How Do You Add Relative Velocities?

Scientific Paper
Title	How Do You Add Relative Velocities?
Read in full	Link to paper
Author(s)	Zbigniew Oziewicz
Keywords	isometry, Einsteinian relative velocity, Minkowskian relative velocity
Published	2004
No. of pages	40

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Abstract

Presented at XXV International Colloquium on Group Theoretical Methods in Physics, Mexico, August 2004. Following Minkowski in 1908, we consider the relative velocity to be the Minkowski space-like vector. We show that the Lorentz boost entails the relative velocity to be ternary: ternary relative velocity is a velocity of a body with respect to interior observer as seen by a preferred exterior observer. The Lorentz boost imply non-associative addition of ternary relative Einsteinian velocities. Within Einstein's special relativity theory, each preferred observer (aether, fixed stars, etc), determine the unique relative velocity among each pair of massive bodies. The special relativity founded on axiom that each pair of reference systems must be related by the Lorentz isometry, needs the preferred reference system in order to have the unique Einstenian relative velocity among each pair of massive bodies. This choice-dependence of relative velocity violate the Relativity Principle that all reference systems must be equivalent.

Overview

This is a mathematical paper — coordinate-free, written in the language of Clifford and Grassmann algebra on Minkowski space — presented at the XXV International Colloquium on Group Theoretical Methods in Physics (Cocoyoc, Mexico, August 2004) and revised in March 2007. Its subject is a question that Oziewicz argues has never been properly answered: what, exactly, is relative velocity, and how do relative velocities compose?

His answer has two parts, and they cut against each other in a way he presents as a genuine dilemma. First, if one accepts the Minkowski axiom that any two reference systems are connected by a Lorentz isometry, then the velocity parameterizing that boost is not a two-argument object at all: it is ternary, depending on a third, preferred observer P in addition to the two bodies whose relative motion is at issue. Different choices of P give different velocities of Bob relative to Alice — which, Oziewicz argues, means Einstein's special relativity silently requires a preferred reference system ("fixed stars, aether, etc.") and therefore violates the Relativity Principle that all reference systems be equivalent. Second, if one insists instead on the Relativity Principle, the relative velocity must be taken as the primitive, choice-free binary notion; but then the resulting transformations are not isometries and do not form the Lorentz group. They form a groupoid, and — the paper's most striking positive result — the addition of binary relative velocities is associative, unlike the Einstein addition, which Ungar showed in 1988 is not.

The departure from the mainstream account is therefore not empirical but structural. Oziewicz does not dispute the Lorentz transformation as a symmetry of the metric of empty spacetime or of Maxwell's equations. He disputes the identification of that symmetry group with the relativity group relating material reference systems, and insists — a distinction he repeats throughout — that "observer-independence, and the Lorentz-invariance, are distinct concepts."

The argument

Reference systems as time-like vector fields

Following Minkowski (1908), a reference system is a normalized future-directed time-like vector field P, with P² = −1. Velocities are space-like Minkowski vectors orthogonal to their observer. This is where Oziewicz breaks with David Hestenes, who in 1974 took the relative velocity to be a Minkowski bivector with magnitude γ_v = −A·B; Oziewicz shows this cannot in general parameterize an arbitrary Lorentz transformation, and coincides with the Einstein definition only when the three-body system is co-planar, P ∧ A ∧ B = 0.

The boost-link problem and ternary velocity

The central technical move is Definition 2.1 together with the "isometry-link" Theorem 2.2. Given three time-like vectors {P, A, B}, the boost equation L_P∧wA = B has a unique space-like solution v = v(P, A, B), and this solution is reciprocal: v(P, A, B) = −v(P, B, A). But it depends on P. "Contrary to popular matrix-statement in many textbooks, a Lorentz boost from A to B is not unique." Oziewicz shows that the familiar 4×4 boost matrix is recovered from his basis-free P-dependent boost only in the special basis where P ≃ (1,0,0,0) — "this explain why someone insists that the 'Lorentz boost is unique!'" Since every space-like vector w admits a two-dimensional submanifold D_w of possible time-like observers, each P in that submanifold generates its own boost L_P∧w.

The conclusion he draws is blunt: within the Minkowski axiom, "to have just one Einstein's velocity of Bob relative to Alice, we need to chose some one and only one reference system to be preferred." He notes that de Abreu and Guerra accept exactly this, abandoning the democratic Relativity Principle in favour of an absolute space — but observes that their absolute velocity is not reciprocal, satisfying v_AG(event, Bob, Aether) = −(γ_v)²v_AG(event, Aether, Bob).

Non-associativity and the Mocanu paradox

Oziewicz then works through the pathologies of the Einstein ⊕-addition. He records the Sommerfeld–Silberstein identity γ_v⊕u = γ_uγ_v(1 + u·v/c²), which shows ⊕ to be an internal operation on the Lobachevsky manifold of velocities, and gives a two-line proof. He then notes two structural defects. The ⊕-inverse coincides with the Galilean inverse, u⁻¹ = −u; combined with non-commutativity this yields the Mocanu paradox, (v ⊕ u)⁻¹ = (v⁻¹) ⊕ (u⁻¹) ≠ (u⁻¹) ⊕ (v⁻¹), "whereas one would expect that the unary inverse operation is an anti-automorphism." And Ungar's 1988 result: for three non-collinear velocities the two bracketings w ⊕ (v ⊕ u) and (w ⊕ v) ⊕ u differ not only in direction — their wedge is a nonzero bivector — but in magnitude, so that "for a system of four or more bodies the ⊕-addition of three non-collinear relative velocities gives the two distinct velocities between two bodies."

Oziewicz rejects the standard rescue that this is merely Thomas rotation. His argument is historical as much as formal: Jackson invoked Thomas precession to supply the factor 2 in spin–orbit doublet separation, but Dirac in 1928 obtained the same factor and the correct spin levels from the Clifford algebra alone, so "no longer did anyone need Thomas's precession except for the non-associative ⊕-addition of velocities." He also notes as a remark on Einstein's 1905 derivation that the reciprocal-velocity property u⁻¹ = −u was "the most important tacit independent assumption," used as an axiom and not derivable from the two stated postulates.

Groupoid relativity

The alternative reverses the order of construction. Instead of relativity transformations ⇒ relative velocities, one axiomatizes the binary relative velocity first and derives the transformations: for A ∈ D_u, the body B = b_uA ≡ γ_u(A + u/c) is the unique body moving with velocity u relative to A, with u/c = B/(−B·A) − A and γ_u = −A·B.

This groupoid boost b_u is not an isometry, and its domain is only a two-dimensional submanifold of observers who can actually measure u, rather than the full four-dimensional module on which a Lorentz boost acts. Because domain and codomain of b_u are disjoint for u ≠ 0, not every pair of morphisms can be composed — the defining feature of a groupoid rather than a group. The inverse velocity is not the Galilean reciprocal: v⁻¹ = −L_P∧vv, so that |v⁻¹| = |v| but v⁻¹·v = −v²/√(1 − v²), and each body "has his own separate zero velocity," v ∘ v⁻¹ ≠ v⁻¹ ∘ v.

The payoff is Section 7. Defining ∘-addition by b_v∘u ≡ b_v ∘ b_u gives γ_v∘u = γ_vγ_u(1 − v·u⁻¹/c²), and the composition is associative — trivially so, Oziewicz notes, because it is composition of morphisms in a category. For collinear motion (equivalently, he proves, for a co-planar three-body system A ∧ B ∧ C = 0) the ∘-addition reduces to the same familiar formula (u + v)/(1 + v·u/c²) as the Einstein addition, so the two theories agree in the one-dimensional case that most experiments probe.

Empirical touchpoints

Oziewicz claims the groupoid theory "predicts the same time-dilation as the relativity-Lorentz-group, however no material rod contraction." For the speed of light he derives c_P/c ≡ L/(−L·P) − P with (c_P/c)² = 1, and stresses that this is stronger than the textbook statement: the magnitude of light speed is independent of every observer, "including all non-inertial, rotating and accelerating observers," while its direction remains observer-dependent, which is what aberration measures. He gives the aberration relation |u|{1 + cosP cosK} = c{cosP + cosK}, and shows that the Doppler formula ν_Q = ν_Pγ_v{1 − (v/c)cosP} is identical in the two theories — so "the Doppler effect does not distinguish the Lorentz group relativity … from groupoid relativity."

Assessment

What is genuinely valuable here is the diagnosis, and it stands independently of whether one accepts the proposed cure. Oziewicz is correct that the Lorentz boost linking two given time-like vectors is not unique unless a further structure is fixed, and correct that the textbook boost matrix conceals this by working in a basis adapted to one particular observer. The distinction he insists on — between the Lorentz group as a symmetry group of the metric and as a putative relativity group of material frames — is a real conceptual distinction that most presentations run together. The non-associativity of Einstein velocity addition is a mathematical fact, not a contested claim, and the observation that Einstein's 1905 derivation quietly assumes reciprocity u⁻¹ = −u as an independent axiom is a fair and rarely-made point. The paper is also unusually honest about where its alternative and the standard theory agree: collinear composition, the Doppler shift, and time dilation are all common ground, and Oziewicz says so.

The difficulties are of two kinds. The first is that the paper's central claim — that ternariness "violates the Relativity Principle" — is an interpretive step, not a theorem, and it is asserted more often than argued. The standard reply, which the paper does not engage in detail, is that the preferred P in Definition 2.1 is not a physical aether but a bookkeeping choice of how to split a general Lorentz transformation into boost-times-rotation; the composite transformation L_P∧v L_P∧u is a Lorentz element regardless of P, and all observers agree on the invariant intervals and on every measurable quantity. That the decomposition is choice-dependent while the group element is not is exactly what "Thomas rotation" names. Oziewicz's rejection of that reading rests largely on the historical argument that Dirac's equation supplied the spin–orbit factor 2 without Thomas precession — but the Dirac equation reproduces the Thomas factor, it does not eliminate the kinematic effect, and Thomas–Wigner rotation is independently measured in spin-precession experiments and is a required correction in accelerator beam dynamics (the Thomas–BMT equation, used routinely in polarized-beam storage rings). A framework that dispenses with it owes an account of those measurements, and none is given.

The second difficulty is that the alternative is presented as structure without dynamics. Groupoid relativity is developed here as kinematics only: there is no field theory, no action, no conserved quantities, and — despite the closing suggestion of "many-body relativistic dynamics without Lorentz/Poincaré invariance" — no demonstration that Maxwell's equations, or any equation of motion, can be written in the groupoid setting. The claim of "no material rod contraction" is stated without derivation and without confrontation with the experiments usually taken to bear on it. The paper's own list of discriminating predictions is deferred to other papers of the author's; within these forty pages, the one asserted difference from the standard theory (rod contraction) is not tested and the three quantities actually computed (Doppler, aberration, light speed) either agree with the standard result or are not compared numerically against it.

Read for what it is — a careful piece of mathematical physics about the algebraic structure of velocity composition — the paper is precise, self-aware and well referenced, and the associativity result for binary velocities is a clean and attractive theorem. Read as a refutation of special relativity, it overstates its reach: it establishes that a certain widely-used parameterization is non-canonical, not that the theory parameterized is wrong.