Remarks on the Foundation of Special Relativity

Scientific Paper
Title	Remarks on the Foundation of Special Relativity
Read in full	Link to paper
Author(s)	Mario Ludovico
Keywords	special relativity, Lorentz Transformation, Aether, mass-energy equivalence, superluminal motion
Published	2006
No. of pages	25

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Abstract

At variance with a largely shared opinion, both the foundation and the logical structure of Special Relativity (SR) have substantially been laid by Hendrik Lorentz and by Henri Poincaré, not by Albert Einstein. Yet, the mathematical generalization of SR comes from Hermann Minkowski, who in 1907 proposed the spacetime reference frame in its current notation, though the first mathematical formulation and use of a spacetime reference frame was clearly made by Poincaré in June 1905. ("Spacetime" is also referred to as "chronotope").

In this paper, which forms a "Special Appendix" to the book "Vacuum, Vortices and Gravitation" (fully and freely readable online at www.mario-ludovico.com), questionable points of Einstein's special relativity are given evidence. In particular, the well-known mass-energy equivalence equation is discussed in the light of Lorentz's theoretical analysis concerning the motion of a material body with respect to the ether. It is in fact remarked that the mass-energy equivalence equation is not an achievement of Einstein's special relativity. That equation is intrinsically inherent in the Lorentz's definition of "transverse mass", when the body's relative speed with respect to the ether is nil.

Perhaps, in a view to attaining - by his own - the "equivalence" relationship between mass and energy previously and differently formulated by Poincaré, Einstein published in September 1905 a very short paper, in which - starting from his precedent paper on special relativity - he "proves" the equation E = mc² through the introduction of an unexpected simplification-approximation of the Lorentz's factor 1/(1-v²/c²)^1/2, which Einsteins equals to 1+v²/2c² cutting the relevant series at the second order term. Should one consider such a formal expedient as logically acceptable and appliable to all special relativity equations, the theory would take (particularly from the experimental standpoint) a "physical" significance remarkably different from that conventionally celebrated.

Accounting for the possible existence of the "ether" (or "plenum" in the author's terminology) the same mass-energy equivalence can be obtained analytically, with no use of relativistic paradigms. It is also observed that the introduction of Minkowski's "chronotope" has actually involved the mass-energy equivalence as an axiom proper to the spacetime paradigm.

Overview

This 25-page essay, dated December 2006 and revised March 2007, is the Special Appendix to Mario Ludovico's book Vacuum, Vortices and Gravitation, and it should be read as part of that larger project: Ludovico works throughout with a physical "plenum" — his preferred term for the ether — out of which matter is built as ring-vortices. The appendix does not attempt to refute the Lorentz transformations as such. Its target is narrower and more historical: the claim that Special Relativity as Einstein formulated it in 1905 is a distinct and necessary theory, rather than a re-presentation of results already obtained by Lorentz and Poincaré within an ether framework, plus two postulates that make the framework unusable.

Ludovico's central positive claim is that the mass–energy relation E = mc² and the Lorentz factor cannot both be achievements of the same theory. He argues that Einstein's September 1905 derivation of the mass–energy equivalence works only because Einstein truncates the binomial series for the Lorentz factor at second order, replacing 1/(1 − β²)^1/2 with 1 + β²/2 — and that if this replacement is legitimate, it must be legitimate elsewhere, in which case Einstein's relativistic kinetic energy collapses to the classical E_k = m_oV²/2. He then shows that an equivalent result can be reached analytically from an ether/plenum picture with no relativistic apparatus at all, by treating the "rest state" of a particle as its oscillation about a fixed point of the plenum. Alongside this he argues that the clock-rate effects taken as confirmations of special relativity follow from the mechanics of clocks under acceleration, and that observed superluminal motion in galactic jets is real rather than apparent.

The argument

Priority: Lorentz, Poincaré and Whittaker

The essay opens with the attribution question. Ludovico quotes Hermann Minkowski's 1907 spacetime paper as the mathematical generalisation of SR while insisting that the first mathematical use of a spacetime frame was Poincaré's Sur la dynamique de l'électron of June 1905. He quotes at length a passage he attributes to Hermann Weyl noting that "almost every idea and formula of the theory had been anticipated by others" — Voigt deriving the transformations formally in 1887 from the wave equation, FitzGerald, Larmor and Lorentz arriving at them by the 1890s including time dilation and length contraction, Poincaré articulating the relativity principle, denying an empirical basis for absolute simultaneity, challenging the ontological significance of the ether and showing the transformations form a group. He notes that Einstein's 1905 paper contains no mention of spacetime, and cites Edmund Whittaker's chapter "The Relativity of Lorentz and Poincaré", which credits the two of them and attributes even E = mc² to Poincaré's 1900 result that the energy of an electromagnetic wave behaves like a fluid of mass density E/c².

Clocks, yard-sticks and an internal inconsistency

Ludovico reconstructs Lorentz's synchronisation argument: two observers at A and B in a system S record the passage of an object P; light takes τ = r/c to travel from A to B, so the event recorded at t_A is "simultaneously" recorded at t_B = t_A + r/c. Lorentz excludes synchronising two clocks at A and carrying one to B. Ludovico identifies what he takes to be an unstated and inconsistent assumption behind this: that within any system measuring rods are rigid and do not change length when moved about, "whereas any kind of clock may in general change its pace if it moves from any point to another of the system". He argues in a footnote that rigid transported yard-sticks are "in a substantial contradiction" with the length-contraction formula, since Lorentz introduced the contraction precisely as a hypothesis about a moving arm of the Michelson–Morley interferometer.

He speculates on the origin of Lorentz's asymmetry: all clocks up to the early twentieth century were regulated against sample pendulums, whose period T = 2π(l/g)^1/2 depends on local gravity, which varies with latitude and altitude and, as gravimetry shows, between points of equal latitude and altitude "because of not fully explained reasons" — the effect Jean Richer observed at Cayenne in 1672. He concludes this is not sufficient to explain the assumption. He also reproduces the argument that the "failure" of Michelson–Morley has been contested, quoting Fabio Tabanelli that early procedures were faulty but "observed fringe shifts were real, albeit much smaller than expected", with diurnal speed-versus-azimuth variation not caused by experimental artefact.

Two systems separating faster than light

The essay's sharpest thought experiment concerns systems in opposite motion. A source at O in frame S emits light in all directions; systems S ′ and S ″ move along +X and −X at, say, 200,000 km/s each with respect to O. Their mutual recession speed is then 400,000 km/s, "remarkably greater than the speed of light", though no direct electromagnetic connection between them is possible while each remains in contact with S. Ludovico observes that relativistic velocity composition gives w = (v − v)/(1 − v²/c²) = 0 for this pair, which he says "makes no sense" as a description of their separation, and that SR provides "no credible explanation... as to the physical fate" of the two systems. In Lorentz–Poincaré relativity, where all speeds are referred to the ether, he holds that the composition of velocities is independent of the speed of light and the difficulty does not arise. He adds the charge that Einstein's second postulate — that light speed does not add with the speed of source or detector — is itself "an implicit assumption that the transmission medium of light is the absolute reference frame".

Measuring relative speed without synchronisation

Ludovico offers two operational methods for assessing the relative speed of S ′ from S. The first is optical and geometrical (his Figure 1): if a transverse length h in S ′ is known, measuring the angle β gives x = h tan β and r = h/sin β, and correcting for the light travel time τ = r/c gives

v = h cos β / (t sin β − h/c) = const, and x = vt = cth cos β / (ct sin β − h).

The second uses the Doppler effect: for emission frequency ψ and observed ψ_v,

ψ_v = ψ (1 − v/c), whence v = c (1 − ψ_v/ψ).

His point is that both give the same values in S and S ′ whatever clocks are used, so that "the problem of synchronisation is a false problem". He faults Christian Møller for asserting that transporting a third clock "clashes against the same fundamental difficulty" without identifying any such difficulty in his text, and charges Massimo Brighi with petitio principii for justifying relativistic synchronisation by appeal to the moving-clock slowdown that is itself at issue.

A mechanical account of clock retardation

The paper's constructive alternative is a mechanical explanation of why moving clocks run slow. Ludovico takes a caesium clock, whose atoms oscillate about a lattice site under a central force described by the harmonic equation m d²s/dt² + ks = 0, with solution s = D cos(t(k/m)^1/2 + φ) and period T = ±4φ(m/k)^1/2. If S ′ accelerates to speed V, each unit mass gains kinetic energy ΔE′ = m(V² − v²)/2, expressible via the Doppler relation, and this is written as an added "active mass"

Δm′ = ΔE′/c² = m(1 − ψ_V/ψ)²/2 = mV²/2c².

Substituting m + Δm′ into the period formula lengthens T, lowering the oscillation frequency, so the clock in S ′ ticks slower — during and after acceleration — and the retardation then stays constant while V does. Ludovico draws three consequences. First, the retardation "does ultimately depend on the effects of different initial accelerations undergone by the relevant systems, and does not depend on their relative speed": from uniform relative speed alone one cannot say which clock is slow. Second, the same account covers gravitational clock effects, since gravity always entails motion; but the change tracks the change in kinetic energy, not the acceleration itself — he gives the example of two clocks on circular orbits of radii R and r with V = v(R/r)^1/2, subject to identical central acceleration, of which the faster is nonetheless the later. Third, since slower clocks in S ′ measure time in different units, "the twins' paradox does not pertain to Einstein's special relativity, for such a case involves relative accelerations". He supports the general move with the analogy that the Ptolemaic system predicted eclipses precisely without thereby being the only adequate theory of them.

Mass energy without relativity

Treating the rest state of a particle as oscillation about a fixed point of the plenum, with rest-state radiation frequency ψ₀, Ludovico writes the motion-induced active mass as

m_V = (m₀/2)(1 − ψ_V/ψ₀)² = (m₀/2)(V/c)²,

so that the total mass with respect to a moving frame is m = m′₀(1 + v²/2c²) and, multiplying by c²,

mc² = m′₀v²/2 + m′₀c², i.e. E_m = mc²,

kinetic energy plus an intrinsic rest energy. He then contrasts this with the relativistic mass equation m = m₀/(1 − v²/c²)^1/2, which he attributes to Lorentz's transverse mass and which Einstein reproduces without citation in §10 of the 1905 paper. A graph in the paper compares the two mass ratios: they are "substantially coincident" up to about v ≈ 0.60c, but at v = c the relativistic curve diverges while his own gives the finite value m_c = 1.5m₀. His conclusion is stated as a dilemma: either E = mc² is well tested, in which case "the Lorentz transformation factor... has no physical significance"; or the Lorentz factor is the achievement of SR, in which case SR cannot be credited with the mass–energy equivalence, which "does certainly conflict with the logical paradigm of Einstein's special relativity".

Spacetime as dimensional collapse

A short section argues that Minkowski's chronotope, by rendering time as the length c·t, changes the physical dimensions of everything else. Speed becomes a length over a length — "a pure number", hence "no more a physical quantity, since it has no physical dimension". Energy, dimensionally mass times speed squared, becomes a numerical multiple of a mass; momentum likewise reduces dimensionally to mass. So in spacetime "energy", "momentum" and "mass" are three terms for one quantity in three conceptual states, and the three dimensions Length, Time and Mass collapse to Length and Mass. Ludovico's point is that the mass–energy equivalence thereby enters the spacetime paradigm as an axiom rather than a result. He adds that acceleration in Minkowski space has the dimension of an inverse length, i.e. of a curvature, and force that of mass times curvature.

Superluminal motion in galactic jets

The closing section holds that the light-speed limit is contradicted by observation. Ludovico cites Pearson et al. on superluminal expansion in 3C273 (1981), Porcas (1983), Davis, Unwin and Muxlow on large-scale superluminal motion in 3C273 (1991), and Biretta, Sparks and Machetto's HST observations of the M87 jet (1999). He notes that the standard explanation applies to jets aligned within about 19 degrees of the line of sight, but that superluminal motion was later observed in jets aligned "almost perpendicular to the line of sight", where he holds the explanation becomes insufficient; apparent speeds of 4 to 9.6 times c are quoted. In his own framework the jets are the visible effect of ring-vortices of plenum travelling through the plenum, and the superluminal motion occurs with respect to the void along the vortex axis and in the ring core. He notes that his other calculations assumed plenum speed at the vortex boundary exceeding 2.5c, and that the astronomical figures suggest it may be "much higher than expected", which "might remarkably modify a few quantitative conclusions" of his analyses. He explicitly declines to argue from quantum entanglement or from Feinberg's tachyons.

Assessment

The historical section is the strongest part of the essay and is largely uncontroversial among historians of physics, whatever the textbooks say: Voigt's 1887 derivation, the FitzGerald–Larmor–Lorentz contraction, Poincaré's 1900 E/c² result and his 1905 group-theoretic paper are all real, and the priority literature (Whittaker most notoriously) has argued this case for seventy years. Ludovico is also right that Einstein's 1905 Annalen paper contains no spacetime and that Minkowski supplied the geometry. Readers should note, though, that the long quotation attributed to Weyl sits oddly with the citation given for it — chapter II, paragraphs 21–22 of Space, Time, Matter (1922) — since it speaks of relativity as "often regarded as the highly original and even revolutionary contribution of a single individual" and closes on Lewis and Tolman in 1909, and the reader is given no page reference to check. The historical case does not depend on it.

The mechanical clock argument is the most substantive original contribution and is genuinely attractive in one respect: it locates the effect in something concrete, the change in the oscillator's energy, and it yields a prediction Ludovico is willing to state plainly — that clock retardation tracks accumulated acceleration history, not instantaneous relative speed. That is a testable position, and it is the same instinct that led Lorentz to a dynamical rather than kinematic reading of the transformations. But the derivation is not carried through. The step from added kinetic energy to an "active mass" Δm′ = ΔE′/c² added to the oscillator is asserted, not derived, and it silently imports precisely the mass–energy equivalence that the paper elsewhere wants to establish independently — a circularity the essay does not acknowledge. The resulting period formula is then read off from the harmonic oscillator, but Equation [9], T = ±4φ(m/k)^1/2, makes the period depend on the arbitrary integration constant φ, which is a slip: the harmonic period is 2π(m/k)^1/2, phase-independent. More seriously, the account is non-relativistic in a way that the data are not: to second order it reproduces the standard result, but the paper's own graph shows its mass ratio reaching only 1.5m₀ at v = c, whereas accelerator practice — the fields required to steer particles at the LEP or Tevatron energies, where γ exceeds 10⁴ — is engineered around the divergent γ and would fail outright on a factor of 1.5. The paper does not address particle accelerators anywhere, and this is the measurement its Equation [15] most directly contradicts.

The two-systems argument conflates two different quantities. That S ′ and S ″ separate at 400,000 km/s as measured in S is not denied by special relativity; the separation rate of two objects in a third frame is not bounded by c, and the velocity-composition formula is not intended to compute it. Ludovico's own arithmetic shows this: he applies the composition formula to v and −v with the sign convention appropriate to a sum rather than a relative velocity, obtains zero, and treats the absurdity as a defect in the theory rather than in the substitution. The correct composition gives 2v/(1 + v²/c²), which for v = 200,000 km/s is about 277,000 km/s — below c. This is the paper's clearest internal error, and it undercuts the section built on it.

The dimensional-collapse observation about Minkowski space is correct as far as it goes and is a familiar point about natural units; whether it shows that mass–energy equivalence is "axiomatic" in spacetime, or merely that the geometry makes an already-derived relation look definitional, the paper does not settle. On superluminal jets, the standard geometric explanation is more robust than Ludovico allows — it requires only that the jet be within roughly arccos(β) of the line of sight, and the apparent speed of 4–9.6c quoted is comfortably within its reach — but his complaint that jets nearly perpendicular to the line of sight have shown superluminal motion would be a real difficulty if the orientation determinations he has in mind are secure; the paper gives no citation for that specific claim, only for the observations themselves.

Finally, the essay's dilemma — either the Lorentz factor or E = mc², not both — rests on reading Einstein's 1905 series truncation as a licence rather than an approximation valid in the stated low-velocity regime. Einstein's short paper derives the mass change for a body emitting radiation and works to first order in v²/c² deliberately, because that suffices for the conclusion; the exact relation follows from the four-momentum without truncation. Ludovico's charge would bite if the truncation were load-bearing for the general result, and he does not show that it is. What survives the criticism is the weaker but still interesting claim with which the paper began: that mass–energy equivalence has multiple independent routes, several of them pre-relativistic, and is therefore poor evidence for special relativity specifically.

See also

• Mario Ludovico
• Special Relativity
• Lorentz Transformation
• Hendrik Lorentz
• Henri Poincaré
• Hermann Minkowski
• Albert Einstein
• Herbert Dingle
• Michelson-Morley Experiment
• Aether
• Length Contraction
• Time Dilation
• Simultaneity
• Franco Selleri