Summation Law for Velocities

• The summation law for velocities is defined as the rule for combining speeds using non-linear formulas that enforce a universal speed limit.
• It employs Lorentz transformations and hyperbolic geometry to account for relativistic effects, revealing group-theoretic and geometric underpinnings.
• Its extension to vector and quantum domains shows non-commutative, non-associative behavior with applications in statistical and cosmological contexts.

The summation law for velocities specifies how velocity quantities are composed in physical theories, distinguishing between the canonical (arithmetic) sum of Newtonian mechanics and the non-linear rules governing velocity addition in relativistic frameworks. In special relativity and generalizations thereof, this law ensures that the resultant velocity never exceeds a limiting speed (classically, the speed of light $c$), encoding core kinematic and causal restrictions. This law underpins modern understandings of Lorentz transformations, the structure of Minkowski spacetime, and the behavior of velocities in both classical, quantum, cosmological, and stochastic contexts.

1. Historical Derivation and Conceptual Basis

The classical (Newtonian) rule for velocity summation is simply $u_\text{tot} = u_1 + u_2$, valid for all $|u_i|$. However, experimental realization of Maxwell's equations and the invariance of the speed of light $c$ mandated a revision: the velocity addition must not yield speeds exceeding $c$. The Einstein–Poincaré velocity addition law resolves this by introducing the non-linear composition formula: $w = \frac{u + v}{1 + u v / c^2}, \qquad |u|,|v|<c$ This law is derivable directly from the Lorentz transformations, but Granik & Granik (Granik et al., 2012) demonstrated that imposing only a universal speed limit $c_+$ on Newtonian trajectories suffices for its emergence—no relativity postulates are needed. The algebraic structure of this law is rooted in symmetry, causality, and group composition properties (see also (Minni, 6 Apr 2025)), which restrict kinematic transformations to Lorentz-type forms with finite limiting velocity $V_L$.

2. Geometric, Group-Theoretic, and Quantum Optical Interpretations

Relativistic velocity addition possesses rich geometric and group-theoretic underpinnings. In one spatial dimension, Kocik (Kocik, 2014) illustrated its origin via geometric diagrams: representing velocities on the real axis and constructing their sum through chords on the unit circle enforces the speed bound naturally by the geometry. This mirrors the tangent addition formula and, at deeper level, the transformation by hyperbolic rapidities: $$v = c \tanh\phi, \quad \phi_\text{tot} = \phi_1 + \phi_2$$ In higher spatial dimensions, the velocity space forms a hyperbolic manifold (Beltrami–Klein model), with group law corresponding to the Lorentz boost composition. Quantum-optical constructions, such as SU(1,1) squeezed states, physically realize Lorentz boosts and the velocity law in the optical domain (Yang et al., 2011), and the addition of velocities is directly mapped to operator composition, encompassing Wigner/Thomas rotation effects.

3. Algebraic Structure: Vector Generalizations and Non-Associativity

In $3+1$ spacetime, the velocity summation law is vectorial and nontrivially incorporates direction: $$\mathbf{w} = \frac{1}{\,1 + \mathbf{u}\!\cdot\!\mathbf{v}/c^2}\,\left[\mathbf{u} + \frac{1}{\gamma_u}\mathbf{v} + \frac{\gamma_u}{1+\gamma_u}\frac{(\mathbf{u}\!\cdot\!\mathbf{v})}{c^2}\,\mathbf{u}\right], \quad \gamma_u=\frac{1}{\sqrt{1-u^2/c^2}}$$ This law is neither commutative nor associative except in collinear cases. Thomas rotation emerges upon composition of non-collinear boosts, inducing a rotation in the velocity space (see (Giulini, 6 Jan 2025Kocik, 2019)). The algebraic structure is a gyrogroup: a quasigroup with identity and a twisted associativity controlled by gyrations (orthogonal rotations).

Dimension	Velocity Addition Law	Associativity	Commutativity	Thomas Rotation
1D (collinear)	$(u + v)/(1 + uv/c^2)$	Yes	Yes	No
3D (general)	See vector formula above	No	No	Yes (non-collinear)

4. Generalizations: Limiting Velocities and Cosmological Contexts

Ignatowsky-type derivations (Minni, 6 Apr 2025) show that, given only relativity, causality, and space-time symmetries, the admissible kinematic transformations are either Galilean (no speed bound) or Lorentz-type (finite limiting speed $V_L$). The velocity addition law takes the general form: $w = \frac{u + v}{1 + u v / V_L^2}$ Recovering Einstein's result for $V_L = c$, but such a law need not be uniquely tied to electromagnetism—any universal maximal speed suffices.

In Robertson–Walker cosmologies, velocity addition laws must accommodate spacetime expansion and metric structure (Klein et al., 2015). For spectroscopic velocities (those defined by Doppler shifts along lightlike separations), the Einstein summation law persists for comoving observers: $$v_{spec,3} = \frac{v_{spec,1} + v_{spec,2}}{1 + v_{spec,1}v_{spec,2}}$$ For Fermi, kinematic, and astrometric velocities, additional scale-factor corrections, dependent on cosmological time and distance, are necessary; only in the Milne universe does the special relativistic law apply universally in all definitions.

5. Probabilistic, Stochastic, and Statistical Aspects

The composition law for velocities extends into statistical physics and stochastic settings. The stable limit law for the sum of $N$ iid random velocities under the Einstein addition law is found by mapping to rapidities, $\sigma(\beta)=\tanh^{-1}(\beta)$, which add linearly (Souza et al., 2024): $$\mathcal{F}(\alpha,\lambda;\beta) = \gamma^2(\beta)\int_{-\infty}^\infty \frac{dk}{2\pi}e^{-ik\sigma(\beta) - |\lambda k|^\alpha}$$ This distribution is closed under the summation law and demonstrates shifts in modal structure and tail behavior with transition from nonrelativistic ($\alpha=2$) to ultrarelativistic ($\alpha<2$) regimes. Empirical data in heavy-ion diffusion and antiproton cooling confirm the utility of these statistical forms.

6. Extension to Quantum Theory, General Bounded Quantities, and Möbius Automorphisms

The summation law is a particular instance of the broader composition law for bounded quantities, formally represented as a Möbius or disk automorphism (Vigoureux, 6 Jan 2026): $a \oplus z = \frac{a + z}{1 + \overline{a}z}$ Specializing to real velocities within $(-c,c)$, this recovers the Einstein formula. This extension is universal: any physical quantity constrained within a unit disk is composed by this law. In quantum theory, its adoption induces novel interpretations for probability composition and state superposition.

7. Velocity Summation in Nonequilibrium Response Theory

While unrelated to relativistic spacetime, the "sum rule for velocities" in underdamped Langevin systems (Yuge, 2010) is a spectral law for the total weight of velocity response: $$\int_{-\infty}^\infty \mathrm{Re}\,\chi_v(\omega)\,d\omega = \frac{1}{m}$$ In overdamped models, this sum rule breaks down due to an instantaneous noise-induced singularity. Maintaining the inertial term is essential for correct spectral strength and proper energy-dissipation accounting.

• Granik & Granik, "Derivation and New Interpretation of the Lorentz Transformations and Einstein's Theorem of Velocity Addition" (Granik et al., 2012)
• "Limiting velocity and generalized Lorentz transformations" (Minni, 6 Apr 2025)
• Kocik, "Geometric diagram for relativistic addition of velocities" (Kocik, 2014)
• "Demonstrating Additional Law of Relativistic Velocities based on Squeezed Light" (Yang et al., 2011)
• "Velocity Addition/Subtraction in Special Relativity" (Giulini, 6 Jan 2025)
• "Making sense of relativistic composition of velocities" (Kocik, 2019)
• "Velocity addition formulas in Robertson-Walker spacetimes" (Klein et al., 2015)
• "Relativistic Lévy processes" (Souza et al., 2024)
• "Superposition of states in quantum theory" (Vigoureux, 6 Jan 2026)
• "Sum rule for response function in nonequilibrium Langevin systems" (Yuge, 2010)
• "Relativistic relative velocities and relativistic acceleration" (Koczan, 2021)

Summary

The relativistic summation law for velocities provides a mathematically rigorous, group-theoretic, and physically necessary constraint on the composition of motion, generalizing classical kinematics, embedding the geometry of spacetime transformations, and enforcing the limiting speed as a structural and causal constant. Its implications pervade fundamental physics, statistical mechanics, quantum theory, and cosmology, with deep connections to algebraic structures such as gyrogroups and Möbius maps. In stochastic and nonequilibrium contexts, analogous summation laws and sum rules underlie spectral and probabilistic relationships essential for energy transfer, response analysis, and statistical inference.