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 imply motion? In Part 1, this question did not arise: the states being compared had the same internal frequency, so their common internal contribution canceled. The observed frequency ratio was determined solely by external motion, allowing the relativistic velocity-addition law to be recovered directly.
We now remove this restriction. Suppose the internal exponents of the two states differ:
\[ \tag{1} \boxed{ a_1(t)\ne a_2(t) }. \] An observed frequency change can then have two distinct causes. One is a change in the internal process itself; the other is the source's external motion relative to the observer:
\[ \tag{2} \boxed{ \text{observed frequency shift} = \text{internal change} + \text{external Doppler mapping} }. \] The plus sign in Eq. (2) should be understood in a logarithmic coordinate. In an ordinary frequency ratio, these contributions multiply rather than add. The main purpose of Part 2 is to separate them and establish the condition under which the measured frequency uniquely determines the external velocity.
This distinction also raises an experimental question. If a particle's internal state depends on the direction of motion or observation, an additional shift could appear in the Ives–Stilwell experiment and its modern counterparts. Differences in initial phase alone, with the same rate of internal rotation, do not produce a spectral shift.
Frequency and time conventions. In this article, \(\omega_{\mathrm{int},k}\) denotes the primary frequency with respect to the internal parameter \(\tau_k\), while \(\omega_{0,k}\) and \(\Omega_{0,k}\) denote the same frequency scale after internal projection, defined with respect to time in the rest frame of the corresponding source's center. The notation \(\Omega_{0,k}\) is retained for normalizing measured frequencies. The frequency \(\omega_k\) refers to the signal at an external receiver. In internal-phase formulas, \(t\) is the argument of the rest-frame time functions being compared; it is not a common laboratory time for two moving sources. Physical transmission of phase to the receiver is treated separately.
1. The General Operator for Two States
In split geometry, the complete-state operator is
\[ \tag{3} \boxed{ J(a,b) = \j^a(-\j)^b = \ep e^{i\pi b} + \em e^{i\pi a} }. \] 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, while \(b\) describes the particle's external motion. 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 the idempotent basis, Eq. (5) takes the 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 exponents are related to ordinary velocities by the model's adopted parameterization:
\[ \tag{7} b_k = \frac{1}{\pi} \arcsin\beta_k, \qquad \beta_k = \frac{v_k}{c} = \sin(\pi b_k). \] For constant frequencies, the internal exponents have the following general form. Here the derivative with respect to \(t\) is taken using the rest-frame time of the corresponding source's center; the frequency \(\omega_{0,k}\) defined below therefore already includes the internal temporal projection:
\[ \tag{8} a_k(t) = \varpi_k t+\varphi_k, \qquad \omega_{0,k} = \pi\dot a_k = \pi\varpi_k. \] The difference between \(a_1\) and \(a_2\) may therefore involve either the initial phases \(\varphi_k\) or the internal angular frequencies \(\omega_{0,k}\). These cases have different observable consequences.
2. What the Condition \(a_1\ne a_2\) Means
First consider a difference in initial phases alone:
\[ \tag{9} \varpi_1=\varpi_2=\varpi, \qquad \varphi_1\ne\varphi_2. \] The exponents then differ, but their derivatives are equal:
\[ \tag{10} a_1(t)\ne a_2(t), \qquad \dot a_1(t)=\dot a_2(t)=\varpi. \] Their frequencies with respect to the rest-frame times of their centers are consequently equal:
\[ \tag{11} \omega_{0,1} = \omega_{0,2} = \pi\varpi. \] A different case arises when the rates of internal rotation 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, unequal internal exponents do not by themselves establish unequal 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 rather than the absolute position within an internal cycle. A constant phase difference must therefore be distinguished from a continuously growing one.
3. Different Phases at the Same Internal Frequency
Under condition (9), the relative internal phase is
\[ \tag{14} \Delta a_{2/1} = a_2-a_1 = \varphi_2-\varphi_1 = \operatorname{const}. \] Its derivative vanishes:
\[ \tag{15} \frac{d}{dt} \Delta a_{2/1} = \dot a_2-\dot a_1 = 0. \] A constant internal phase difference at equal \(\omega_{0,k}\) therefore produces no separate spectral shift. Frequency equality already refers to the states after internal projection. In a consistent longitudinal arrangement, the observed frequency ratio is determined solely by the external Doppler factors:
\[ \tag{16} \boxed{ \frac{\omega_2}{\omega_1} = \frac{D_2}{D_1} }. \] The result of Part 1 therefore remains unchanged:
\[ \tag{17} \boxed{ \beta_{2/1} = \frac{ \beta_2-\beta_1 }{ 1-\beta_1\beta_2 } }. \] The difference \(\varphi_2-\varphi_1\) may appear when two waves are coherently superposed, because it determines their interference pattern. It does not, however, change the spectral-line position as long as \(\dot a_1=\dot a_2\).
4. Different Internal Frequencies and Projections
Now consider the general case \(\dot a_1\ne\dot a_2\). The frequencies of the compared states with respect to the rest-frame times of their centers are
\[ \tag{18} \omega_{0,1} = \pi\dot a_1, \qquad \omega_{0,2} = \pi\dot a_2. \] To relate these frequencies to primary internal dynamics, introduce the internal parameters \(\tau_k\). In the model's kinematic realization, they are associated with normalized internal velocities and the corresponding factors:
\[ \tag{19} \eta_k = \frac{v_{\mathrm{int},k}}{c}, \qquad \gamma_{a,k} = \frac{1}{ \sqrt{1-\eta_k^2} }. \] Denote each state's internal temporal projection coefficient by \(P_{m,k}\). To retain consistency with Part 1, we also keep the symbol \(\mathcal D_{a,k}\):
\[\tag{20} P_{m,k}=\mathcal D_{a,k}=\frac{d\tau_k}{dt}=\frac{1}{\gamma_{a,k}}. \] The same phase satisfies \(d(\pi a_k)=\omega_{\mathrm{int},k}d\tau_k=\omega_{0,k}dt\). The rest-frame scale can therefore be written as
\[\tag{21} \boxed{ \Omega_{0,k}=\omega_{0,k}=\pi\dot a_k =P_{m,k}\omega_{\mathrm{int},k}. } \] The inequality \(a_1\ne a_2\) does not by itself require \(P_{m,1}\ne P_{m,2}\). Projected frequencies may differ because of the primary frequencies, projection coefficients, or both. These differences may also compensate: different \(\omega_{\mathrm{int},k}\) need not give different \(\Omega_{0,k}\). Since \(\omega_{0,k}=\pi\dot a_k\) is already defined with respect to the time \(t\), external to the internal process, it must not be multiplied by \(P_{m,k}\) again.
The relation \(P_{m,k}=1/\gamma_{a,k}\) is adopted as the model's internal temporal projection, motivated by relativistic clock comparisons. It does not follow from the orthogonality of idempotent channels alone. A literal interpretation as a transverse Doppler effect would require specifying the signal source, receiver, ray direction, and reference frames separately. If the primary process is a light wave, \(\tau_k\) cannot automatically be treated as the photon's proper time: it is an internal model parameter whose physical meaning requires separate justification.
5. The Full Observed Frequency
Introduce the directional Doppler factor for each source's external motion relative to a common receiver. The following formula applies to light propagating in vacuum along a common axis. The sign of \(\beta_k\) is chosen so that positive velocity increases the received frequency:
\[ \tag{22} D_k = \sqrt{ \frac{1+\beta_k} {1-\beta_k} }. \] Provided that the emitted signal's frequency in the source rest frame is \(\Omega_{0,k}\), the observed angular frequency is
\[\tag{23} \boxed{ \omega_k=D_k\Omega_{0,k}=D_k\omega_{0,k} =D_kP_{m,k}\omega_{\mathrm{int},k}. } \] The ratio of the two observed frequencies is therefore
\[ \tag{24} \frac{\omega_2}{\omega_1} = \frac{\Omega_{0,2}}{\Omega_{0,1}} \frac{D_2}{D_1}. \] Introduce the rest-frame frequency ratio. It can be expressed either through the already projected frequencies or through the primary frequencies and projection coefficients:
\[\tag{25} \boxed{ I_{2/1}=\frac{\Omega_{0,2}}{\Omega_{0,1}} =\frac{\dot a_2}{\dot a_1} =\frac{P_{m,2}\omega_{\mathrm{int},2}} {P_{m,1}\omega_{\mathrm{int},1}}. } \] As in Part 1, denote the external relative factor by
\[ \tag{26} D_{2/1} = \frac{D_2}{D_1}. \] The central decomposition of Part 2 then becomes
\[ \tag{27} \boxed{ \frac{\omega_2}{\omega_1} = I_{2/1}D_{2/1} }. \] One observed frequency shift contains two independent factors. Without knowing \(I_{2/1}\), one therefore cannot uniquely determine how much of the measured ratio comes from an internal-state change and how much from external motion.
Scope of the Doppler description. A periodic phase in \(J\) does not by itself specify how that phase is detected. For a light signal traveling from a source to a stationary receiver, the general factor is \(D(\beta,\theta)=[\gamma(1-\beta\cos\theta)]^{-1}\), with the angle defined in the receiver frame. Equation (22) uses the longitudinal case \(\theta=0\). The internal mapping \(P_m\) and the external Doppler effect are different operations; the product in (23) does not follow from perpendicular phase planes. A hidden internal periodicity additionally requires a physical coupling to the received signal.
6. Isolating External Motion
Equation (27) gives the external relative factor by normalization against the internal ratio:
\[ \tag{28} \boxed{ D_{2/1} = \frac{ \omega_2/\omega_1 }{ I_{2/1} } = \frac{ \omega_2\Omega_{0,1} }{ \omega_1\Omega_{0,2} } }. \] In the adopted longitudinal arrangement, velocity is uniquely recovered from the directional Doppler factor using
\[ \tag{29} \boxed{ \beta = \frac{D^2-1}{D^2+1} }. \] Substituting (28) gives 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 equals \(D_2/D_1\). Expanding the Doppler factors explicitly therefore gives
\[ \tag{31} \left(D_{2/1}\right)^2 = \frac{ (1+\beta_2)(1-\beta_1) }{ (1-\beta_2)(1+\beta_1) }. \] The inverse transformation (29) again yields the relativistic velocity-difference law:
\[ \tag{32} \boxed{ \beta_{2/1} = \frac{ \beta_2-\beta_1 }{ 1-\beta_1\beta_2 } }. \] Likewise, sequential addition first requires isolating each state's normalized external factor:
\[ \tag{33} D_k = \frac{\omega_k}{\Omega_{0,k}}, \qquad D_{1\oplus2} = D_1D_2. \] Applying the inverse transformation gives
\[ \tag{34} \boxed{ \beta_{1\oplus2} = \frac{ \beta_1+\beta_2 }{ 1+\beta_1\beta_2 } }. \] Different rest-frame frequencies therefore do not alter the law for adding or subtracting external velocities. When two sources are compared, their velocities and ray directions must be specified in a consistent common frame. The product \(D_1D_2\) describes sequential collinear composition, rather than simply adding two measured laboratory velocities. Without normalization against \(\Omega_{0,k}\), an internal change is incorrectly included as part of the motion.
7. The Full Logarithmic Coordinate
The multiplicative decomposition (27) becomes additive after taking logarithms. Define the full observed shift as
\[ \tag{35} Q_{2/1} = \ln \frac{\omega_2}{\omega_1}. \] Then
\[ \tag{36} Q_{2/1} = \ln I_{2/1} + \ln D_2 - \ln D_1. \] As shown in Part 1, the logarithm of the external Doppler factor is the rapidity:
\[ \tag{37} q_k = \ln D_k = \operatorname{artanh}\beta_k. \] The full shift is therefore
\[ \tag{38} \boxed{ Q_{2/1} = \ln I_{2/1} + \operatorname{artanh}\beta_2 - \operatorname{artanh}\beta_1 }. \] In this coordinate, 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 isolated by subtracting the internal contribution:
\[ \tag{40} \boxed{ q_{2/1} = Q_{2/1} - \ln I_{2/1} }. \] Then \(\beta_{2/1}=\tanh q_{2/1}\), and the velocity-difference formula again takes the form (32).
8. Effective Velocity for an Unknown Internal State
Suppose the entire observed frequency ratio is mistakenly interpreted as an external Doppler factor:
\[ \tag{41} R_{\mathrm{full}} = \frac{\omega_2}{\omega_1} = I_{2/1}D_{2/1}. \] The measurement then yields an effective quantity rather than the true relative velocity:
\[ \tag{42} \boxed{ \beta_{\mathrm{eff}} = \frac{ R_{\mathrm{full}}^2-1 }{ R_{\mathrm{full}}^2+1 } }. \] Substituting the decomposition (41) gives
\[ \tag{43} \boxed{ \beta_{\mathrm{eff}} = \frac{ I_{2/1}^{\,2}D_{2/1}^{\,2}-1 }{ I_{2/1}^{\,2}D_{2/1}^{\,2}+1 } }. \] Even without external relative motion, when \(D_{2/1}=1\), different internal states can produce a nonzero effective velocity:
\[ \tag{44} D_{2/1}=1 \quad\Longrightarrow\quad \beta_{\mathrm{eff}} = \frac{ I_{2/1}^{\,2}-1 }{ I_{2/1}^{\,2}+1 }. \] A single frequency ratio therefore cannot uniquely determine velocity if the source's intrinsic internal frequency is unknown or changing. Determining the motion requires either a known internal reference or an independent measurement of \(I_{2/1}\).
9. Relative Phase, Beats, and Averaging
First compare the relative internal phase using matched rest-frame time arguments. This expression concerns the sources' reference phase functions, rather than the two signals directly at the receiver:
\[ \tag{45} \boxed{ \Delta a_{2/1}(t) = a_2(t)-a_1(t) }. \] For the linear phases (8), we obtain
\[ \tag{46} \Delta a_{2/1}(t) = (\varpi_2-\varpi_1)t + (\varphi_2-\varphi_1). \] The magnitude of this relative phase's rate of change gives the beat frequency of the reference functions:
\[\tag{47} \boxed{ \omega_{\mathrm{beat},0} =\left|\pi\frac{d}{dt}\Delta a_{2/1}\right| =|\Omega_{0,2}-\Omega_{0,1}|. } \] Three quantities must be distinguished here. A constant phase difference controls the interference pattern; a frequency difference produces beats; and a frequency ratio enters the internal factor \(I_{2/1}\) and affects velocity recovery.
Actual signal superposition requires frequencies received by one instrument and a common receiver time \(t_{\mathrm{rec}}\). If the signals are coherent and their frequencies and geometry remain constant over the observation interval, the cross term contains
\[\tag{48} \cos(\Delta\omega\,t_{\mathrm{rec}}+\Delta\phi_{\mathrm{rec}}), \qquad \Delta\omega=\omega_2-\omega_1. \] If the observation time is \(T_{\mathrm{obs}}\) and the frequency difference satisfies
\[ \tag{49} \boxed{ |\Delta\omega| T_{\mathrm{obs}} \gg 2\pi }, \] then the interference term undergoes many oscillations during the measurement and averages nearly to zero:
\[\tag{50} \left\langle \cos(\Delta\omega\,t_{\mathrm{rec}}+\Delta\phi_{\mathrm{rec}}) \right\rangle_{T_{\mathrm{obs}}}\approx0. \] Elementary particles' internal frequencies may be extremely high compared with the inverse duration of ordinary observations. Large values of \(\omega_1\) and \(\omega_2\) do not imply a large difference, however: two enormous frequencies can be very close. Averaging is governed by the product \(|\Delta\omega|T_{\mathrm{obs}}\), rather than either absolute frequency.
Under condition (49), temporal averaging suppresses the rapidly oscillating cross term. Resolving two spectral components separately also depends on their linewidths, the instrument bandwidth, and measurement duration. The rest-frame frequency ratio itself does not disappear under this averaging:
\[ \tag{51} \boxed{ \text{rapidly oscillating interference term} \longrightarrow 0, \qquad I_{2/1} = \frac{\Omega_{0,2}}{\Omega_{0,1}} \ne 0 }. \] High frequency therefore obscures the instantaneous phase pattern without eliminating a systematic frequency shift. After correct internal normalization, it does not affect the law for adding or subtracting 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 exactly describes the differences between the original states' exponents. The exponent \(b_2-b_1\), however, is not the exponent associated with the physical relativistic relative velocity. The reason is that \(b\) is related to velocity through \(\arcsin\beta\), whereas external velocities transform according to a fractional-linear law.
The physically recovered relative velocity is
\[ \tag{53} \beta_{2/1} = \frac{ \beta_2-\beta_1 }{ 1-\beta_1\beta_2 }. \] Its corresponding external exponent is
\[ \tag{54} \boxed{ b_{2/1} = \frac{1}{\pi} \arcsin \left( \frac{ \beta_2-\beta_1 }{ 1-\beta_1\beta_2 } \right) }. \] Two objects must therefore be distinguished:
\[ \tag{55} \boxed{ K_{2/1} = \j^{a_2-a_1} (-\j)^{b_2-b_1} } \quad \text{— algebraic ratio,} \] \[ \tag{56} \boxed{ J_{2/1} = \j^{\Delta a_{2/1}} (-\j)^{b_{2/1}} } \quad \text{— physical relative operator.} \] In the idempotent basis, the physical relative operator takes the form
\[ \tag{57} J_{2/1} = \ep e^{i\pi b_{2/1}} + \em e^{i\pi\Delta a_{2/1}}. \] The first component contains the recovered external relative velocity, while the second contains the relative internal phase under the adopted comparison convention. Here “physical relative operator” means a parameterization of these two characteristics, rather than an already derived complete Lorentz operator transforming a field or spinor. It does not replace relativistic composition with the simple exponent difference \(b_2-b_1\).
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 equal frequencies and velocities.
\[ \tag{59} \dot a_1=\dot a_2, \qquad \beta_1=\beta_2 \quad\Longrightarrow\quad J_{2/1} = \j^{\varphi_2-\varphi_1}. \] Only a constant relative phase remains; there is neither external relative velocity nor internal beating.
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 beating occurs even though the external relative velocity is zero.
Identical 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 considered in Part 1.
Different internal and external states simultaneously.
\[ \tag{62} I_{2/1}\ne1, \qquad D_{2/1}\ne1 \quad\Longrightarrow\quad \frac{\omega_2}{\omega_1} = I_{2/1}D_{2/1}. \] Only after separating these factors can the internal change and external relative velocity be recovered independently.
12. Relating Internal Frequency to Energy and Mass
For mass-associated periodicity, the primary phase-energy scale must be distinguished from the rest energy. The primary frequency corresponds to
\[\tag{63} \mathcal E_{\mathrm{ph},k}=\hbar\omega_{\mathrm{int},k}. \] Following “Mass as a Geometric Projection of a Closed Wave”, the rest energy is defined through a separate physical mapping:
\[\tag{64} E_{0,k}=P_{m,k}\mathcal E_{\mathrm{ph},k}=m_kc^2. \] It is therefore the frequency after mass projection that is identified with the Compton frequency:
\[\tag{65} \boxed{ \omega_{C,k}=\Omega_{0,k}=\omega_{0,k} =P_{m,k}\omega_{\mathrm{int},k} =\frac{m_kc^2}{\hbar}. } \] For two states' primary frequencies, with nonzero projection coefficients, we obtain
\[\tag{66} \frac{\omega_{\mathrm{int},2}}{\omega_{\mathrm{int},1}} =\frac{m_2}{m_1}\frac{P_{m,1}}{P_{m,2}}. \] After accounting for the projections, the internal factor (25) becomes
\[\tag{67} \boxed{ I_{2/1} =\frac{P_{m,2}\omega_{\mathrm{int},2}} {P_{m,1}\omega_{\mathrm{int},1}} =\frac{\omega_{C,2}}{\omega_{C,1}} =\frac{m_2}{m_1}. } \] The mass ratio already includes the mass projections. Multiplying it again by \(\mathcal D_{a,2}/\mathcal D_{a,1}=P_{m,2}/P_{m,1}\) would count the same transition from primary frequency to rest energy twice. The electron realization adopts \(P_m=\alpha_{\mathrm{fs}}\). Extending this value to other particle types requires separate justification.
The identifications (65)–(67) apply only to mass-associated periodicity. Observing it through a light signal requires a separate coupling model. For a spectral transition, the rest-frame frequency is determined by the energy-level difference, rather than the particle's entire rest energy. The general formulas (23)–(30) also apply to spectral signals when \(\Omega_{0,k}\) denotes the corresponding line frequency, but the equality \(I_{2/1}=m_2/m_1\) is not generally used in that case.
The following sections concern actual spectral lines. Their rest-frame frequency is \(\nu_0=(E_i-E_f)/h\); it is not identified with the Compton frequency \(m c^2/h\). The standard longitudinal Doppler factors act on the already established line frequency; no additional internal coefficient is applied to it again.
13. The Standard Ives–Stilwell Arrangement
The Ives–Stilwell experiment [1] separates the first-order longitudinal Doppler shift from the second-order frequency change. Radiation from the same fast beam is observed along and opposite to its direction of motion. In the standard case, both lines correspond to the same transition with a single rest-frame frequency \(\nu_0\).
In an idealized longitudinal frequency-measurement arrangement observing one beam along and opposite to its motion, the received frequencies are
\[ \tag{68} \nu_+ = \nu_0\gamma(1+\beta), \qquad \nu_- = \nu_0\gamma(1-\beta), \] where
\[ \tag{69} \gamma = \frac{1}{ \sqrt{1-\beta^2} }. \] The half-difference of the lines isolates the first-order longitudinal effect:
\[ \tag{70} \frac{ \nu_+-\nu_- }{2} = \gamma\nu_0\beta. \] Their mean eliminates the linear terms:
\[ \tag{71} \boxed{ \overline{\nu} = \frac{ \nu_++\nu_- }{2} = \gamma\nu_0 }. \] At low velocities,
\[ \tag{72} \gamma \approx 1+\frac{\beta^2}{2}, \] and the relative shift of the line-pair center is
\[ \tag{73} \boxed{ \frac{ \overline{\nu}-\nu_0 }{ \nu_0 } \approx \frac{\beta^2}{2} }. \] A constant phase difference does not change the selected line's frequency. If \(a_+\) and \(a_-\) denote the phase exponents of this particular transition with respect to the source's rest-frame time, then
\[ \tag{74} \dot a_+ = \dot a_- \quad\Longrightarrow\quad \nu_{0,+} = \nu_{0,-} = \nu_0. \] Equality of the intrinsic frequencies for forward and backward observation allows the remaining shift to be interpreted as the standard second-order effect.
14. Possible Internal Asymmetry
As a separate hypothesis, suppose that the spectral-transition frequency, defined with respect to the source's rest-frame time, depends on the emission direction:
\[ \tag{75} \nu_{0,+} \ne \nu_{0,-}. \] The two lines then take the form
\[ \tag{76} \nu_+ = \nu_{0,+}\gamma(1+\beta), \qquad \nu_- = \nu_{0,-}\gamma(1-\beta). \] Introduce the mean intrinsic frequency
\[ \tag{77} \overline{\nu}_0 = \frac{ \nu_{0,+}+\nu_{0,-} }{2} \] and the dimensionless internal asymmetry
\[ \tag{78} \delta_a = \frac{ \nu_{0,+}-\nu_{0,-} }{ \nu_{0,+}+\nu_{0,-} }. \] Hence
\[ \tag{79} \nu_{0,+} = \overline{\nu}_0(1+\delta_a), \qquad \nu_{0,-} = \overline{\nu}_0(1-\delta_a). \] Substituting (79) into (76) and adding the two lines gives
\[ \tag{80} \boxed{ \overline{\nu} = \frac{ \nu_++\nu_- }{2} = \gamma\overline{\nu}_0 \left( 1+\beta\delta_a \right) }. \] For \(|\beta|\ll1\) and small \(|\delta_a|\), the relative shift is
\[ \tag{81} \boxed{ \frac{ \overline{\nu} - \overline{\nu}_0 }{ \overline{\nu}_0 } \approx \frac{\beta^2}{2} + \beta\delta_a }. \] Equation (81) contains 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 be of the same order as the standard effect, approximately
\[ \tag{83} \beta\delta_a \sim \frac{\beta^2}{2} \quad\Longrightarrow\quad \boxed{ \delta_a \sim \frac{\beta}{2} }. \] The inequality \(a_1\ne a_2\) does not by itself imply (75). Directional dependence of the line frequency must be derived separately from the model or introduced as a testable hypothesis. Conventional theory does not predict such dependence for the same transition of an unperturbed isotropic source; angular dependence of intensity does not itself imply different frequencies.
15. Separating the Two Effects Experimentally
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 if the sign of \(\delta_a\) remains unchanged:
\[ \tag{85} \beta \longrightarrow -\beta \quad\Longrightarrow\quad \beta\delta_a \longrightarrow -\beta\delta_a. \] Measurements with opposite beam directions therefore allow even and odd combinations of the observed shift to be formed. Denoting the measured relative shifts by \(S(\beta)\) and \(S(-\beta)\), we have
\[ \tag{86} S_{\mathrm{even}} = \frac{ S(\beta)+S(-\beta) }{2} \approx \frac{\beta^2}{2}, \] \[ \tag{87} S_{\mathrm{odd}} = \frac{ S(\beta)-S(-\beta) }{2} \approx \beta\delta_a. \] This decomposition makes the proposed directional internal asymmetry testable. The absence of an odd component places an upper bound on \(|\delta_a|\); detecting one would require excluding ordinary systematic causes, including unequal observation geometries, electric and magnetic fields, collisional shifts, and differences between spectral lines.
Possible directional dependence of the asymmetry itself must also be considered. If \(\delta_a(-\beta)=-\delta_a(\beta)\), the product \(\beta\delta_a\) becomes even and cannot be isolated by simply reversing the velocity. Experimental tests must therefore rely on a specifically derived law \(\delta_a(\beta)\), as well as symmetry.
16. Results of Part 2
In Part 1, a common projected frequency scale served as the reference. Part 2 distinguishes initial phase, primary internal frequency, temporal projection, and the external Doppler shift. The comparison formulas require knowledge of the rest-frame frequencies \(\Omega_{0,k}\); equal initial phases are unnecessary.
The complete separation procedure is
\[ \tag{88} \boxed{ \begin{aligned} J_1,J_2 &\longrightarrow \omega_1,\omega_2 \\ &\longrightarrow I_{2/1} = \frac{\Omega_{0,2}}{\Omega_{0,1}} \\ &\longrightarrow D_{2/1} = \frac{ \omega_2/\omega_1 }{ I_{2/1} } \\ &\longrightarrow \beta_{2/1} = \frac{ D_{2/1}^{\,2}-1 }{ D_{2/1}^{\,2}+1 } \\ &\longrightarrow b_{2/1} = \frac{1}{\pi} \arcsin\beta_{2/1} \\ &\longrightarrow J_{2/1} = \j^{a_2-a_1} (-\j)^{b_{2/1}}. \end{aligned} } \] The central result can be expressed 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
An observed frequency is not, by itself, a velocity. If the intrinsic internal states differ, the same frequency ratio contains both internal and external factors. Only after removing the internal ratio \(I_{2/1}\) can the remaining Doppler factor be uniquely converted to an external relative velocity.
Different internal exponents do not always imply a spectral difference. A constant \(a_2-a_1\) specifies a relative phase. Changes in the rest-frame frequency \(\Omega_0=P_m\omega_{\mathrm{int}}\) are determined jointly by the primary frequency and temporal projection; at constant \(P_m\), they correspond to changes in \(\dot a\).
Extremely high internal frequencies do not violate the velocity-composition law. If the frequency difference is large compared with the inverse observation time, the rapidly oscillating interference term averages out. The internal frequency ratio nevertheless remains and must be accounted for when extracting velocity from spectral data.
\[ \tag{90} \boxed{ \text{to determine external motion} \quad \frac{\omega_2}{\omega_1} \;\longrightarrow\; \frac{ \omega_2/\omega_1 }{ I_{2/1} } \;\longrightarrow\; \beta_{2/1} }. \] A hypothesis of particular interest is directional dependence of a spectral-transition frequency in the source rest frame. In an idealized Ives–Stilwell arrangement, it would add a contribution to the standard second-order effect. This is an independent testable hypothesis, rather than a consequence of phase orthogonality or the inequality \(a_1\ne a_2\) alone.
The operator's mathematical origin is discussed in “From Euler's Formula to Split Geometry.” The connection between internal frequency and mass is developed in “Particle Mass as a Geometric Projection,” while the successive separation of internal and external motion is applied in “The Geometric Origin of the Square of the Fine-Structure Constant and the Parameters of the Bohr Atom.”
Materials used
- H. E. Ives, G. R. Stilwell. An Experimental Study of the Rate of a Moving Atomic Clock. Journal of the Optical Society of America, 28, 215–226 (1938).
- Ted Jacobson. Phys410 Notes, Demos and Supplements. University of Maryland, 2011. Раздел от 8 ноября: частота наблюдателя и релятивистский эффект Доплера.