Zero Velocity Must be Relative
| Scientific Paper | |
|---|---|
| Title | Zero Velocity Must be Relative |
| Read in full | Link to paper |
| Author(s) | Zbigniew Oziewicz, William S Page |
| Keywords | velocity, relativity theory, reference frame, monad versus tetrad, observer, relativity groupoid |
| Published | 2010 |
| No. of pages | 34 |
Read the full paper here
Abstract
The central concept of the relativity theory is a relative velocity. Relativity of the velocity means that the velocity is not absolute concept. The velocity of positive-mass body is not an intrinsic property of this body; it depends on the free choice of the reference system, it is reference-dependent. We stress that the zero velocity must be relative. Every reference system possess the own zero-velocity relative to exactly this system. Many zeroth-velocities contradict to group structure, and therefore the relativity theory in terms of relative velocities must be formulated within the groupoid structure, which is not a group. There is a dichotomy: two different concepts of a positive mass body, a tetrad versus a monad.
Overview
This draft paper by Zbigniew Oziewicz (Universidad Nacional Autónoma de México) and William S. Page pushes the relativity of velocity to what the authors regard as its logical conclusion. If velocity is never intrinsic to a body but always a function of an ordered pair of bodies — a source-body doing the observing and a target-body observed — then zero velocity cannot be an exception. The velocity of the Sun relative to the Sun and the velocity of a bus relative to that bus are, they insist, different objects, and identifying them is the hidden mistake that forces relativity into a group structure it does not deserve.
Their positive proposal is that the collection of all relative velocities forms an associative groupoid — a category in which every morphism is invertible, with as many identity elements (zero velocities) as there are reference bodies, and in which not every pair of morphisms may be composed. This is offered as a rival to Abraham Ungar's well-known coset loop of Einstein velocity addition, which is a single-identity, non-associative structure. Alongside this runs a second thesis, a "conceptual dichotomy" about what a reference system is: Einstein's coordinate tetrad (four scalar fields on spacetime) versus the Euler-Minkowski monad (a single timelike vector field, the reference fluid). The paper argues these are not equivalent, that they yield different definitions of relative velocity, and hence "distinct relativity theories". Notably, the authors also drop the term "special relativity" — they hold that coordinates are irrelevant to physics — and they deny that the velocity of light is a primary concept of relativity at all, since light is massless and therefore cannot serve as a reference body.
The argument
Velocity as a two-body function
The first axiom is a definitional identification: a material body, an observer, and a reference system are synonymous. Every velocity is then written with its source and target retained, v(street → bus), or vST. From this the authors immediately draw a consequence about composability: the velocity of the Sun relative to the Earth and the velocity of Mars relative to Jupiter simply cannot be added — the sum would be meaningless. Composition of relative velocities is therefore a partial operation. They also object to writing composition with "+", since order matters: vC←B ∘ vB←S = vC←S, read right to left "as in Arabic".
Because a set with a partially defined product and many units is not a group, they reach for the Brandt groupoid (introduced 1926, "the same year quantum mechanics was born"). The contrast with the Lorentz group is made structurally: the Lorentz group is an isometry group whose isometries permute the whole module of vector fields derF, so the group has exactly one object; the relativity groupoid "possesses as many objects as there are massive bodies in mutual motions".
Monad versus tetrad
Einstein in 1905 identified a physical reference system with a coordinate system — scalar fields attached to an observer-body, later called a tetrad or vierbein. Minkowski in 1908 instead defined a reference system as a timelike vector field on spacetime, a construction the authors trace to Euler's 1754 material derivative in fluid mechanics and which was later named a perfect fluid by Eckart (1940) and a monad field by Zel'manov (1976). In adapted coordinates the reference fluid satisfies At ≡ 1 and AxA ≡ 0, i.e. A = (∂/∂t)xA.
The authors argue the two are genuinely inequivalent. A tetrad need not contain a timelike vector at all; it may contain two (Gödel's metric); it may be built from null vectors plus spacelike ones — "thus where is a mass?" A tetrad with one timelike vector plus three spacelike vectors is, on their view, the same body merely rotated, so the spacelike triple is redundant and rotation is better encoded in the covariant derivative of the monad. Crucially, monads exclude lightlike vector fields as reference systems, which is exactly what puts light outside the domain of the non-isometric groupoid. They enlist Léon Brillouin's posthumous Relativity Reexamined (1970) on the same point — "A badly needed distinction between mathematical sets of coordinates and physical frames of reference" — quoting Brillouin's insistence that a frame of reference "is a heavy laboratory, built on a rigid body of tremendous mass", and his question of where the mass is in a coordinate system.
Relativity of space, not of time
A long section argues that Galileo's 1632 insight is really about the relativity of place. If "to be in the same place" requires an arbitrary choice of outside body, then three-dimensional space "is an illusion" — a "ghost-space". Galilean spacetime is a simultaneity bundle over one-dimensional time with no preferred space, and they explicitly correct Trautman: a fibre over a time-moment is a set of simultaneous events, not a set of places. A body's space is not a fibre but a quotient, Space ≡ Space-time / material-body. Each observer-monad V gives a surjection πV from four-dimensional spacetime onto a three-dimensional quotient space of places, and two events are at the same place for that observer iff π maps them to the same point. This is illustrated with a bus and a street: for the driver, departure and near-arrival are the same place; for the crowd on the street, departure and a latecomer's arrival are.
The provocative corollary is that relativity of time is not obligatory. Relative time is metric-dependent proper time only, and absolute simultaneity is "perfectly compatible" with Einstein and Minkowski as one synchronisation convention among many — radio synchronisation, they note, gives a metric-free simultaneity. "The primary concern of relativity theory is the obligatory relativity of space."
The main definition and the groupoid
Working with unit timelike fields (S2 = B2 = −1), the paper recovers the Lorentz gamma factor intrinsically: there is a unique ωSB with S·ωSB = 0 and B = γSB(S + ωSB), whence γSB = −S·B = γBS and γ2 = 1/(1 − ω2), with γSS ≡ 1. Introducing a section sA of the projection πA (so that π ∘ s = id and s ∘ π is idempotent), and pulling back algebras of functions, they define the relative velocity as a vector field tangent to the observer's quotient space:
vAB ≡ s*A ∘ (B / (−A·B)) ∘ π*A ∈ derFA, vAA = 0A.
Because vAA and vBB live in different algebras derFA and derFB, they are not the same zero — which the authors call "the main objective of the present paper". The objects of the groupoid are the unit timelike fields; the morphisms are the relative velocities; each has an inverse, but vSB = (vBS)−1 is emphatically not −vBS, since the two are tangent to different spaces. A composition formula for velocities on quotient spaces is stated, carrying a denominator 1 − s*S(ωBC·ωBS)/c2 of the familiar Einstein-addition form. The conclusion lists three properties: a separate zero per system with vBA ∘ 0B = vBA = 0A ∘ vBA; non-commuting composition; and composability restricted to matching source and target.
Assessment
The paper's genuine contribution is conceptual bookkeeping, and it is a real contribution. Ungar's discovery that Einstein velocity addition is non-associative and needs a gyrogroup was, as the authors fairly say, an odd thing to notice eighty years late; their diagnosis — that the anomaly is an artefact of collapsing all the zero velocities into one, and disappears once velocities are indexed by source and target — is elegant and mathematically well posed. Category-theoretic and groupoid reformulations of kinematics are a legitimate and active area, and the observation that a relative velocity naturally lives in derFA, the derivations of the observer's own quotient-space algebra, is more than notation: it explains why vAB and vBA cannot be related by a simple sign. The monad/tetrad discussion is likewise substantive; the point that a tetrad carries three redundant spacelike legs when only the timelike direction defines the body is well taken, and the Gödel-metric and null-tetrad counterexamples show the two notions really do come apart.
Against this, the article is explicitly a draft and reads like one. Section 8 contains a bare "???" where the source and target maps of the groupoid should be pinned down, and a marginal note "Explain connection" survives in the text near Figure 3. Several equations are cross-referenced to "(??)" — broken references that leave the metric postulates underlying the composition theorem unstated in this document. Key results, including the associative addition on spacetime from which the quotient-space composition theorem is "deduced", are not derived here but referred to Oziewicz 2005 and 2007. The scalar a(v) introduced in equation (54) is left undetermined — "it is not obvious that the scalar a must be obligatorily a ≡ 1" — which is precisely the coefficient that would decide whether the resulting transformation is the Lorentz one; leaving it open leaves the physical content unfixed.
More substantively, the paper's claims about what is and is not obligatory in relativity are stated rather than argued against measurement. That light "must not be considered a primary concept of relativity theory" is defensible as a foundational preference — relativity can be derived from isotropy and the group property without the light postulate — but the authors do not engage the fact that the numerical value entering γ is fixed empirically by light propagation, nor with the Michelson-Morley and Kennedy-Thorndike class of experiments that fix it. Similarly, the suggestion that absolute simultaneity is merely one convention understates the constraint: conventionality of one-way synchronisation is standard, but the two-way invariance and the observed transverse Doppler and time-dilation results (muon lifetimes, Ives-Stilwell, Hafele-Keating) restrict which conventions are consistent, and the paper does not say how a "radio-simultaneity" absolute convention would accommodate them. Finally, and most importantly for a physics readership, no new empirical prediction is offered. The groupoid formalism is presented as a better description of the same phenomena — the authors are careful to say their definition "does not depend on the existence or absence of a privileged reference system or æther" — so on its own terms this is a foundations-and-formalism paper, and should be judged as one. Judged that way it is largely sound; judged as physics it is, as yet, untestable.