View source for Ternary Relative Velocity

{{Infobox paper
| title = Ternary Relative Velocity
| url = [http://www.naturalphilosophy.org/pdf/abstracts/abstracts_133.pdf Link to paper]
| author = [[Zbigniew Oziewicz]]
| keywords = [[Special relativity]], [[groupoid category]], [[isometry]], [[Minkowski geometry]], [[Non-Euclidean geometry]], [[Lorentz metric]], [[ternary relative velocity]], [[binary relative velocity]]
| published = 2007
| num_pages = 19
}}

'''Read the full paper''' [http://www.naturalphilosophy.org/pdf/abstracts/abstracts_133.pdf here]

==Abstract==

It is proved that the Lorentz boost entails the relative velocity to be ternary: the ternary relative velocity is a velocity of a body with respect to an interior observer, as seen by a preferred exterior-observer. The Lorentz-boost imply non-associative addition of ternary relative velocities. Within Einstein's special relativity theory, each preferred observer (fixed stars, aether, etc), determine the unique relative velocity among each pair of massive bodies. Therefore, the special relativity founded on axiom, that each pair of reference systems must be related by Lorentz isometry, needs a preferred reference system in order to have the unique Einstein's 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. This astonishing conflict of the Lorentz relativity group, with the Relativity Principle, can be resolved in two alternative ways. Either, abandon the Relativity Principle in favor of a preferred reference system. Or, within the Relativity Principle, replace the Lorentz relativity group by the relativity groupoid, with the choice-free binary relative velocities (not parametrizing isometry). The axiomatic definition of the kinematical unique binary relative velocity as the choice-free Minkowski space-like vector, leads to the groupoid structure of the set of all deduced relativity transformations (instead of the Lorentz relativity group), with the associative addition of binary relative velocities. Observer-independence and the Lorentz-group-invariance are distinct concepts. This suggest the possibility of formulating many-body relativistic dynamics without Lorentz/Poincare invariance.

==Overview==

Presented at the 2007 Physical Interpretation of Relativity Theory conference in Moscow, this is a mathematician's attack on relativity from an angle almost nobody else in the dissident literature uses. Oziewicz does not dispute the Lorentz transformation, the constancy of ''c'', or any experiment. He disputes a definition — the definition of ''relative velocity'' — and argues that the standard one is not well posed. His claim is that if relative velocity is defined by the Lorentz boost, then it is not a function of two bodies but of three: Alice, Bob, and a third, preferred observer ''P'' with respect to whom the boost is taken. He calls this the ''ternary'' relative velocity.

If that is right, the consequence is severe for the received view. To have exactly one velocity of Bob relative to Alice, one must single out one reference system as preferred — "the concept of the preferred reference system, an [[Aether]], is built in the Einstein velocities" — and that singling-out is exactly what the Relativity Principle forbids. Oziewicz states the resulting choice bluntly: either abandon the Relativity Principle in favour of a preferred frame, or abandon the Lorentz relativity ''group'' in favour of what he calls a relativity ''groupoid'', built on a choice-free ''binary'' relative velocity. He develops the second alternative and notes that it makes a testable difference.

The paper is written "in the ancient Aristotelian spirit," taking relative velocity as the primary concept and treating time and space as its attributes, in deliberate contrast to what Oziewicz calls the "sacred" Felix Klein view in which the group must come first.

==The argument==

===Binary relative velocity===

Following Minkowski (1908), a reference system is identified with a normalised time-like vector field, ''P''<sup>2</sup> = &minus;1. For a non-zero space-like Minkowski vector ''w'', Definition 1.1 attaches two hyperboloids: the potential ''observers'' of ''w'', ''O''<sub>''w''</sub> = {''A''<sup>2</sup> = &minus;1, ''A''·''w'' = 0}, and the potential ''possessors'' of ''w'', ''S''<sub>''w''</sub> = {''B''<sup>2</sup> = &minus;1, ''B''·''w'' = ''w''<sup>2</sup>}. The Heaviside–FitzGerald–Lorentz factor enters through ''w'' = γ<sub>''v''</sub>''v''/''c''.

Axiom 1.2 then defines the ''binary'' relative velocity: given ''A'' ∈ ''O''<sub>''u''</sub> with ''u''<sup>2</sup> < ''c''<sup>2</sup>, there exists exactly one massive body ''B'' moving with velocity ''u'' relative to ''A'', and

: ''u''/''c'' = ϖ(''A'', ''B'')/''c'' = ''B''/(&minus;''B''·''A'') &minus; ''A'',  with γ<sub>''u''</sub> = &minus;''A''·''B''.

This velocity is a function of the ordered pair (''A'', ''B'') alone — "choice-free," in Oziewicz's phrase — and is a Minkowski space-like vector, not the bivector of the Hestenes theory.

===Ternary relative velocity===

Minkowski's definition-axiom of special relativity (Definition 2.1) says only that any two reference systems must be connected by a Lorentz isometry. Oziewicz observes that this axiom never mentions relative velocity, and asks what velocity it actually determines.

Each Minkowski bivector ''P'' ∧ ''Q'' generates an isometry ''L''<sub>''P''∧''Q''</sub> ∈ ''O''(1,3). Definition 2.2 then defines the Einstein, isometric or ternary velocity ''v'' as the space-like vector for which the boost ''L''<sub>''P''∧''v''</sub> carries ''A'' into ''B''. The isometry-link theorem (Theorem 2.3, Oziewicz 2005, 2006) states that for three time-like vectors {''P'', ''A'', ''B''} the equation ''L''<sub>''P''∧''w''</sub>''A'' = ''B'' has a unique solution ''w'' = ''v''(''P'', ''A'', ''B''), given in closed form and looking "like a kind of subtraction of absolute/binary velocities":

: ''v''(''P'', ''A'', ''B'') = ''P''·(''A'' + ''B'')[(''P''·''B'')ϖ(''P'', ''B'') &minus; (''P''·''A'')ϖ(''P'', ''A'')] / [(''P''·''A'')<sup>2</sup> + (''P''·''B'')<sup>2</sup> &minus; 1 &minus; ''A''·''B''].

Two corollaries sharpen the point. When the three bodies are co-planar, ''P'' ∧ ''A'' ∧ ''B'' = 0, the expression reduces to the familiar velocity-difference formula [ϖ(''P'',''B'') &minus; ϖ(''P'',''A'')] / [1 &minus; ϖ(''P'',''B'')·ϖ(''P'',''A'')/''c''<sup>2</sup>] of de Abreu and Guerra. And Theorem 2.5 states that the scalar magnitudes of the binary and ternary velocities coincide ''if and only if'' the system is co-planar — so the difference between the two concepts is invisible in the textbook one-dimensional case and appears only for three genuinely non-coplanar bodies.

Oziewicz anticipates the obvious objection: the textbook Lorentz boost is a fixed matrix parameterised by one velocity, so how can it be non-unique? His answer is that the matrix form presupposes a particular basis, ''P'' ≃ (1,0,0,0), and that his basis-free boost reduces to it in that basis. "This explain why someone insists that the 'Lorentz boost is unique!'" Without a chosen ''P'' there is a whole bunch of Lorentz links from ''A'' to ''B'', generated by different bivectors, "but there is no relative velocity among massive bodies."

His Main Conclusion (2.6) is conditional and clearly stated: ''if'' there must be one and only one relative velocity between each pair of massive bodies, ''and'' each pair must be related by an isometry, ''then'' a preferred reference system must be chosen.

===The groupoid alternative===

The alternative keeps the Relativity Principle and drops the group. Take the binary relative velocity as primitive and derive the transformations from it; the resulting set of transformations forms a ''groupoid category'' — a category in which every morphism has a two-sided inverse, a group being the special case with a single object. Oziewicz argues that a choice-free binary velocity cannot parameterise a Lorentz isometry, one reason being that its domain is restricted to the two-dimensional submanifold ''O''<sub>''w''</sub> → ''S''<sub>''w''</sub>. Addition of binary velocities is associative (Table 2); addition of ternary velocities is not.

Crucially, this is not presented as a purely formal alternative. Oziewicz states two differences of prediction. The groupoid theory "predicts exactly the same time-dilation as the relativity-Lorentz-group, however no material rod contraction." And in moving-media electrodynamics, for ''u'' × ''B'' = 0, the combination ''E''&prime;·''E'' &minus; γ''E''<sup>2</sup> equals &minus;[γ<sup>2</sup>/(γ+1)](''u''·''E''/''c'')<sup>2</sup> in Lorentz-group relativity but exactly zero in groupoid relativity — an experimental test he refers to proposals by Gladyshev and colleagues.

===Non-associativity, the Mocanu paradox, and Thomas rotation===

Section 5 collects the peculiarities of ternary addition. The ⊕-inverse is the reciprocal velocity ''u''<sup>&minus;1</sup> = &minus;''u'', as with absolute time. This produces the Mocanu paradox (1986): the inverse is an ⊕-automorphism, (''v'' ⊕ ''u'')<sup>&minus;1</sup> = ''v''<sup>&minus;1</sup> ⊕ ''u''<sup>&minus;1</sup>, whereas one expects a unary inverse to be an ''anti''-automorphism, (''f'' ∘ ''g'')<sup>&minus;1</sup> = ''g''<sup>&minus;1</sup> ∘ ''f''<sup>&minus;1</sup>. And Ungar's 1988 result gives non-associativity: ''w'' ⊕ (''v'' ⊕ ''u'') and (''w'' ⊕ ''v'') ⊕ ''u'' are neither collinear nor equal in magnitude, so for four or more bodies the addition of three non-collinear relative velocities yields two distinct velocities between the same pair.

Oziewicz rejects the standard reading of this as Thomas rotation. His argument is historical as well as formal: Jackson invoked the Thomas precession to explain the factor 2 in spin–orbit doublet separation, but [[Paul Dirac]] in 1928 obtained the same factor and the correct spin levels from the Clifford algebra and the Dirac equation without it, so "no longer did anyone need Thomas's precession except for the non-associative ⊕-addition of velocities." He also notes, in the Final Remarks, that Einstein's 1905 derivation of the Lorentz group tacitly used the reciprocal-velocity axiom as an independent assumption, one "not related to the verbal Einstein's two postulates."

The paper closes with a chain the author plainly regards as the heart of the matter: no absolute space ⇒ no Einstein relative velocities ⇒ no Einstein special relativity ⇒ no asymmetric biological ageing of twins.

==Assessment==

This is among the most technically serious anti-relativity arguments on this wiki, and it deserves to be judged on its mathematics. The isometry-link theorem is a genuine result: given three time-like vectors, the boost carrying ''A'' to ''B'' really does depend on which bivector plane is used, and the closed form (2.2) really does reduce to the textbook expression only in the co-planar case. The observation that the "unique" boost matrix is unique only relative to a chosen basis is correct and is the kind of thing a physicist trained on matrices can easily miss. Oziewicz is also scrupulous about the logical status of his conclusion: it is stated as an implication from two premises he lists, and he explicitly invites readers to prefer a different definition of relative velocity — "Reader do not need to like the definition of a ternary relative velocity." That is a rarer intellectual manner than the subject usually produces.

The difficulties are these. First, the whole argument turns on a premise that is doing more work than it appears to: that there ''must'' be one and only one relative velocity between each pair of bodies, understood as a vector transforming in a particular way. The standard reply is that the physically meaningful quantity is the invariant γ = &minus;''A''·''B'', which is a function of ''A'' and ''B'' alone and is fully choice-free, while the ''direction'' assigned to a relative velocity is basis-dependent for the same reason that "north" is — a fact usually filed under Wigner rotation rather than under a violation of the Relativity Principle. Oziewicz does not engage this reply directly; he dismisses the Thomas-rotation reading in a paragraph, and his Dirac-equation argument establishes only that the spin–orbit factor 2 has another derivation, not that the kinematic rotation is absent.

Second, the paper announces experimental consequences without pursuing them. The claim that groupoid relativity predicts time dilation but "no material rod contraction" is stated in one sentence and never derived, and no comparison is offered with the measurements usually taken to bear on contraction — the Michelson–Morley null result and its modern optical-cavity successors, or the observed lifetimes and transverse energy distributions of relativistic beams. Likewise the moving-media prediction (4.2) is given, but no experiment is analysed and the Gladyshev proposals are cited rather than evaluated. A prediction referenced is not a prediction tested.

Third, the closing chain — no absolute space, therefore no asymmetric ageing of twins — is rhetoric, not derivation. Differential ageing on the two arms of a twin circuit follows from the proper-time integral along each world line, an invariant quantity that does not require any preferred frame; nothing earlier in the paper touches it. Given that the muon-lifetime and atomic-clock-transport measurements are exactly of this kind, the last line asserts more than the preceding sixteen pages establish.

What remains after those deductions is still substantial: a precise demonstration that "relative velocity," as parameterised by a Lorentz boost, is a three-argument function, and a coherent categorical alternative in which it is a two-argument one. Whether nature prefers the groupoid is left, appropriately, as an experimental question.

==See also==

* [[Zbigniew Oziewicz]]
* [[Special Relativity]]
* [[Lorentz Transformation]]
* [[Hermann Minkowski]]
* [[Aether]]
* [[Twin Paradox]]
* [[Herbert Dingle]]
* [[Rodrigo de Abreu]]
* [[Vasco Guerra]]
* [[Simultaneity]]
* [[Speed of Light]]
* [[Length Contraction]]
* [[Time Dilation]]
* [[Paul Dirac]]

[[Category:Scientific Paper|ternary relative velocity]]

[[Category:Relativity|ternary relative velocity]]

[[Category:Aether|ternary relative velocity]]

[[Category:Philosophy of Science|ternary relative velocity]]