding exponential is
\[ \tag{31} e^{\eta\j} = \cosh\eta + \j\sinh\eta. \] Since
\[ \tag{32} \cosh\eta=\gamma, \qquad \sinh\eta=\gamma\beta, \] we obtain the standard form of the hyperbolic boost:
\[ \tag{33} \boxed{ e^{\eta\j} = \gamma(1+\j\beta) }. \] The circular parameter \(b\) and the hyperbolic rapidity \(\eta\) describe the same external motion with different coordinates:
\[ \tag{34} \boxed{ \beta = \sin(\pi b) = \tanh\eta }, \] \[ \tag{35} \boxed{ \gamma = \frac{1}{\cos(\pi b)} = \cosh\eta }, \] \[ \tag{36} \boxed{ \gamma\beta = \tan(\pi b) = \sinh\eta }. \] The phase power \((-\j)^b\) cannot be identified with the hyperbolic exponential \(e^{\eta\j}\). The former is a unit rotation in the complex idempotent plane, and the latter is a hyperbolic boost. They are related through a common physical velocity \(\beta\), but represent different mathematical functions.Mathematical operations.
9. Why mass does not depend on external velocity
In the complete operator (8), the internal and external phases are separated:
\[ \tag{37} J(a,b) = \underbrace{\ep e^{i\pi b}}_{\text{external motion}} + \underbrace{\em e^{i\pi a}}_{\text{internal state}}. \] Changing the velocity changes the exponent \(b\), but does not in itself require a change in the internal exponent \(a\). Therefore, acceleration redistributes the external energy representation between \(E_0\) and \(pc\), without transforming the rest energy into a function of the laboratory velocity:
\[ \tag{38} E_0=mc^2=\operatorname{const}, \qquad E=\gamma E_0, \qquad pc=\gamma\beta E_0. \] The origin of the rest energy itself from internal periodicity is discussed in detail in the article "Particle Mass as a Geometric Projection". This paper begins with the already formed quantity \(E_0\) and examines only its external relativistic representation.
10. The Speed of Light Limit
For a massive particle, \(E_0>0\). As the external velocity approaches the speed of light
\[ \tag{39} \beta\to1, \qquad b\to\frac12, \qquad \cos(\pi b)\to0, \qquad \gamma\to\infty. \] Therefore
\[ \tag{40} \boxed{ m>0, \quad v\to c \quad\Longrightarrow\quad E=\gamma mc^2\to\infty }. \] A massive particle does not reach \(v=c\) at finite energy. If the state moves exactly at the speed of light and its energy is finite, then formula (17) implies
\[ \tag{41} \beta=1, \qquad b=\frac12, \qquad E_0=E\cos\left(\frac{\pi}{2}\right)=0. \] Therefore, the light branch satisfies the conditions
\[ \tag{42} \boxed{ m=0, \qquad E=|p|c }. \] This does not mean that a massive particle turns into a wave under ordinary acceleration. The massive and light branches are distinct classes of state: for \(m>0\), achieving \(v=c\) requires infinite energy, while the light branch initially has \(E_0=0\).
11. Internal and external factors cannot be mixed
External Lorentz factor
\[ \tag{43} \gamma = \frac{1}{\sqrt{1-\beta^2}} \] describes the transformation of already formed rest energy during the motion of the center of mass. It is determined by the external exponent \(b\) and does not explain the origin of mass.
In the accepted electron model, the fine structure constant \(\alpha_{\mathrm{fs}}\) is considered as the coefficient of the transverse Doppler projection of the internal frequency. This internal coefficient and the external factor \(\gamma\) have different origins and perform different functions:
\[ \tag{44} \boxed{ \begin{aligned} \alpha_{\mathrm{fs}} &\quad\text{— the internal projection that forms the rest energy,} \\[1mm] \gamma &\quad\text{— the external factor of the total energy and momentum.} \end{aligned} } \] In the bound state of the hydrogen atom, a composite internal projection \(\alpha_{\mathrm{fs}}^2\) arises. Its geometric meaning and connection with the parameters of the Bohr atom are discussed in a separate paper. Neither \(\alpha_{\mathrm{fs}}\) nor \(\alpha_{\mathrm{fs}}^2\) should be identified with the external Lorentz factor of a freely moving particle.
12. Final Geometric Scheme
The Complete State Operator
\[ \tag{45} J(a,b) = \j^a(-\j)^b = \ep e^{i\pi b} + \em e^{i\pi a} \] separates the intrinsic periodicity and the extrinsic motion between two orthogonal planes. The external phase determines the velocity and the inverse Lorentz factor:
\[ \tag{46} \sin(\pi b)=\beta, \qquad \cos(\pi b)=\frac1\gamma. \] After physically mapping these projections, we obtain
\[ \tag{47} \frac{pc}{E}=\sin(\pi b), \qquad \frac{E_0}{E}=\cos(\pi b), \] and then
\[ \tag{48} E=\gamma E_0, \qquad pc=\gamma\beta E_0, \qquad E^2-p^2c^2=E_0^2. \] Thus, the Lorentz factor appears as the reciprocal of the fixed projection of unit external rotation. The energy invariant, in turn, arises as the conservation of the length of the corresponding energy vector. Split geometry connects circular phase, hyperbolic rapidity, and standard relativistic kinematics without confusing the external law of motion with the internal origin of mass.
Materials used
- Wikipedia. Lorentz factor.