View source for Relativity Groupoid Instead of Relativity Group

{{Infobox paper
| title = Relativity Groupoid Instead of Relativity Group
| url = [http://www.naturalphilosophy.org/pdf/abstracts/abstracts_140.pdf Link to paper]
| author = [[Zbigniew Oziewicz]]
| keywords = [[associative addition of binary relative velocities]], [[groupoid category]]
| published = 2007
| volume = 4
| number = 5
| num_pages = 11
| pages = 739-749
}}

'''Read the full paper''' [http://www.naturalphilosophy.org/pdf/abstracts/abstracts_140.pdf here]

==Abstract==

<em>International Journal of Geometric Methods in Modern Physics</em>, V4, N5 (2007) 739-749. The Lorentz covariance and invariance are acepted to be the cornerstone of the physical theory. Observer-dependence within the relativity groupoid, and the Lorentz-covariance withinh the Lorentz relativity group, are different concepts. Laws of Physics could be observer-free, rather than to be Lorentz-invariant. In 1908 Minkowski introduced space-like binary velocity-field of a medium, relative to an observer. Hestenes in 1974 introduced a relative velocity as a Minkowski bivector. Here we propose binary relative velocity as a traceless nilpotent endomorphism in a operator algebra. Each concept of a binary relative velocity made possible the replacement of the Lorentz relativity group by the relativity groupoid. The relativity groupoid is a category of massive bodies in mutual relative motions, where a binary relative velocity is interpreted as a categorical morphism with the associative addition. This associative addition is to be contrasted with non-associative addition of ternary relative velocities in an isometric special relativity. We consider an algebra of many time-plus-space splits, as an operator algebra generated by observers-idempotents. The Lorentz covariance and invariance are acepted to be the cornerstone of the physical theory. Observer-dependence within the relativity groupoid, and the Lorentz-covariance withinh the Lorentz relativity group, are different concepts. Laws of Physics could be observer-free, rather than to be Lorentz-invariant. In 1908 Minkowski introduced space-like binary velocity-field of a medium, relative to an observer. Hestenes in 1974 introduced a relative velocity as a Minkowski bivector. Here we propose binary relative velocity as a traceless nilpotent endomorphism in a operator algebra. Each concept of a binary relative velocity made possible the replacement of the Lorentz relativity group by the relativity groupoid. The relativity groupoid is a category of massive bodies in mutual relative motions, where a binary relative velocity is interpreted as a categorical morphism with the associative addition. This associative addition is to be contrasted with non-associative addition of ternary relative velocities in an isometric special relativity. We consider an algebra of many time-plus-space splits, as an operator algebra generated by observers-idempotents.

==Overview==

This is a mathematical-physics paper, published in the ''International Journal of Geometric Methods in Modern Physics'', proposing that the algebraic home of relativity is not a ''group'' but a ''groupoid'' — a category in which the objects are massive bodies and the arrows are relative velocities. The motivation is a point about the logical order of concepts: in Newton's account, space and time come first and velocity is derived from them, whereas Oziewicz follows Galileo in taking ''relative motion between massive bodies'' as the primitive, with space and time derived. "Galilean space has no reality without the bodies that 'it contains'," he writes; the paper's opening section sets this reading of Galileo against the Cartesian and Newtonian picture of space as an absolute substratum.

The technical claim that carries the argument is a distinction between two kinds of relative velocity. In [[Special Relativity]] as normally formulated, the relative velocity of two bodies is the parameter of a Lorentz boost — and because the boost group is defined relative to a chosen frame, this makes velocity a ''ternary'' relation: the velocity of body ''q'' with respect to body ''p'' as seen by a third, preferred, exterior observer. Oziewicz instead adopts a ''binary'' relative velocity, defined directly between an ordered pair of bodies and independent of any third. Minkowski used such an object in 1908 and Hestenes reintroduced it in 1974 as a Minkowski bivector; the paper's contribution is to represent it as a traceless nilpotent operator, and to show that with this representation velocity composition becomes associative — in explicit contrast to the well-known non-associativity of Einstein's ternary velocity addition. The concluding provocation is that "Laws of Physics could be observer-free, rather than Lorentz-invariant."

==The argument==

===Observers as idempotents===

The construction begins by identifying each massive body ''p'' with an idempotent operator, ''p''<sup>2</sup> = ''p''. The motivation is the time-plus-space split: an idempotent acting on the module of vector fields decomposes it as (ker ''p'') ⊕ (im ''p'') = (space) ⊕ (time). Choosing one body — the Earth, say — as the reference system therefore requires no coordinates and no basis; Oziewicz stresses that "no measuring devices, rods and clocks are involved." An observer in this sense is a rank-one projector, and he notes that seen as an operator it "looks like a pure state in quantum mechanics."

Where a metric ''g'' is available, an observer is required to be metric-compatible, ''p''* ∘ ''g'' = ''g'' ∘ ''p'', which forces the form ''p'' = ''P'' ⊗ (''gP'') / ''g''(''P'',''P'') for a timelike vector field ''P''. Oziewicz remarks that this compatibility "could be tested experimentally rather than postulated ''a priori''", and that such an observer-idempotent is essentially the energy-momentum endomorphism of a non-viscous relativistic fluid in Tolman's sense. The simultaneity of the observer is then the Einstein–Minkowski proper-time one-form ''gP''.

===Velocity as a morphism===

The relativity groupoid ϖ has massive bodies as objects and binary relative velocities as arrows, one arrow for each ordered pair — a ''pair groupoid''. Every body carries its own identity arrow, its zero velocity relative to itself, and Oziewicz insists these are not interchangeable: "the zero velocity of the Earth relative to Earth must not be identified with the zero velocity of the Sun relative to the Sun." This is precisely why a groupoid rather than a group is needed: a group has a single universal neutral element, and velocity addition is in any case a ''partial'' operation, since not every pair of relative velocities is composable.

The arrow from ''p'' to ''q'' is defined by ϖ(''p'',''q'') = ''qp''/tr(''qp'') − ''p'', accompanied by trace axioms including tr(''pq'') = tr(''qp'') ≥ 1 and tr(''pqr'') = tr(''qpr''). Two properties are asserted for it: it is nilpotent, ϖ(''p'',''q'')<sup>2</sup> = 0, and traceless. A separate postulate supplies the magnitude,

: (|ϖ(''p'',''q'')| / ''c'')<sup>2</sup> = 1 − 1/tr(''pq'')

so the trace functional carries the metric information. Because the groupoid admits accelerated as well as inertial bodies, Oziewicz notes that it "goes beyond boundary of the special relativity" — there is no need to treat constant and variable velocities separately.

===The algebra of observers===

Formal linear combinations of objects and arrows are postulated to form an associative algebra Obs(ϖ), and from the trace relations Oziewicz deduces a full multiplication table: object times object, object times arrow, arrow times object, and arrow times arrow (his equations 9–12). Associativity of this algebra is then the engine of the main theorem. Writing ''u'' = ϖ(''p'',''q'') and ''v'' = ϖ(''q'',''r''), the identity ''r''(''qp'') = (''rq'')''p'' is expanded both ways, and comparing the two expansions yields the composition law

: (1 − ''v''·''u''<sup>−1</sup>/''c''<sup>2</sup>)(''v'' ∘ ''u'') = ''u'' + (1 − ''u''<sup>2</sup>/''c''<sup>2</sup>)''vp'' + (1/''c'')(''v''·''u''<sup>−1</sup>)''p''.

The formula is explicitly ''p''-dependent — it depends on the observer who is the source of the first arrow — but the composition it defines is associative, as follows from considering four bodies, ''p''(''qrs'') = (''pqr'')''s''. The inverse arrow ϖ(''q'',''p'') = ϖ(''p'',''q'')<sup>−1</sup> gives the velocity of ''p'' relative to ''q'', with the two composites ϖ ∘ ϖ<sup>−1</sup> landing on the two different identities 0<sub>''q''</sub> and 0<sub>''p''</sub> — again the groupoid, not group, structure.

===The Galilean limit===

Two bodies share the same simultaneity if and only if ''pq'' = ''p'', equivalently tr(''pq'') = 1, and in the Galilean algebra every trace of every string of objects is 1. The relations collapse to ϖ(''p'',''q'') = ''q'' − ''p'', velocity becomes an exactly skew-symmetric function of its two arguments, and composition reduces to plain vector addition, (''q'' − ''p'') + (''r'' − ''q'') = ''r'' − ''p''. Oziewicz observes that this is the ''c'' → ∞ limit of the relativistic law, since ''vp'' = ''v'' in that regime.

He closes by noting that kinematics of the relativity groupoid is governed by an associative Frobenius operator algebra whereas its dynamics would require the non-associative Frölicher–Richardson algebra, and that electromagnetics of moving bodies within the groupoid differs from the isometric formulation — a topic deferred.

==Assessment==

Checked against its own axioms, the paper's algebra holds up. The two properties asserted of ϖ(''p'',''q'') = ''qp''/tr(''qp'') − ''p'' can be verified directly from the postulates given: with ''A'' = ''qp''/tr(''qp''), the trace relations force tr(''qpqp'') = tr(''qp'')<sup>2</sup> and tr(''pqp'') = tr(''qp''), from which ''A''<sup>2</sup> = ''A'' and ''pA'' = ''p'' follow, and hence (''A'' − ''p'')<sup>2</sup> = ''p'' − ''pA'' = 0. Tracelessness is immediate, tr(''A'') − tr(''p'') = 1 − 1 = 0. The magnitude postulate is also correctly calibrated: setting tr(''pq'') = γ turns it into ''v''<sup>2</sup>/''c''<sup>2</sup> = 1 − 1/γ<sup>2</sup>, which is the standard relation, so the trace functional really is the Lorentz factor and not a free parameter. The metric-compatible observer ''p'' = ''P'' ⊗ ''gP''/''g''(''P'',''P'') genuinely satisfies ''p''<sup>2</sup> = ''p'' and tr ''p'' = 1 as claimed, and the ''c'' → ∞ limit of the composition law does reduce to ''v'' ∘ ''u'' = ''u'' + ''v'' once ''vp'' = ''v'' is imposed. The arithmetic is correct, and the paper's background claim about Einstein's addition — that composing ternary relative velocities is non-associative — is the standard gyrogroup result of Ungar, correctly cited.

What is genuinely attractive here is the diagnosis rather than any new prediction. Oziewicz has put a finger on a real awkwardness: a group has one identity, but there is no single universal zero velocity, and velocity composition is only partially defined. A groupoid is the natural structure for exactly that situation, and the observation that the usual relative velocity is covertly a three-body notion is a sharp one. His identification of an observer with a rank-one projector rather than a coordinate frame is also a clean way of saying that a reference body, not a coordinate system, is what physics actually uses.

The difficulties are of interpretation rather than of algebra. The headline contrast — associative composition here, non-associative there — is less decisive than it is made to sound. Composition of morphisms in ''any'' category is associative by axiom, so once one has decided to model velocities as arrows in a groupoid the associativity is not a discovery but a definition; the content lies entirely in the explicit formula (18) being well defined and reducing correctly. And the non-associativity of Einstein addition is not an inconsistency in relativity: composition of Lorentz transformations is perfectly associative, and the non-associativity appears only in the ''parameterization'' by velocity vectors, because a boost followed by a boost is a boost composed with a Thomas rotation. The gyrogroup formalism handles this without difficulty and correctly predicts Thomas precession, which is measured — most familiarly in the spin–orbit fine structure of atomic spectra and in the geodetic precession measured by Gravity Probe B. Oziewicz's binary velocity buys associativity by declining to parameterize the boost, and it is not shown here that anything measured by that rotation is thereby better accounted for.

There is also a residual dependence of the kind the paper set out to remove. Equation (18) is explicitly observer-dependent — the composed velocity depends on ''p'', the source of the first arrow. That is a weaker dependence than the ternary case, since ''p'' is one of the bodies rather than an exterior fourth party, but "binary" is doing some work in the abstract that the formula does not quite deliver on its own terms. Finally, the paper is a programme rather than a completed theory: dynamics is signposted as requiring a different, non-associative algebra and is not carried out; electromagnetics within the groupoid is declared to differ from the standard formulation but is explicitly placed "beyond the scope of this note"; and the one point at which the framework is said to touch experiment — whether observers are metric-compatible — is raised without any indication of what such a test would look like. What is offered is a reformulation whose empirical consequences remain, on its own account, unwritten.

==See also==

* [[Zbigniew Oziewicz]]
* [[Hermann Minkowski]]
* [[Lorentz Transformation]]
* [[Simultaneity]]
* [[Speed of Light]]
* [[Albert Einstein]]
* [[Isaac Newton]]

[[Category:Scientific Paper|relativity groupoid instead relativity group]]

[[Category:Relativity|relativity groupoid instead relativity group]]