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Velocity Addition/Subtraction in Special Relativity

Domenico Giulini

TL;DR

The paper develops a comprehensive, geometry-driven treatment of velocity addition/subtraction in Special Relativity, clarifying how Einstein addition arises from polar decompositions of Lorentz boosts and how the Thomas rotation encodes non-commutativity and non-associativity. It introduces a covariant, invariant definition of relative velocity via the boost-link theorem, defining a ternary link-velocity that depends on a reference state but remains Lorentz-equivariant. By contrasting this SR framework with Galilei-Newton spacetime, it highlights how boosts form a non-abelian structure in SR, while Galilean boosts form an abelian subgroup, and it connects polar decomposition to semi-direct-product structures. The work presents explicit, accessible derivations of the boost compositions, Thomas rotation, and the link-velocity, emphasizing geometric intuition, parallel transport on state space, and a clean comparison with Newtonian spacetime. The results deepen understanding of the algebraic and geometric underpinnings of relativistic velocity addition and its dependence on reference frames, with clear implications for relativistic kinematics and its Newtonian limit.

Abstract

We reconsider velocity addition/subtraction in Special Relativity and re-derive its well-known non-commutative and non-associative algebraic properties in a self contained way, including various explicit expressions for the Thomas angle, the derivation of which will be seen to be not as challenging as often suggested. All this is based on the polar-decomposition theorem in the traditional component language, in which Lorentz transformations are ordinary matrices. In the second part of this paper we offer a less familiar alternative geometric view, that leads to an invariant definition of the concept of relative velocity between two states of motion, which is based on the boost-link-theorem, of which we also offer an elementary proof that does not seem to be widely known in the relativity literature. Finally we compare this to the corresponding geometric definitions in Galilei-Newton spacetime, emphasising similarities and differences.

Velocity Addition/Subtraction in Special Relativity

TL;DR

The paper develops a comprehensive, geometry-driven treatment of velocity addition/subtraction in Special Relativity, clarifying how Einstein addition arises from polar decompositions of Lorentz boosts and how the Thomas rotation encodes non-commutativity and non-associativity. It introduces a covariant, invariant definition of relative velocity via the boost-link theorem, defining a ternary link-velocity that depends on a reference state but remains Lorentz-equivariant. By contrasting this SR framework with Galilei-Newton spacetime, it highlights how boosts form a non-abelian structure in SR, while Galilean boosts form an abelian subgroup, and it connects polar decomposition to semi-direct-product structures. The work presents explicit, accessible derivations of the boost compositions, Thomas rotation, and the link-velocity, emphasizing geometric intuition, parallel transport on state space, and a clean comparison with Newtonian spacetime. The results deepen understanding of the algebraic and geometric underpinnings of relativistic velocity addition and its dependence on reference frames, with clear implications for relativistic kinematics and its Newtonian limit.

Abstract

We reconsider velocity addition/subtraction in Special Relativity and re-derive its well-known non-commutative and non-associative algebraic properties in a self contained way, including various explicit expressions for the Thomas angle, the derivation of which will be seen to be not as challenging as often suggested. All this is based on the polar-decomposition theorem in the traditional component language, in which Lorentz transformations are ordinary matrices. In the second part of this paper we offer a less familiar alternative geometric view, that leads to an invariant definition of the concept of relative velocity between two states of motion, which is based on the boost-link-theorem, of which we also offer an elementary proof that does not seem to be widely known in the relativity literature. Finally we compare this to the corresponding geometric definitions in Galilei-Newton spacetime, emphasising similarities and differences.
Paper Structure (27 sections, 14 theorems, 224 equations, 3 figures)

This paper contains 27 sections, 14 theorems, 224 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Velocity addition/subtraction: the traditional story
  3. A reminder on the elementary text-book derivation
  4. Lorentz transformations in the traditional matrix representation
  5. The generalised orthogonal group and its Lie algebra
  6. Restriction to four dimensions and Lorentzian signature
  7. Polar decomposition of boost products
  8. On the magnitude of Thomas rotation
  9. On the algebraic structure of Einstein addition
  10. Velocity subtraction: the modern (geometric) story
  11. States of motion and the non-naturalness of polar decoposition
  12. The kinematical setting
  13. Defining link velocity
  14. Solving the boost-link problem
  15. Base-point dependence of the link-velocity
  16. ...and 12 more sections

Key Result

Theorem 4

Any Lorentz transformation eq:LorentzMatrix-1 that is 'proper', i.e. satisfies $\det(L)=+1$ and 'orthochronous', i.e. $c \geq 1$, can be composed into the product of a rotation and a boost. This decomposition is just the "polar decomposition" with respect to the standard Euclidean metric in $\mathbb We further have $\boldsymbol{\mathbf{\beta}}=\underline{\boldsymbol{\mathbf{D}}}\boldsymbol{\mathbf

Figures (3)

  • Figure 1: Non commutativity of velocity addition due to Thomas rotation. Shown ist the addition of two perpendicular velocities of equal magnitude $\beta=0.8$ so that $\gamma_*^{-1}=0.6$. The Thomas rotation $T(\boldsymbol{\mathbf{\beta}}_1,\boldsymbol{\mathbf{\beta}}_2)$ rotates by an positive angle $\theta$ in the oriented plane spanned by the ordered pair $\{\boldsymbol{\mathbf{a}},\boldsymbol{\mathbf{b}}\}$. Since $\boldsymbol{\mathbf{a}}$ is proportional to $\boldsymbol{\mathbf{\beta}}_2\oplus\boldsymbol{\mathbf{\beta}}_1$ and $\boldsymbol{\mathbf{b}}$ to $\boldsymbol{\mathbf{\beta}}_1\oplus\boldsymbol{\mathbf{\beta}}_2$, this rotation is in the clockwise -- i.e. negative -- orientation with respect to the ordered pair $\{\boldsymbol{\mathbf{\beta}}_1,\boldsymbol{\mathbf{\beta}}_2\}$.
  • Figure 2: Hierarchy of algebraic structures. Einstein addition $\oplus$ endows the open ball $\mathring{B}_1(\mathbb{R}^3)$ with the structure of a loop, which is just short of being a group by its failure to satisfy associativity. (Picture source: https://commons.wikimedia.org/wiki/File:Magma_to_group3.svg. Picture attribution: Tomruen, CC0, via Wikimedia Commons)
  • Figure 3: Graphs of $\gamma(\gamma_*)$ and $\gamma(\varphi)$ showing how $\gamma(s,s_1,s_2)$ decreases for increasing "tilt" of $s$ against $\mathrm{Span}\{s_1,s_2\}$.

Theorems & Definitions (58)

  • Definition 1
  • Definition 2
  • Remark 3
  • Theorem 4
  • proof
  • Definition 5
  • Theorem 6
  • proof
  • Definition 7
  • Definition 8
  • ...and 48 more