2026-07-29
The sum and difference of velocities as a composition of Doppler projections
Part 1. The same internal state
What exactly is added when velocities are added? In classical mechanics, the answer seems obvious: just add or subtract two numbers. But at higher velocities, this rule breaks down. Two velocities, each less than the speed of light, can, when added together, yield a value greater than \(c\). Nature does not allow such a result, and an additional denominator unexpectedly appears in the relativistic formula.
We can accept this denominator as a ready-made rule. But we can ask a deeper question: is there a variable of motion that truly adds up linearly, and the familiar velocity is obtained from it only after the inverse transformation?
In this paper, such a variable is the logarithm of the Doppler projection of the particle's intrinsic frequency. The main idea of the article is the following: velocities do not add up directly. Each external motion is first converted to a Doppler scale, the scales are multiplied or divided, and then the result is converted back to velocity.
\[ \tag{1} \boxed{ \text{velocity} \;\longrightarrow\; \text{Doppler scale} \;\longrightarrow\; \text{composition} \;\longrightarrow\; \text{velocity} } \] In the concept of wave electricity, the application of the Doppler effect is not a formal device. The particle operator contains internal rotation and, therefore, an intrinsic natural frequency. It is this physically defined periodicity that is subject to Doppler projection during external motion.
The first part considers the clearest case: two particles or two states being compared have exactly the same internal state, that is, \(a_1(t)=a_2(t)\). Due to this, the internal component is common and cancels out during comparison. In the second part, this limitation will be lifted.
1. Full State Operator
The mathematical basis of the model is the split geometry operator
\[ \tag{2} \boxed{ J(a,b) = \j^{a}(-\j)^{b} = \ep e^{i\pi b} + \em e^{i\pi a} }. \] The idempotents \(\ep\) and \(\em\) define two independent complex planes:
\[ \tag{3} \ep^{2}=\ep, \qquad \em^{2}=\em, \qquad \ep\em=0, \qquad \ep+\em=1. \] The parameter \(a\) describes the internal state of the particle, and the parameter \(b\) describes its external motion. Therefore, one operator combines two independent phase processes without mixing them with each other.
For the two states under consideration, we write
\[ \tag{4} J_1 = \j^{a}(-\j)^{b_1}, \qquad J_2 = \j^{a}(-\j)^{b_2}. \] The common exponent \(a\) in formula (4) denotes a stronger condition than the simple equality of internal frequencies:
\[ \tag{5} a_1(t)=a_2(t)=a(t), \qquad \dot a_1(t)=\dot a_2(t)=\dot a(t). \] Consequently, both the position in the inner cycle and the speed of internal rotation are the same. The difference between the states is contained only in the external indices \(b_1\) and \(b_2\).
The relationship between the external index and the speed is determined by the parameterization adopted in the model.
\[ \tag{6} b_k = \frac{1}{\pi}\arcsin\beta_k, \qquad \beta_k = \frac{v_k}{c}, \qquad \beta_k = \sin(\pi b_k), \quad k=1.2. \] For \(|\beta_k| < 1\), the exponent \(b_k\) remains finite, and the velocity of each initial state is less than \(c\). The goal of this article is to obtain a composite external exponent from \(b_1\) and \(b_2\) without violating this constraint.
2. Why the Doppler Effect Can Be Applied to the Operator
The Doppler effect is only meaningful when there is a periodic process whose frequency can be measured by the observer. If the operator \(J\) described only the position without internal periodicity, using the Doppler coefficient would be artificial.
In the model under consideration, the periodic process is present from the start. Let
\[ \tag{7} a(t)=\varpi t, \qquad \omega_0=\pi\varpi=\pi\dot a. \] Then the internal component of the operator has the form
\[ \tag{8} \em e^{i\pi a(t)} = \em e^{i\omega_0 t}. \] Thus, \(\omega_0\) is the angular frequency of the real internal rotation in the plane \(\{\em,i\em\}\). External motion does not create this frequency, but rather changes its observed projection.
\[ \tag{9} \boxed{ \omega_{\mathrm{obs}} = \mathcal D\,\omega_0 }. \] It is the presence of \(\omega_0\) that provides the physical basis for using the Doppler effect. We are not transforming an abstract number \(v\), but the frequency of an existing internal process. The velocity is then reconstructed based on the magnitude of the frequency shift.
For states (4), the original frequency is the same:
\[ \tag{10} \omega_{0,1} = \omega_{0,2} = \omega_0. \] This equality will be key: the ratio of the observed frequencies will depend only on the difference in external motions.
3. Internal and External Motions
The operator components belong to two mutually orthogonal idempotent planes. In the physical interpretation of the model, this means that internal rotation and external motion are considered as two successive orthogonal processes:
\[ \tag{11} \mathbf v_{\mathrm{int}} \perp \mathbf v_{\mathrm{ext}}. \] These are not the usual Cartesian components of a single velocity vector measured in the same plane. We are talking about two levels of motion of the total state: internal rotation, associated with the exponent \(a\), and external displacement, associated with the exponent \(b\).
Therefore, the total projection of the observed frequency is written as a successive product of the internal transverse and external directional projections. We denote the normalized velocity of the internal motion by
\[ \tag{12} \eta = \frac{v_{\mathrm{int}}}{c}, \qquad \gamma_a = \frac{1}{\sqrt{1-\eta^2}}. \] For the external motion of each state, we have
\[ \tag{13} \gamma_k = \frac{1}{\sqrt{1-\beta_k^2}}. \] For the chosen observation direction, we write the combined Doppler coefficient in the form
\[ \tag{14} \mathcal D_k = \underbrace{\frac{1}{\gamma_a}}_{ \substack{ \text{internal}\\ \text{transverse projection} }} \underbrace{\frac{1}{\gamma_k(1-\beta_k)}}_{ \substack{ \text{external}\\ \text{directional projection} }}. \] The sign of \(\beta_k\) indicates the direction of motion. With the accepted notation, a positive \(\beta_k\) corresponds to the direction in which the observed frequency increases. When the direction of \(\beta_k\to-\beta_k\) changes, the coefficient automatically inverts: \(D(-\beta_k)=1/D(\beta_k)\).
The observed internal frequency of each state is
\[ \tag{15} \omega_k = \mathcal D_k\omega_0. \] The perpendicularity of the internal and external motions is taken into account in the absolute projection of each state through a successive product of factors. However, since the internal state of \(J_1\) and \(J_2\) is the same, the common factor \(1/\gamma_a\) will later cancel out.
4. External Doppler scale
We transform the external part of formula (14). Using the definition of \(\gamma_k\), we obtain
\[ \tag{16} \begin{aligned} \frac{1}{\gamma_k(1-\beta_k)} &= \frac{\sqrt{1-\beta_k^2}}{1-\beta_k} \\[1mm] &= \frac{ \sqrt{(1-\beta_k)(1+\beta_k)} }{ 1-\beta_k } \\[1mm] &= \sqrt{ \frac{1+\beta_k}{1-\beta_k} }. \end{aligned} \] We introduce two separate notations:
\[ \tag{17} \mathcal D_a = \frac{1}{\gamma_a}, \qquad D_k = \sqrt{ \frac{1+\beta_k}{1-\beta_k} }. \] Then the total coefficient and the observed frequency take a transparent form:
\[ \tag{18} \boxed{ \mathcal D_k = \mathcal D_aD_k }, \qquad \boxed{ \omega_k = \omega_0\mathcal D_aD_k }. \] The quantity \(\mathcal D_a\) characterizes the overall internal state, while \(D_k\) characterizes only the external directed motion. Therefore, for operations on external velocities, it is necessary to use the normalized external scale \(D_k\), and not the full coefficient \(\mathcal D_k\).
This distinction is especially important during addition. If we multiply the full coefficients, the internal factor will erroneously appear twice:
\[ \tag{19} \mathcal D_1\mathcal D_2 = \mathcal D_a^2D_1D_2. \] But the internal state does not add to itself: it is the common background of both external motions. Therefore, normalization is performed first.
\[ \tag{20} \boxed{ D_k = \frac{\mathcal D_k}{\mathcal D_a} = \frac{\omega_k}{\omega_0\mathcal D_a} }. \] 5. Velocity Difference from the Ratio of Doppler Scales
Before introducing the logarithm and rapidity, we obtain the relative velocity directly from the Doppler scales. This order of derivation is important: first, we see where the relativistic denominator comes from, and only then introduce the coordinate in which the resulting transformation is written linearly.
Let's determine the external velocity of the second state relative to the first. The transition from state 1 to state 2 consists of two operations. First, the inverse factor \(D_1^{-1}\) eliminates the Doppler projection of the first motion, then the factor \(D_2\) introduces the projection of the second motion. Therefore, the relative scale is
\[ \tag{21} \boxed{ D_{2/1} = \frac{D_2}{D_1} }, \qquad \boxed{ D_{2/1}^{2} = \frac{ (1+\beta_2)(1-\beta_1) }{ (1-\beta_2)(1+\beta_1) } }. \] Now let's show how to dopplerThe velocity is uniquely restored to the Rovsky scale. For arbitrary motion, the definition of \(D^2=(1+\beta)/(1-\beta)\) can be solved for \(\beta\). Multiplying both sides by \(1-\beta\) and collecting terms with \(\beta\), we obtain
\[ \tag{22} \begin{aligned} D^2 &= \frac{1+\beta}{1-\beta}, \\[1mm] D^2(1-\beta) &= 1+\beta, \\[1mm] D^2-1 &= \beta(D^2+1), \\[1mm] \boxed{ \beta = \frac{D^2-1}{D^2+1} }. \end{aligned} \] This formula allows us to go from the ratio of frequency scales directly to the relative velocity. Let's substitute expression (21) into it. To avoid losing signs, we'll denote the numerator of the fraction for \(D_{2/1}^{2}\) by \(A\), and the denominator by \(B\). Then \((A/B-1)/(A/B+1)=(A-B)/(A+B)\), and therefore
\[ \tag{23} \begin{aligned} \beta_{2/1} &= \frac{ D_{2/1}^{2}-1 }{ D_{2/1}^{2}+1 } \\[1mm] &= \frac{ (1+\beta_2)(1-\beta_1) - (1-\beta_2)(1+\beta_1) }{ (1+\beta_2)(1-\beta_1) + (1-\beta_2)(1+\beta_1) }. \end{aligned} \] Let's expand both pairs of parentheses. In the numerator, the constant terms and the products \(\beta_1\beta_2\) cancel out, and in the denominator, the linear terms cancel out. What remains is
\[ \tag{24} \begin{aligned} \beta_{2/1} &= \frac{ 2\beta_2-2\beta_1 }{ 2-2\beta_1\beta_2 } \\[1mm] &= \boxed{ \frac{ \beta_2-\beta_1 }{ 1-\beta_1\beta_2 } }. \end{aligned} \] Thus, an additional denominator arises already during the reverse transition from the ratio of Doppler scales to velocity. To obtain it, neither the logarithm nor hyperbolic functions needed to be introduced beforehand.
6. Logarithmic Doppler Coordinate
Now that the formula for the velocity difference has been directly obtained, we can determine which coordinate makes this composition linear. Doppler scales are multiplied during successive transitions, and divided during relative comparisons. The logarithm transforms these operations into addition and subtraction, respectively. Therefore, we introduce
\[ \tag{25} \boxed{ q_k = \ln D_k }. \] Substituting the definition of the external Doppler scale, we find
\[ \tag{26} \begin{aligned} q_k &= \ln \sqrt{ \frac{1+\beta_k}{1-\beta_k} } \\[1mm] &= \frac{1}{2} \ln \frac{1+\beta_k}{1-\beta_k} \\[1mm] &= \operatorname{artanh}\beta_k. \end{aligned} \] The coordinate \(q\) is known in relativistic kinematics as rapidity. Here, it is not introduced as a predetermined change of variable, but arises from the Doppler scale of motion. The inverse transformation is \(\beta_k=\tanh q_k\).
For relative motion, dividing the scales becomes subtracting their logarithms:
\[ \tag{27} \begin{aligned} q_{2/1} &= \ln D_{2/1} \\ &= \ln D_2-\ln D_1 \\ &= q_2-q_1. \end{aligned} \] Therefore, the relative velocity can be written as
\[ \tag{28} \begin{aligned} \beta_{2/1} &= \tanh q_{2/1} \\ &= \tanh(q_2-q_1) \\ &= \frac{ \tanh q_2-\tanh q_1 }{ 1-\tanh q_1\tanh q_2 }. \end{aligned} \] Since \(\tanh q_k=\beta_k\), we again arrive at the independently obtained result.
\[ \tag{29} \boxed{ \beta_{2/1} = \frac{ \beta_2-\beta_1 }{ 1-\beta_1\beta_2 } }. \] In dimensional form, the formula takes the familiar form.
\[ \tag{30} \boxed{ v_{2/1} = \frac{ v_2-v_1 }{ 1-\dfrac{v_1v_2}{c^2} } }. \] Using the initial exponents of the split operators, the result is written directly as
\[ \tag{31} \boxed{ \beta_{2/1} = \frac{ \sin(\pi b_2)-\sin(\pi b_1) }{ 1- \sin(\pi b_1) \sin(\pi b_2) } }. \] The denominator in formulas (29)–(31) was not introduced separately as a correction to the classical difference. It appeared during the inverse mapping of the difference in Doppler coordinates to conventional velocity.
7. Sum of Velocities
Now let us consider two successive external motions in the same direction. Their Doppler scales are multiplied:
\[ \tag{32} \boxed{ D_{1\oplus2} = D_1D_2 }. \] After taking the logarithm, the product becomes the sum:
\[ \tag{33} \begin{aligned} q_{1\oplus2} &= \ln(D_1D_2) \\ &= \ln D_1+\ln D_2 \\ &= q_1+q_2. \end{aligned} \] Converting to velocity yields
\[ \tag{34} \beta_{1\oplus2} = \tanh(q_1+q_2). \] Using the formula for the sum of hyperbolic tangents, we obtain
\[ \tag{35} \boxed{ \beta_{1\oplus2} = \frac{ \beta_1+\beta_2 }{ 1+\beta_1\beta_2 } }. \] In dimensional form:
\[ \tag{36} \boxed{ v_{1\oplus2} = \frac{ v_1+v_2 }{ 1+\dfrac{v_1v_2}{c^2} } }. \] Through parameters \(b_1\) and \(b_2\):
\[ \tag{37} \boxed{ \beta_{1\oplus2} = \frac{ \sin(\pi b_1)+\sin(\pi b_2) }{ 1+ \sin(\pi b_1) \sin(\pi b_2) } }. \] And here, the additional denominator is a consequence of the reverse transition from the linear coordinate \(q_1+q_2\) to the limited velocity \(\tanh(q_1+q_2)\).
8. Unified Composition Rule
The sum and difference can be combined in a single sequence of transformations. Addition is performed for the upper sign, and subtraction for the lower sign:
\[ \tag{38} \boxed{ D_{12}^{(\pm)} = D_2D_1^{\,\pm1} }. \] The logarithmic coordinate makes this operation linear:
\[ \tag{39} \boxed{ q_{12}^{(\pm)} = q_2\pm q_1 }. \] After the inverse transformation, we obtain a single law
\[ \tag{40} \boxed{ \beta_{12}^{(\pm)} = \tanh(q_2\pm q_1) = \frac{ \beta_2\pm\beta_1 }{ 1\pm\beta_1\beta_2 } }. \] Thus, addition and subtraction of velocities are two variants of the same operation on Doppler scales: product and ratio. In the coordinate \(q\), they turn into the usual sum and difference.
9. Return to the external exponent \(b\)
The resulting velocity must be returned to the geometry of the original operator. By definition (6), the composite external exponent is equal to
\[ \tag{41} \boxed{ b_{12}^{(\pm)} = \frac{1}{\pi} \arcsin \left( \beta_{12}^{(\pm)} \right) }. \] Substituting formula (40), we find
\[ \tag{42} \boxed{ b_{12}^{(\pm)} = \frac{1}{\pi} \arcsin \left[ \frac{ \sin(\pi b_2) \pm \sin(\pi b_1) }{ 1 \pm \sin(\pi b_1) \sin(\pi b_2) } \right] }. \] When comparing two states, the common internal factor \(\j^a\) is not involved in calculating the relative external velocity. The normalized external relative operator has the form
\[ \tag{43} J_{\mathrm{ext},12}^{(\pm)} = (-\j)^{b_{12}^{(\pm)}}. \] However, the complete resulting state of the particle must still contain its total internal rotation. Therefore, after calculating the external composition, the factor \(\j^a\) is re-added:
\[ \tag{44} \boxed{ J_{12}^{(\pm)} = \j^a (-\j)^{b_{12}^{(\pm)}} }. \] This distinction eliminates a possible misunderstanding. When calculating the external velocity, the internal part is canceled, but the particle itself does not lose its internal state. Formula (43) describes only the external transformation, while formula (44) describes the reconstructed full state.
In an idempotent basis, the result again has the original structure:
\[ \tag{45} J_{12}^{(\pm)} = \ep e^{i\pi b_{12}^{(\pm)}} + \em e^{i\pi a}. \] Both components remain unit phase factors in their planes. Therefore, the velocity composition does not take the state outside the normalized geometry of the operator.
10. Why the speed of light is not exceeded
The speed limit becomes especially clear in the Doppler coordinate. Any finite \(q\) corresponds to
\[ \tag{46} -1 < \tanh q < 1. \] Since the sum or difference of two finite coordinates is also finite, we have
\[ \tag{47} |q_1| < \infty, \quad |q_2| < \infty \quad\Longrightarrow\quad \left| \beta_{12}^{(\pm)} \right| < 1. \] Hence,
\[ \tag{48} \boxed{ |v_1| < c, \quad |v_2| < c \quad\Longrightarrow\quad \left| v_{12}^{(\pm)} \right| < c }. \] To obtain \(|\beta|=1\), we would need \(|q|\to\infty\), that is, an infinite Doppler scale:
\[ \tag{49} \beta\to1 \quad\Longleftrightarrow\quad D(\beta)\to\infty \quad\Longleftrightarrow\quad q\to\infty. \] Therefore, the speed of light acts not as an artificially established limit of addition, but as the boundary of the inverse mapping \(\beta=\tanh q\).
11. Verification of the obtained law
Small speeds. If \(|\beta_1|\ll1\) and \(|\beta_2|\ll1\), then the product \(\beta_1\beta_2\) can be neglected:
\[ \tag{50} \beta_{12}^{(\pm)} \approx \beta_2\pm\beta_1. \] Thus, classical addition is an approximation of relativistic composition at low velocities.
Identical velocities. If both states move identically, their relative velocity is zero:
\[ \tag{51} \beta_2=\beta_1 \quad\Longrightarrow\quad \beta_{2/1}=0. \] One stationary state. When \(\beta_1=0\), the outer scale of the first state is unity:
\[ \tag{52} D_1=1, \qquad q_1=0, \qquad \beta_{2/1}=\beta_2. \] Opposite motions. From the definition of the Doppler scale, it follows
\[ \tag{53} D(-\beta) = \frac{1}{D(\beta)}, \qquad q(-\beta)=-q(\beta). \] Therefore, a change in the direction of motion automatically replaces addition with subtraction and vice versa.
Associativity of successive composition. For three motions.
\[ \tag{54} q_{1\oplus2\oplus3} = q_1+q_2+q_3. \] The usual associativity of coordinate addition \(q\) means the associativity of the one-dimensional relativistic composition of velocities.
12. Numerical example.
Let two successive motions have the same velocities.
\[ \tag{55} \beta_1=\beta_2=0.6. \] Classical addition would yield \(1.2c\), which exceeds the speed of light. The Doppler composition yields
\[ \tag{56} \begin{aligned} \beta_{1\oplus2} &= \frac{0.6+0.6}{1+0.6\cdot0.6} \\[1mm] &= \frac{1.2}{1.36} \\[1mm] &\approx 0.88235. \end{aligned} \] Therefore, the resulting velocity is approximately \(0.882c\), not \(1.2c\). In this case, the Doppler coordinates actually add up linearly:
\[ \tag{57} q_1=q_2 = \operatorname{artanh}(0.6) \approx 0.69315, \qquad q_{1\oplus2} \approx 1.38629. \] The inverse transformation returns the same result:
\[ \tag{58} \tanh(1.38629) \approx 0.88235. \] This example shows the source of the nonlinearity: the values of \(q\) are added, while the velocity is the limited hyperbolic tangent projection of the resulting sum.
13. What exactly was obtained in this article?
The work uses the well-known relativistic formula for the directional Doppler effect.
\[ \tag{59} D(\beta) = \sqrt{ \frac{1+\beta}{1-\beta} }. \] Therefore, the obtained result should be accurately called the Doppler derivation of the velocity addition law. It reveals the internal mathematical structure of the law and relates it to the frequency of the operator \(J\), but is not an independent derivation of the entire special theory of relativity.
The independence of the proposed interpretation lies elsewhere. The Doppler effect here is applied to the internal rotation of the particle, which is already contained in the exponent \(a\). The same internal frequency of the two states serves as a common reference against which the external Doppler scales are determined.
The complete sequence is of the form
\[ \tag{60} \boxed{ \begin{aligned} J_1,J_2 &\longrightarrow \omega_1,\omega_2 \\ &\longrightarrow D_1,D_2 \\ &\longrightarrow q_1=\ln D_1, \quad q_2=\ln D_2 \\ &\longrightarrow q_{12}^{(\pm)} = q_2\pm q_1 \\ &\longrightarrow \beta_{12}^{(\pm)} = \tanh q_{12}^{(\pm)} \\ &\longrightarrow b_{12}^{(\pm)} = \frac{1}{\pi} \arcsin\beta_{12}^{(\pm)} \\ &\longrightarrow J_{12}^{(\pm)} = \j^a(-\j)^{b_{12}^{(\pm)}}. \end{aligned} } \] Conclusion
In the usual velocity coordinate, the relativistic composition appears nonlinear. In the Doppler coordinate, its structure becomes simple: successive frequency scales are multiplied, their logarithms are added, and the difference in motion is obtained by dividing the scales and subtracting the logarithms.
\[ \tag{61} \boxed{ q_{12}^{(\pm)} = q_2\pm q_1 }. \] The additional denominator of the velocity addition law arises not as an external correction, but when returning from the linear Doppler coordinate to the bounded variable \(\beta\):
\[ \tag{62} \boxed{ \beta_{12}^{(\pm)} = \frac{ \beta_2\pm\beta_1 }{ 1\pm\beta_1\beta_2 } }. \] In the geometry of the operator \(J=\j^a(-\j)^b\), this conclusion receives physical support: the internal rotation frequency \(\omega_0=\pi\dot a\) undergoes the Doppler transform, and the external motion is encoded by the exponent \(b\). Under the condition \(a_1=a_2\), the internal part serves as a common reference and cancels out, so the relative external velocity is determined only by the parameters \(b_1\) and \(b_2\).
The mathematical origin of this operator is discussed in more detail in the article "From Euler's Formula to Split Geometry". The separation of internal and external motions is also used in the papers "Particle Mass as a Geometric Projection" and "Geometric Origin of the Squared Fine Structure Constant and the Parameters of the Bohr Atom".
In the second part, we will consider the general case of \(a_1(t)\ne a_2(t)\). Then, the observed frequency shift contains not only the external relative motion, but also the difference in internal phases, internal frequencies, and their transverse projections. This results in a complete relative operator that simultaneously describes the external velocity and the beating of internal rotations.
Materials used
- Wikipedia. Velocity-addition formula.