2026-07-29
Sum and difference of velocities as a composition of Doppler projections
Part 2. Various internal states
Does an observed frequency shift always indicate motion? In the first part, this question did not arise: the states being compared had the same intrinsic frequency, so their common intrinsic component canceled out. The ratio of the observed frequencies was determined only by external motion, and from it the relativistic law of velocity addition was directly reconstructed.
Now the constraint is removed. Let the intrinsic values of the two states differ:
\[ \tag{1} \boxed{ a_1(t)\ne a_2(t) }. \] In this case, the observed frequency change can have two different causes. The first is associated with a change in the internal process itself, the second with the external motion of the source relative to the observer:
\[ \tag{2} \boxed{ \text{observed frequency shift} = \text{internal change} + \text{external Doppler projection} }. \] The plus sign in formula (2) should be understood in logarithmic coordinates. In a typical frequency ratio, these two contributions are not added, but multiplied. The main goal of the second part is to separate them and show under what conditions the external velocity can truly be uniquely reconstructed from the measured frequency.
This distinction also raises an experimental question. If the internal state of a particle depends on the direction of motion or observation, then an additional shift may manifest itself in the Ives-Stilwell experiment and its modern analogues. But if only the initial phases differ with the same internal rotation rate, no spectral shift occurs.
1. General Operator of Two States
The operator of the full state in split geometry has the form
\[ \tag{3} \boxed{ J(a,b) = \j^a(-\j)^b = \ep e^{i\pi b} + \em e^{i\pi a} }. \] The idempotents \(\ep\) and \(\em\) define two independent complex planes:
\[ \tag{4} \ep^2=\ep, \qquad \em^2=\em, \qquad \ep\em=0, \qquad \ep+\em=1. \] The parameter \(a\) describes the internal state, and the parameter \(b\) describes the external motion of the particle. In general, the two states being compared are written as
\[ \tag{5} \boxed{ J_1 = \j^{a_1}(-\j)^{b_1}, \qquad J_2 = \j^{a_2}(-\j)^{b_2} }. \] In an idempotent basis, formula (5) takes a particularly transparent form:
\[ \tag{6} J_k = \ep e^{i\pi b_k} + \em e^{i\pi a_k}, \qquad k=1,2. \] The external indicators are related to the standard velocities by the parameterization adopted in the model.
\[ \tag{7} b_k = \frac{1}{\pi} \arcsin\beta_k, \qquad \beta_k = \frac{v_k}{c} = \sin(\pi b_k). \] We present the internal indicators in the most general form for constant frequencies.
\[ \tag{8} a_k(t) = \varpi_k t+\varphi_k, \qquad \omega_{0,k} = \pi\dot a_k = \pi\varpi_k. \] Therefore, the difference between \(a_1\) and \(a_2\) can be contained both in the initial phases \(\varphi_k\) and in the internal angular frequencies \(\omega_{0,k}\). These two cases have different observable consequences.
2. What does the condition \(a_1\ne a_2\) mean?
First, let's consider the difference in the initial phases only:
\[ \tag{9} \varpi_1=\varpi_2=\varpi, \qquad \varphi_1\ne\varphi_2. \] Then the exponents are indeed not the same, but their derivatives are the same:
\[ \tag{10} a_1(t)\ne a_2(t), \qquad \dot a_1(t)=\dot a_2(t)=\varpi. \] Therefore, the natural internal frequencies are also the same:
\[ \tag{11} \omega_{0,1} = \omega_{0,2} = \pi\varpi. \] Another case arises if the internal rotation rates differ:
\[ \tag{12} \varpi_1\ne\varpi_2 \quad\Longrightarrow\quad \dot a_1\ne\dot a_2 \quad\Longrightarrow\quad \omega_{0,1}\ne\omega_{0,2}. \] Thus, the inequality of internal exponents alone does not prove a difference in internal frequencies:
\[ \tag{13} \boxed{ a_1\ne a_2 \quad\not\Rightarrow\quad \dot a_1\ne\dot a_2 }. \] A spectral instrument measures the rate of phase change, not the absolute position in the inner loop. Therefore, a constant phase difference and a continuously increasing phase difference must be considered separately.
3. Different Phases at the Same Intrinsic Frequency
When condition (9) is satisfied, the relative internal phase is
\[ \tag{14} \Delta a_{2/1} = a_2-a_1 = \varphi_2-\varphi_1 = \operatorname{const}. \] Its derivative vanishes:
\[ \tag{15} \frac{d}{dt} \Delta a_{2/1} = \dot a_2-\dot a_1 = 0. \] Therefore, the constant internal phase difference does not create a separate spectral shift. With identical internal transverse projections, the ratio of the observed frequencies is still determined only by the external Doppler scales:
div> \[ \tag{16} \boxed{ \frac{\omega_2}{\omega_1} = \frac{D_2}{D_1} }. \] Therefore, the result of the first part remains unchanged:
\[ \tag{17} \boxed{ \beta_{2/1} = \frac{ \beta_2-\beta_1 }{ 1-\beta_1\beta_2 } }. \] The difference \(\varphi_2-\varphi_1\) can appear during the coherent addition of two waves, since it determines their interference pattern. But it does not change the position of the spectral line until \(\dot a_1=\dot a_2\).
4. Different Internal Frequencies and Projections
Now let's consider the general case of \(\dot a_1\ne\dot a_2\). The internal angular frequencies are equal to
\[ \tag{18} \omega_{0,1} = \pi\dot a_1, \qquad \omega_{0,2} = \pi\dot a_2. \] In addition to differences in the frequencies themselves, their internal transverse projections may also differ. Let us denote the normalized velocities of internal motions by
\[ \tag{19} \eta_k = \frac{v_{\mathrm{int},k}}{c}, \qquad \gamma_{a,k} = \frac{1}{ \sqrt{1-\eta_k^2} }. \] We write the internal transverse factor of each state as
\[ \tag{20} \mathcal D_{a,k} = \frac{1}{\gamma_{a,k}}. \] It is convenient to combine the natural frequency and its intrinsic projection into a single observable intrinsic scale:
\[ \tag{21} \boxed{ \Omega_{0,k} = \omega_{0,k}\mathcal D_{a,k} = \frac{ \pi\dot a_k }{ \gamma_{a,k} } }. \] The inequality \(a_1\ne a_2\) does not in itself require that \(\mathcal D_{a,1}\ne\mathcal D_{a,2}\). The difference in transverse factors is an additional physical feature of the model. If the internal velocities are the same, then \(\mathcal D_{a,1}=\mathcal D_{a,2}\), and the ratio of the internal scales reduces to the ratio of the natural frequencies.
5. Total Observable Frequency
For the external motion of each state, we introduce a directional Doppler scale
\[ \tag{22} D_k = \sqrt{ \frac{1+\beta_k} {1-\beta_k} }. \] The total observed angular frequency is a sequential projection of the internal and external motions:
\[ \tag{23} \boxed{ \omega_k = \omega_{0,k} \mathcal D_{a,k} D_k = \Omega_{0,k}D_k }. \] Therefore, the ratio of the two observed frequencies is
\[ \tag{24} \frac{\omega_2}{\omega_1} = \frac{\Omega_{0,2}}{\Omega_{0,1}} \frac{D_2}{D_1}. \] Introduce the internal ratio
\[ \tag{25} \boxed{ I_{2/1} = \frac{\Omega_{0,2}}{\Omega_{0,1}} = \frac{\dot a_2}{\dot a_1} \frac{ \mathcal D_{a,2} }{ \mathcal D_{a,1} } }. \] We denote the external relative scale, as in the first part, by
\[ \tag{26} D_{2/1} = \frac{D_2}{D_1}. \] Then the central decomposition of the second part takes the form
\[ \tag{27} \boxed{ \frac{\omega_2}{\omega_1} = I_{2/1}D_{2/1} }. \] A single observed frequency shift contains two independent factors. Therefore, without knowing \(I_{2/1}\), it is impossible to unambiguously determine which part of the measured ratio is due to a change in internal state and which is due to external motion.
6. Extracting External Motion
From formula (27), the external relative scale is found by normalizing to the internal ratio:
\[ \tag{28} \boxed{ D_{2/1} = \frac{ \omega_2/\omega_1 }{ I_{2/1} } = \frac{ \omega_2\Omega_{0,1} }{ \omega_1\Omega_{0,2} } }. \] The velocity is uniquely reconstructed from any directional Doppler scale using the formula
\[ \tag{29} \boxed{ \beta = \frac{D^2-1}{D^2+1} }. \] Substituting (28), we obtain the full measurement formula for the external relative velocity:
\[ \tag{30} \boxed{ \beta_{2/1} = \frac{ \left( \dfrac{ \omega_2\Omega_{0,1} }{ \omega_1\Omega_{0,2} } \right)^2 -1 }{ \left( \dfrac{ \omega_2\Omega_{0,1} }{ \omega_1\Omega_{0,2} } \right)^2 +1 } }. \] After internal normalization, the expression in parentheses is equal to \(D_2/D_1\). Therefore, fully revealing the Doppler scales, we have
\[ \tag{31} \left(D_{2/1}\right)^2 = \frac{ (1+\beta_2)(1-\beta_1) }{ (1-\beta_2)(1+\beta_1) }. \] The inverse transformation (29) again leads to a relativistic velocity difference:
\[ \tag{32} \boxed{ \beta_{2/1} = \frac{ \beta_2-\beta_1 }{ 1-\beta_1\beta_2 } }. \] Similarly, for sequential addition, it is necessary to first identify the normalized external scales of each state:
\[ \tag{33} D_k = \frac{\omega_k}{\Omega_{0,k}}, \qquad D_{1\oplus2} = D_1D_2. \] After the inverse transformation, we obtain
\[ \tag{34} \boxed{ \beta_{1\oplus2} = \frac{ \beta_1+\beta_2 }{ 1+\beta_1\beta_2 } }. \] Thus, different internal frequencies do not change the law of the sum or difference of external velocities ifEach observed frequency is pre-normalized to its own internal scale. Without such normalization, the internal change is erroneously included in the result as part of the external motion.
7. The Complete Logarithmic Coordinate
The multiplicative expansion (27) becomes additive after taking the logarithm. We introduce the complete observed shift.
\[ \tag{35} Q_{2/1} = \ln \frac{\omega_2}{\omega_1}. \] Then
\[ \tag{36} Q_{2/1} = \ln I_{2/1} + \ln D_2 - \ln D_1. \] As shown in the first part, the logarithm of the outer Doppler scale is equal to the rapidity:
\[ \tag{37} q_k = \ln D_k = \operatorname{artanh}\beta_k. \] Therefore, the total shift is
\[ \tag{38} \boxed{ Q_{2/1} = \ln I_{2/1} + \operatorname{artanh}\beta_2 - \operatorname{artanh}\beta_1 }. \] It is in this coordinate that the symbolic equality (2) becomes a rigorous mathematical formula:
\[ \tag{39} \boxed{ \text{total shift} = \text{internal shift} + \text{external shift} }. \] The external relative rapidity is determined by subtracting the internal contribution:
\[ \tag{40} \boxed{ q_{2/1} = Q_{2/1} - \ln I_{2/1} }. \] After this \(\beta_{2/1}=\tanh q_{2/1}\), and the velocity difference formula again takes the form (32).
8. Effective Velocity with an Unknown Internal State
Suppose that the entire observed frequency ratio is mistaken for the external Doppler scale:
\[ \tag{41} R_{\mathrm{full}} = \frac{\omega_2}{\omega_1} = I_{2/1}D_{2/1}. \] Then the measurement will yield not the true relative velocity, but an effective value.
\[ \tag{42} \boxed{ \beta_{\mathrm{eff}} = \frac{ R_{\mathrm{full}}^2-1 }{ R_{\mathrm{full}}^2+1 } }. \] Substituting expansion (41), we find
\[ \tag{43} \boxed{ \beta_{\mathrm{eff}} = \frac{ I_{2/1}^{,2}D_{2/1}^{,2}-1 }{ I_{2/1}^{,2}D_{2/1}^{,2}+1 } }. \] Even in the absence of external relative motion, when \(D_{2/1}=1\), the difference in internal states will create a non-zero effective velocity:
\[ \tag{44} D_{2/1}=1 \quad\Longrightarrow\quad \beta_{\mathrm{eff}} = \frac{ I_{2/1}^{,2}-1 }{ I_{2/1}^{,2}+1 }. \] Therefore, the velocity cannot be unambiguously determined from the frequency ratio alone if the source's internal frequency is unknown or varies. To determine the motion, either a known internal reference or an independent measurement of \(I_{2/1}\) is required.
9. Relative Phase, Beats, and Averaging
The total relative internal state is determined by the difference in the indices
\[ \tag{45} \boxed{ \Delta a_{2/1}(t) = a_2(t)-a_1(t) }. \] For linear phases (8), we obtain
\[ \tag{46} \Delta a_{2/1}(t) = (\varpi_2-\varpi_1)t + (\varphi_2-\varphi_1). \] The rate of change of the relative phase determines the angular frequency of the beats:
\[ \tag{47} \boxed{ \omega_{\mathrm{beat}} = \pi \frac{d}{dt} \Delta a_{2/1} = \omega_{0.2}-\omega_{0.1} }. \] Here, three quantities must be distinguished. The constant phase difference controls the interference pattern, the frequency difference creates the beats, and the frequency ratio is included in the internal factor \(I_{2/1}\) and affects the velocity recovery.
Let two coherent signals be added together in a conventional scalar observation. Their cross-interference term contains
\[ \tag{48} \cos \left( \Delta\omega\,t + \Delta\varphi \right), \qquad \Delta\omega = \omega_{0,2}-\omega_{0,1}. \] If the observation time is \(T_{\mathrm{obs}}\), and the frequency difference satisfies the condition
\[ \tag{49} \boxed{ |\Delta\omega| T_{\mathrm{obs}} \gg 2\pi }, \] then, during the measurement, the interference term undergoes numerous oscillations and averages out to practically zero:
\[ \tag{50} \left\langle \cos \left( \Delta\omega\,t + \Delta\varphi \right) \right\rangle_{T_{\mathrm{obs}}} \approx 0. \] The internal frequencies of elementary particles can be extremely high compared to the inverse of the normal observation time. However, the large values of \(\omega_1\) and \(\omega_2\) do not necessarily imply that their difference is also large: two enormous frequencies can be very close. The averaging is determined by the product \(|\Delta\omega|T_{\mathrm{obs}}\), not by the absolute value of each frequency.
When condition (49) is met, fast internal oscillations do not manifest as slow beats and are recorded as independent spectral components. However, the frequency ratio does not disappear:
\[ \tag{51} \boxed{ \text{fast interference term} \longrightarrow 0, \qquad I_{2/1}= \frac{\Omega_{0,2}}{\Omega_{0,1}} \ne 0 }. \] Therefore, high frequency obscures the instantaneous phase pattern but does not eliminate the systematic frequency shift. After proper internal normalization, it does not affect the law of the sum or difference of external velocities; without normalization, its ratio remains in the measured result.
10. Algebraic and Physical Relative Operators
The direct algebraic ratio of the two operators (5) is
\[ \tag{52} \begin{aligned} K_{2/1} &= J_2J_1^{-1} \\[1mm] &= \j^{a_2-a_1} (-\j)^{b_2-b_1}. \end{aligned} \] This operator accurately describes the difference in the exponents of the two initial states. However, the exponent \(b_2-b_1\) is not an exponent of the physical relativistic relative velocity. The reason is that \(b\) is related to velocity via \(\arcsin\beta\), whereas external velocities transform according to a fractional-linear law.
The physically reconstructed relative velocity is
\[ \tag{53} \beta_{2/1} = \frac{ \beta_2-\beta_1 }{ 1-\beta_1\beta_2 }. \] It corresponds to the external exponent
\[ \tag{54} \boxed{ b_{2/1} = \frac{1}{\pi} \arcsin \left( \frac{ \beta_2-\beta_1 }{ 1-\beta_1\beta_2 } \right) }. \] Therefore, it is necessary to distinguish two objects:
\[ \tag{55} \boxed{ K_{2/1} = \j^{a_2-a_1} (-\j)^{b_2-b_1} } \quad \text{— an algebraic relation,} \] \[ \tag{56} \boxed{ J_{2/1} = \j^{\Delta a_{2/1}} (-\j)^{b_{2/1}} } \quad \text{— a physical relative operator.} \] In an idempotent basis, the physical relative operator has View
\[ \tag{57} J_{2/1} = \ep e^{i\pi b_{2/1}} + \em e^{i\pi\Delta a_{2/1}}. \] The first component contains the reconstructed external relative velocity, and the second contains the actual internal relative phase. This separation preserves the geometric meaning of both processes and does not replace the relativistic composition with a simple difference of exponents \(b_2-b_1\).
11. Special Cases and Checks
Completely identical states.
\[ \tag{58} a_1=a_2, \qquad b_1=b_2 \quad\Longrightarrow\quad J_{2/1}=1. \] Different constant phases at the same frequencies and speeds.
\[ \tag{59} \dot a_1=\dot a_2, \qquad \beta_1=\beta_2 \quad\Longrightarrow\quad J_{2/1} = \j^{\varphi_2-\varphi_1}. \] In this case, only a constant relative phase remains, but there is no external relative velocity or internal beats.
Different internal frequencies without external relative motion.
\[ \tag{60} \beta_1=\beta_2, \qquad \dot a_1\ne\dot a_2 \quad\Longrightarrow\quad D_{2/1}=1, \qquad I_{2/1}\ne1. \] An internal frequency shift or beat is observed, although the external relative velocity is zero.
The same internal states at different velocities.
\[ \tag{61} I_{2/1}=1 \quad\Longrightarrow\quad \frac{\omega_2}{\omega_1} = D_{2/1}. \] This is the case in the first part of the article.
Simultaneously different internal and external states.
\[ \tag{62} I_{2/1}\ne1, \qquad D_{2/1}\ne1 \quad\Longrightarrow\quad \frac{\omega_2}{\omega_1} = I_{2/1}D_{2/1}. \] Only after separating these factors can we independently reconstruct the internal change and external relative velocity.
12. Relationship of Internal Frequency to Energy and Mass
In the concept of wave electricity, internal frequency is not an arbitrary parameter, but an energy characteristic of the state:
\[ \tag{63} E_k = \hbar\omega_{0,k}. \] If rest energy is related to mass by the relation
\[ \tag{64} E_k = m_kc^2, \] then the natural internal frequency is
\[ \tag{65} \boxed{ \omega_{0,k} = \frac{m_kc^2}{\hbar} }. \] For two states, we obtain
\[ \tag{66} \frac{ \omega_{0,2} }{ \omega_{0,1} } = \frac{m_2}{m_1}. \] Taking into account the internal transverse projections, factor (25) takes the form
\[ \tag{67} \boxed{ I_{2/1} = \frac{m_2}{m_1} \frac{ \mathcal D_{a,2} }{ \mathcal D_{a,1} } }. \] Consequently, the internal factor may reflect a real difference in the energies or masses of the states being compared. It should not be automatically attributed to external motion. For the same particle in the same internal state, the ratio of masses and internal projections is unity, and the formulas transform into the results of the first part.
13. Standard Scheme of the Ives-Stilwell Experiment
The Ives-Stilwell experiment [1] allows one to separate the first-order longitudinal Doppler shift from the second-order frequency change.order. The radiation of the same fast beam is observed in the direction of its motion and against it. In the standard case, both lines belong to the same transition with the same natural frequency \(\nu_0\).
For an approaching and receding source, the observed frequencies are equal
\[ \tag{68} \nu_+ = \nu_0\gamma(1+\beta), \qquad \nu_- = \nu_0\gamma(1-\beta), \] where
\[ \tag{69} \gamma = \frac{1}{ \sqrt{1-\beta^2} }. \] The half-difference of the lines highlights the first-order longitudinal effect:
\[ \tag{70} \frac{ \nu_+-\nu_- }{2} = \gamma\nu_0\beta. \] Their average value eliminates the linear terms:
\[ \tag{71} \boxed{ \overline{\nu} = \frac{ \nu_++\nu_- }{2} = \gamma\nu_0 }. \] At low speeds
\[ \tag{72} \gamma \approx 1+\frac{\beta^2}{2}, \] and the relative displacement of the center of the pair of lines is
\[ \tag{73} \boxed{ \frac{ \overline{\nu}-\nu_0 }{ \nu_0 } \approx \frac{\beta^2}{2} }. \] A constant difference in internal phases does not affect the position of the spectral lines, if
\[ \tag{74} \dot a_+ = \dot a_- \quad\Longrightarrow\quad \nu_{0,+} = \nu_{0,-} = \nu_0. \] It is the equality of the natural frequencies of the counter and co-current observations that allows us to interpret the remaining shift as a standard second-order effect.
14. Possible Internal Asymmetry
Now assume that the observed natural frequency of the internal process depends on the direction:
\[ \tag{75} \nu_{0,+} \ne \nu_{0,-}. \] Then the two lines look like
\[ \tag{76} \nu_+ = \nu_{0,+}\gamma(1+\beta), \qquad \nu_- = \nu_{0,-}\gamma(1-\beta). \] Introduce the average natural frequency
\[ \tag{77} \overline{\nu}_0 = \frac{ \nu_{0,+}+\nu_{0,-} }{2} \] and the dimensionless internal asymmetry
\[ \tag{78} \delta_a = \frac{ \nu_{0,+}-\nu_{0,-} }{ \nu_{0,+}+\nu_{0,-} }. \] From here
\[ \tag{79} \nu_{0,+} = \overline{\nu}_0(1+\delta_a), \qquad \nu_{0,-} = \overline{\nu}_0(1-\delta_a). \] Substituting formula (79) into (76) and adding the two lines, we obtain
\[ \tag{80} \boxed{ \overline{\nu} = \frac{ \nu_++\nu_- }{2} = \gamma\overline{\nu}_0 \left( 1+\beta\delta_a \right) }. \] For \(|\beta|\ll1\) and small \(|\delta_a|\), the relative displacement is
\[ \tag{81} \boxed{ \frac{ \overline{\nu} - \overline{\nu}_0 }{ \overline{\nu}_0 } \approx \frac{\beta^2}{2} + \beta\delta_a }. \] Formula (81) shows two distinct contributions:
\[ \tag{82} \underbrace{ \frac{\beta^2}{2} }_{ \text{standard second-order effect} } + \underbrace{ \beta\delta_a }_{ \text{internal directional asymmetry} }. \] For the additional contribution to have the same order of magnitude as the standard effect, approximately
\[ \tag{83} \beta\delta_a \sim \frac{\beta^2}{2} \quad\Longrightarrow\quad \boxed{ \delta_a \sim \frac{\beta}{2} }. \] However, the inequality \(a_1\ne a_2\) by itself does not lead to formula (75). The directional dependence \(\dot a_+\ne\dot a_-\) must be separately derived from the model geometry or introduced as a testable hypothesis.
15. How to experimentally separate the two effects
The standard relativistic contribution is an even function of velocity:
\[ \tag{84} \beta \longrightarrow -\beta \quad\Longrightarrow\quad \frac{\beta^2}{2} \longrightarrow \frac{\beta^2}{2}. \] The additional term \(\beta\delta_a\) is odd while the sign of \(\delta_a\) remains unchanged:
\[ \tag{85} \beta \longrightarrow -\beta \quad\Longrightarrow\quad \beta\delta_a \longrightarrow -\beta\delta_a. \] Therefore, measurements with opposite beam directions allow us to form even and odd combinations of the observed shift. If we denote the measured relative displacements by \(S(\beta)\) and \(S(-\beta)\), then
\[ \tag{86} S_{\mathrm{even}} = \frac{ S(\beta)+S(-\beta) }{2} \approx \frac{\beta^2}{2}, \] \[ \tag{87} S_{\mathrm{odd}} = \frac{ S(\beta)-S(-\beta) }{2} \approx \beta\delta_a. \] This decomposition turns the assumption of directional internal asymmetry into a testable consequence. The absence of an odd component sets an upper bound for \(|\delta_a|\), and its detection would require ruling out the usual systematic causes: inequalities in observation geometry, electric and magnetic fields, collisional shifts, and differences in spectral lines.
It is also important to consider the possible dependence of the asymmetry itself on direction. If \(\delta_a(-\beta)=-\delta_a(\beta)\), the product \(\beta\delta_a\) will become even and will not be separated by a simple change in the sign of the velocity. Therefore, an experimental verification should rely not only on symmetry, but also on a specifically derived law \(\delta_a(\beta)\).
16. What was obtained in the second part
In the first part, the same internal state served as a common frequency reference. In the second part, it is shown that for \(a_1\ne a_2\), it is necessary to separately take into account the relative phase, the ratio of internal frequencies, and the external Doppler projection.
The complete separation sequence has view
\[ \tag{88} \boxed{ \begin{aligned} J_1,J_2 &\longrightarrow \omega_1,\omega_2 \\ &\longrightarrow I_{2/1} = \frac{\Omega_{0.2}}{\Omega_{0.1}} \\ &\longrightarrow D_{2/1} = \frac{ \omega_2/\omega_1 } { I_{2/1} } \\ &\longrightarrow \beta_{2/1} = \frac{ D_{2/1}^{,2}-1 }{ D_{2/1}^{,2}+1 } \\ &\longrightarrow b_{2/1} = \frac{1}{\pi} \arcsin\beta_{2/1} \\ &\longrightarrow J_{2/1} = \j^{a_2-a_1} (-\j)^{b_{2/1}}. \end{aligned} } \] The central result of the paper can be written in two complementary ways:
\[ \tag{89} \boxed{ \frac{\omega_2}{\omega_1} = I_{2/1}D_{2/1} }, \qquad \boxed{ \ln \frac{\omega_2}{\omega_1} = \ln I_{2/1} + q_{2/1} }. \] Conclusion
The observed frequency is not itself a velocity. If the internal eigenstates differ, the same frequency ratio simultaneously contains internal and external factors. Only after removing the internal ratio \(I_{2/1}\) does the remaining Doppler scale unambiguously transform into an external relative velocity.
However, a difference in internal indices does not always imply a spectral difference. The constant \(a_2-a_1\) specifies only the relative phase. The observed internal shift occurs only when \(\dot a_1\ne\dot a_2\) or when the internal transverse projections differ.
Very high internal frequencies do not violate the velocity composition law. If the frequency difference is large compared to the reciprocal of the observation time, the rapidly oscillating interference term averages out. However, the internal frequency ratio does not vanish and must be taken into account when extracting velocity from spectral data.
\[ \tag{90} \boxed{ \text{to determine external motion} \quad \frac{\omega_2}{\omega_1} \;\longrightarrow\; \frac{ \omega_2/\omega_1 }{ I_{2/1} } \;\longrightarrow\; \beta_{2/1} }. \] Of particular interest is the possible dependence of the internal frequency on the observation direction. In the Ives-Stilwell scheme, it would provide an additional contribution to the standard second-order effect. This assumption does not follow automatically from the inequality \(a_1\ne a_2\), but can be formulated as an independent testable hypothesis of the model.
The mathematical origin of the operator is discussed in the article "From Euler's Formula to Split Geometry." The relationship between internal frequency and mass is discussed in detail in the article "Particle Mass as a Geometric Projection," and the consistent separation of internal and external motions is applied in the paper "Geometric Origin of the Squared Fine Structure Constant and the Parameters of the Bohr Atom."
Materials used
- An Experimental Study of the Rate of a Moving Atomic Clock. HERBERT E. IVES and G. R. STILWEL. [PDF]