2026-07-16
Mass as a geometric projection of a closed wave
Zero Electron Scale, Internal Time, and the Cause of Zero Photon Mass
How a Particle's Proper Time Changes the Dynamic Scale
The Large Hadron Collider offers a useful relativistic example to begin our discussion of mass. Its circumference is approximately \(26.659\) meters, and its geometric mean radius is about \(4.24\) kilometers. For a laboratory observer, a particle moving nearly at the speed of light completes one revolution in approximately \(88.9\) microseconds.
\[\tag{1} C_{\mathrm{LHC}}=26,659\ {\rm m}, \qquad R_{\mathrm{LHC}}=\frac{C_{\mathrm{LHC}}}{2\pi} \approx4.24\ {\rm km}. \] The LHC accelerates protons and ions, not electrons. If a proton has a velocity of \(v=\beta c\), then the laboratory rotation time \(T_{\mathrm{lab}}\) and the proper time \(\Delta\tau_p\), accumulated by a moving proton between two successive passes of a single point on the ring, are related by the Lorentz factor:
\[\tag{2} T_{\mathrm{lab}} =\frac{C_{\mathrm{LHC}}}{\beta c}, \qquad \Delta\tau_p =\frac{T_{\mathrm{lab}}}{\gamma}, \qquad \gamma=\frac{1}{\sqrt{1-\beta^2}}. \] At an injection energy of 450 GeV, the proton's Lorentz factor is approximately 479.6. One laboratory rotation takes 88.9 μs, while the proton's own clock ticks only about 185 ns. The dynamical path obtained by multiplying the laboratory velocity by this proper time is approximately \(55.6\) m:
\[\tag{3} \gamma_{450}\approx479.6, \qquad \Delta\tau_p\approx185\ {\rm ns}, \qquad v\Delta\tau_p =\frac{C_{\mathrm{LHC}}}{\gamma_{450}} \approx55.6\ {\rm m}. \] At a proton energy of \(6.8\) TeV, the same dynamic path decreases to approximately \(3.68\) m:
\[\tag{4} \gamma_{6.8\,\mathrm{TeV}}\approx7247, \qquad v\Delta\tau_p \approx3.68\ {\rm m}. \] If we mentally choose the factor \(\gamma_0=1/\alpha_{\mathrm{fs}}\approx137\), then the corresponding dynamic radius would be approximately \(137\) times smaller than the laboratory radius:
\[\tag{5} R_{\tau} =\frac{R_{\mathrm{LHC}}}{\gamma_0} =\alpha_{\mathrm{fs}}R_{\mathrm{LHC}} \approx31\ {\rm m}. \] Formulas (3)–(5) do not mean that the accelerator's typical geometric radius is literally reduced to a few meters. The radius is directed perpendicular to the instantaneous velocity, and a rotating system does not have a single global inertial rest frame. Here, only a dynamic scale is defined, constructed from the proper time of one revolution. It is this limited analogy that is then transferred to the internal phase geometry of the electron.
1. One Observer from Two Perspectives
Consider one internal cycle and two events: \(A\) — the beginning of the cycle, and \(B\) — the return of the wave to its original phase state. First, the mental observer associates the clock with the local internal motion and measures the proper interval \(T_0\). The same measurement principle is then applied from an external position in the rest frame of the center of the localized particle and yields the interval \(T_C\):
\[\tag{6} T_0=\tau_0(B)-\tau_0(A), \qquad T_C=t(B)-t(A). \] One real observer is not simultaneously in two places: we are talking about a sequential mental comparison of identical clocks and one pair of phase events. The internal observer locally does not notice the slowing of his own clock. The difference is revealed only when compared with external time:
\[\tag{7} T_0=\frac{T_C}{\gamma_0}, \qquad d\tau_0=\frac{dt}{\gamma_0}. \] This description is useful as a physical motivation, but further conclusions should not depend on the subjective word "sees." Therefore, let us move from two observation positions to the more precise concept of phase-geometric projection.
2. From Observation to Projection
Let us denote the internal, not yet externally mapped geometric level by \(\mathcal G_0\), and the external rest level of the localized state by \(\mathcal G_1\). They are connected by the mass projection operator \(\Pi_m\):
\[\tag{8} \boxed{ \Pi_m:\mathcal G_0\longrightarrow\mathcal G_1 }. \] We denote the coefficient of this projection by \(P_m\). The kinematic analogy of the previous section motivates its connection with the ratio of internal and external time:
\[\tag{9} \boxed{ P_m =\frac{d\tau_0}{dt} =\frac{1}{\gamma_0} }. \] The operator \(\Pi_m\) is not the usual Lorentz transformation of the total energy of a moving particle. It is a separate physical mapping of the model: the internal phase geometry of a closed wave is translated into the global frequency and rest energy of a localized state.
3. Primary State Operator
The mathematical basis is two mutually orthogonal idempotents:
\[\tag{10} \ep^2=\ep, \qquad \em^2=\em, \qquad \ep\em=0, \qquad \ep+\em=1. \] Their difference forms a hyperbolic unit:
\[\tag{11} \j=\ep-\em, \qquad \j^2=1. \] The complete normalized state is written by the operator
\[\tag{12} \boxed{ J(a,b) =\j^a(-\j)^b =\ep e^{i\pi b}+\em e^{i\pi a} }. \] The parameter \(a\) specifies the internal phase, and \(b\) specifies the external motion of the center of the already formed particle:
\[\tag{13} a=\varpi t, \qquad \omega=\pi\varpi, \qquad b=\frac{\arcsin\beta}{\pi}, \qquad \beta=\frac{v}{c}. \] When separating geometric levels, the same internal phase can be written either in internal proper time or after external projection:
\[\tag{14} \varphi =\omega_{\mathrm{int}}\tau_0 =\omega_C t. \] At the observable level, the usual notation \(a=\varpi t\) is retained, while the zero level uses its own phase time \(\tau_0\). The operator norm remains unity:
\[\tag{15} \boxed{ J\overline J=\ep+\em=1 }. \] The unit norm expresses the conservation of the complete state, but does not in itself specify the numerical values of mass, radius, or energy.
4. Wave Closure and Particle Emergence
For a center at rest relative to an external observer, \(b=0\):
\[\tag{16} J(a,0) =\ep+\em e^{i\varphi}. \] A state is associated with a geometric velocity of constant norm, and the spatial geometry is obtained by integration:
\[\tag{17} J(t) \longrightarrow V(t)=cJ(t) \longrightarrow \mathbf R(t)=\int V(t)\,dt. \] A periodic internal component creates a phase-dynamic radius
\[\tag{18} \boxed{ r_0=\frac{c}{\omega_{\mathrm{int}}}, \qquad r_0\omega_{\mathrm{int}}=c }. \] If, after a full period, the phase and position relative to the localization center are restored, a stable closed cycle arises:
\[\tag{19} \omega_{\mathrm{int}}T_0=2\pi, \qquad \mathbf R(T_0)=\mathbf R(0). \] \[\tag{20} \boxed{ \text{particle} =\text{stable localized state of a closed wave} }. \] There is no separate material point inside, mechanically rotating and radiating like a classical charge. The entire phase structure is closed. Therefore, \(r_0\) is primarily the scale of the internal phase cycle, and not the experimentally measured radius of the charge distribution.
5. The Zeroth Geometric Scale of the Electron
Let's call \(r_0\) the zeroth proper phase-geometric scale of the electron. The word "zero" means that this level exists before the external mass projection, before the Compton map, and before any possible deeper radial splitting of the internal geometry.
In the first unsplit approximation, the phenomenological identification is accepted:
\[\tag{21} \boxed{ r_0\equiv r_e }. \] Here \(r_e\) is the classical radius of the electron. In standard physics, this is a combination of \(e\), \(m_e\), and \(c\), not the directly measured size of the electron. In the proposed model, it receives an additional interpretation as the zero phase scale.
6. The Postulate of Internal Kinematics
For the electronic implementation of the model, we adopt an additional postulate: the local internal process has a near-light tangential velocity such that its Lorentz factor is equal to the inverse of the fine structure constant:
\[\tag{22} \boxed{ \gamma_0=\frac{1}{\alpha_{\mathrm{fs}}} }. \] The corresponding speed is less than the speed of light:
\[\tag{23} \boxed{ \beta_0 =\sqrt{1-\alpha_{\mathrm{fs}}^2}, \qquad v_0=c\sqrt{1-\alpha_{\mathrm{fs}}^2} \approx0.99997336c }. \] Therefore, the internal local observer has a finite proper time:
\[\tag{24} \boxed{ d\tau_0 =\alpha_{\mathrm{fs}}dt }. \] The velocity \(v_0\) characterizes the internal tangential kinematics and is not the external velocity of the center \(v\), specified by the parameter \(b\). The total norm of wave motion in the map \(V=cJ\) remains equal to \(c\).
7. Mass Projection Coefficient
Comparing (9) and (24), we obtain for the electron:
\[\tag{25} \boxed{ P_m =\frac{d\tau_0}{dt} =\frac{1}{\gamma_0} =\alpha_{\mathrm{fs}} }. \] This equality gives the projection coefficient a kinematic meaning. However, the number \(1/137.035dots\) has not yet been derived from a single algebra \(J\). It is postulated that a stable electron wave chooses precisely this value of the internal factor \(\gamma_0\).
8. Two Levels of the Same Geometry
The zero level contains the internal proper time, internal frequency, and zero scale. The firstThe level contains the rest frame time of the center, the Compton frequency, and the external Compton scale:
\[\tag{26} \boxed{ \begin{aligned} \mathcal G_0&=(\tau_0,\omega_{\mathrm{int}},r_0),\\ \mathcal G_1&=(t,\omega_C,r_C),\\ \mathcal G_1&=\Pi_m(\mathcal G_0). \end{aligned} }. \] These are not two different electrons or two matter rings. These are two ways of phase-geometrically representing a single closed state: before and after its mapping to the external mass channel.
9. Internal Frequency Projection
The phase of the cycle does not depend on the method of its parameterization:
\[\tag{27} d\varphi =\omega_{\mathrm{int}}d\tau_0 =\omega_Cdt. \] Hence the frequency projection law follows:
\[\tag{28} \boxed{ \omega_C =P_m\omega_{\mathrm{int}} }. \] For an electron:
\[\tag{29} \boxed{ \omega_C =\alpha_{\mathrm{fs}}\omega_{\mathrm{int}} }. \] Formula (29) is the model's mapping law. It should not be confused with the frequency \(E/\hbar\) of an arbitrarily moving particle. Here \(\omega_C\) is the global rest frequency of a localized electron, and \(\omega_{\mathrm{int}}\) is the frequency of its primary internal phase process.
10. Projection of the Geometric Scale
We associate the external phase scale with the first level
\[\tag{30} r_C=\frac{c}{\omega_C}. \] Using (18) and (28), we obtain:
\[\tag{31} r_C =\frac{c}{P_m\omega_{\mathrm{int}}} =\frac{r_0}{P_m}. \] Therefore:
\[\tag{32} \boxed{ r_0=P_mr_C }. \] For an electron, the first outer level is identified with the reduced Compton length:
\[\tag{33} \boxed{ r_0=r_e, \qquad r_C=\overline\lambda_C, \qquad r_e=\alpha_{\mathrm{fs}}\overline\lambda_C }. \] Thus, the classical and Compton scales are related not by a literal contraction of a single material ring, but by a mutually inverse mapping of frequency and phase radius.
11. Phase Energy Scale
The primary phase energy scale is associated with the intrinsic frequency:
\[\tag{34} \boxed{ \mathcal E_{\mathrm{ph}} =\hbar\omega_{\mathrm{int}} }. \] For an electron, it is related to the Compton frequency through the projection coefficient:
\[\tag{35} \mathcal E_{\mathrm{ph}} =\frac{\hbar\omega_C}{\alpha_{\mathrm{fs}}}. \] The quantity \(\mathcal E_{\mathrm{ph}}\) is not yet called the total directly accessible energy of the electron. It is a zero-level frequency energy scale. To interpret it as a real internal energy, it would be necessary to separately construct the coupling dynamics and explain the transition to the observed energy \(m_ec^2\).
12. Rest Energy as a Mass Projection
The basic physical postulate of the mass map is that the rest energy is equal to the projection of the primary phase scale:
\[\tag{36} \boxed{ E_0 =P_m\mathcal E_{\mathrm{ph}} }. \] Taking into account (28) and (34):
\[\tag{37} E_0 =P_m\hbar\omega_{\mathrm{int}} =\hbar\omega_C. \] Identifying this energy with the rest energy of the localized state, we obtain:
\[\tag{38} \boxed{ E_0 =\hbar\omega_C =mc^2 }. \] Then the internal phase scale can be expressed in terms of the observed mass: \(\mathcal E_{\mathrm{ph}}=m_ec^2/\alpha_{\mathrm{fs}}\).
The mass in this formula is independent of the arbitrary position of the observer. It belongs to the global rest frame of the center of the closed wave and is an invariant characteristic of the localized state.
13. Mass as a Frequency Projection
From (38) it follows:
\[\tag{39} \boxed{ m =\frac{\hbar\omega_C}{c^2} =\frac{P_m\hbar\omega_{\mathrm{int}}}{c^2} }. \] Through the zero phase-geometric scale:
\[\tag{40} \boxed{ m =\frac{P_m\hbar}{cr_0} }. \] For an electron:
\[\tag{41} \boxed{ m_e =\frac{\alpha_{\mathrm{fs}}\hbar}{cr_e} =\frac{\hbar}{c\overline\lambda_C} }. \] Mass acquires the following geometric meaning: it is the energetic equivalent of the external frequency projection of the internal closed wave. It is not a separate substance within the particle and is not identified with the entire phase scale \(\mathcal E_{\mathrm{ph}}\).
14. The geometric meaning of the fine structure constant
For the electronic realization, several ratios are combined by a single coefficient:
\[\tag{42} \boxed{ \alpha_{\mathrm{fs}} =P_m =\frac{d\tau_0}{dt} =\frac{1}{\gamma_0} =\frac{\omega_C}{\omega_{\mathrm{int}}} =\frac{r_e}{\overline\lambda_C} =\frac{E_0}{\mathcal E_{\mathrm{ph}}} }. \] The fine structure constant relates internal and external time, zeroth and Compton scales, internal and observed frequencies, as well as the phase energy scale and rest energy.
However, the equality \(r_e/\overline\lambda_C=\alpha_{\mathrm{fs}}\) is also known from the standard definitions of these quantities. Therefore, formula (42) is a geometric interpretation of a known ratio, and not an independent calculation of the number \(\alpha_{\mathrm{fs}}\).
15. External Motion of a Formed Particle
After the emergence of a stable closed state, its center can move in external space. This motion is described by the parameter \(b\):
\[\tag{43} \sin(\pi b)=\beta=\frac{v}{c}, \qquad \cos(\pi b)=\sqrt{1-\beta^2}=\frac{1}{\gamma}. \] It is necessary to distinguish between the constant internal factor \(\gamma_0\), which is involved in the mass projection, and the external factor \(\gamma(v)\), which depends on the motion of the center:
\[\tag{44} \boxed{ \gamma_0=\frac{1}{\alpha_{\mathrm{fs}}} \quad\ne\quad \gamma(v)=\frac{1}{\sqrt{1-v^2/c^2}} }. \] The sequence of mappings is
\[\tag{45} \boxed{ \mathcal G_0 \xrightarrow{\;\Pi_m\;} (m,E_0) \xrightarrow{\;b\;} (E,p) }. \] External motion changes the total energy and momentum, but not the proper mass:
\[\tag{46} E=\gamma mc^2, \qquad pc=\gamma\beta mc^2. \] \[\tag{47} \boxed{ E^2-p^2c^2=m^2c^4 }. \] 16. Why acceleration doesn't turn an electron into a photon?
For a massive particle, the outer limit \(v\to c\) requires an unlimited increase in energy:
\[\tag{48} E=\frac{mc^2}{\sqrt{1-\beta^2}} \longrightarrow\infty \qquad \text{for} \qquad \beta\longrightarrow1. \] Increasing the parameter \(b\) moves the center of the closed wave, but does not in itself change its topology:
\[\tag{49} \boxed{ b\uparrow \;\not\Rightarrow\; \text{opening of the internal wave} }. \] Therefore, an electron cannot be converted into a photon simply by acceleration. At any finite energy, its center moves slower than light, and the internal localized cycle is preserved.
17. Wave Uncoupling and Zero Photon Mass
A particle and a photon differ not only in speed, but primarily in the mode of wave existence. In a closed mode, the phase cyclically returns to its own center. In an open mode, it becomes the transport phase of free propagation:
\[\tag{50} \boxed{ \text{cyclic phase relative to the center} \longrightarrow \text{transport phase of a free wave} }. \] A free photon has no rest frame, constant internal radius, or natural frequency of its localized state. Its energy and momentum satisfy
\[\tag{51} E_\gamma=\hbar\omega_\gamma, \qquad p_\gamma=\frac{E_\gamma}{c}. \] Therefore:
\[\tag{52} E_\gamma^2-p_\gamma^2c^2=0 =m_\gamma^2c^4, \] \[\tag{53} \boxed{ m_\gamma=0 }. \] Zero photon mass does not mean zero energy. The photon frequency determines the propagation energy, but not the rest energy. The mass projection \(\Pi_m\) refers to a stable closed state and does not apply to a free transport wave without its own center.
18. Field, Radiation, and Complete Disconnection
Three regimes must be distinguished. A closed internal wave manifests as a particle. Its external continuation creates a field but does not destroy the source localization. The separated field change propagates as free radiation.
\[\tag{54} \boxed{ \begin{aligned} \text{particle:}\quad &\text{closed wave}, &&m_0>0, \\[2mm] \text{field:}\quad &\text{extension of the source state}, &&\text{not a separate particle}, \\[2mm] \text{radiation:}\quad &\text{free open wave}, &&m_0=0. \end{aligned} } \] Ordinary radiation does not mean the conversion of an electron into a photon. The electron maintains an internal closure, and the free wave carries away the difference in state energies:
\[\tag{55} E_\gamma=E_i-E_f=\hbar\omega_\gamma. \] A complete transformation of the electron structure is only possible within an interacting system with conservation of charge, energy, and momentum. An example is annihilation:
\[\tag{56} e^-+e^+ \longrightarrow \gamma_1+\gamma_2. \] 19. What follows from the model, and what is accepted as a hypothesis
From the mapping \(V=cJ\) and the periodic internal phase, we obtain the phase scale \(r_0=c/\omega_{\mathrm{int}}\). After adopting the projection law \(\omega_C=P_m\omega_{\mathrm{int}}\), the mutually inverse transformations of frequency and radius mathematically follow:
\[\tag{57} \boxed{ \omega_C=P_m\omega_{\mathrm{int}}, \qquad r_C=\frac{r_0}{P_m} }. \] For the electron the following are postulated:
\[\tag{58} \boxed{ \gamma_0=\frac{1}{\alpha_{\mathrm{fs}}}, \qquad P_m=\alpha_{\mathrm{fs}}, \qquad E_0=P_m\mathcal E_{\mathrm{ph}} }. \] With these postulates, we obtain:
\[\tag{59} \boxed{ r_e=\alpha_{\mathrm{fs}}\overline\lambda_C, \qquad \omega_C=\alpha_{\mathrm{fs}}\omega_{\mathrm{int}}, \qquad m_e=\frac{\alpha_{\mathrm{fs}}\hbar}{cr_e} }. \] Until deduced from a single algebra \(J\): numerical value \(\alpha_{\mathrm{fs}}\); The reason for the stable choice of \(\gamma_0=1/\alpha_{\mathrm{fs}}\); the zero-loop stabilization mechanism; the full energy dynamics of the internal phase scale; the operator of free photon de-coupling and production.
A possible direction for further development is to relate the coefficient \(P_m\) to the deep idempotent splitting of the internal phase plane. Until such a conclusion is reached, the equality \(P_m=\alpha_{\mathrm{fs}}\) should be explicitly called a postulate of the electron model.
Conclusion
The original object is a normalized wave. Its closure creates a localized state with a zero intrinsic phase-geometric scale \(r_0\) and an intrinsic frequency \(\omega_{\mathrm{int}}\). Near-light internal kinematics motivates the time projection coefficient \(P_m=1/\gamma_0\).
For an electron, \(\gamma_0=1/\alpha_{\mathrm{fs}}\). Then the operator \(\Pi_m\) converts the internal frequency to the Compton frequency, and the zero scale to the reduced Compton length:
\[\tag{60} \boxed{ \begin{aligned} \omega_C&=\alpha_{\mathrm{fs}}\omega_{\mathrm{int}},\\ r_e&=\alpha_{\mathrm{fs}}\overline\lambda_C. \end{aligned} }. \] The energy of this external frequency is the rest energy, and its equivalent is mass:
\[\tag{61} \boxed{ mc^2 =\hbar\omega_C =P_m\hbar\omega_{\mathrm{int}} }. \] \[\tag{62} \boxed{ \text{rest mass} =\text{energy equivalent of the external frequency projection of a closed wave} }. \] A free photon does not have such a closed internal level and rest frame. Therefore, its frequency determines the propagation energy but does not create a rest mass. The difference between an electron and a photon is determined not by the velocity alone, but by the geometric regime of the wave: stable closure or free propagation.
Materials Used
- CERN. Large Hadron Collider: structure and main parameters.
- CERN. CERN Accelerator Complex and Injection Energy.
- Wikipedia. Fine-structure constant.
- C. W. Berenda. The Problem of the Rotating Disk. Physical Review, 62, 280–298.