Electromagnetic Waves, Inertial Transformations and Compton Effect: Difference between revisions
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| url = [http://www.naturalphilosophy.org/pdf/abstracts/abstracts_1857.pdf Link to paper] | | url = [http://www.naturalphilosophy.org/pdf/abstracts/abstracts_1857.pdf Link to paper] | ||
| author = [[Biagio Buonaura]] | | author = [[Biagio Buonaura]] | ||
| keywords = [[Electromagnetic Waves]], [[Inertial Transformations]], | | keywords = [[Electromagnetic Waves]], [[Inertial Transformations]], '''Compton effect''' | ||
| published = 2007 | | published = 2007 | ||
| journal = [[Apeiron]] | | journal = [[Apeiron]] | ||
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The Inertial Transformations (IT) are a new set of transformations of the space and time variables providing an alterative (but empirically equivalent) approach to the Theory of Special Relativity (TSR). With the IT, the one way velocity of light is isotropic only in a privileged reference frame, S0. In this new theory only a weak form of relativity principle holds. We apply the IT to the collision of an energetic photon with an electron (Compton effect). A theoretical description of the dual quantum mechanical photon having both wave and particle properties is required. From the undulatory point of view we use the IT to deduce the e.m. wave equations in an inertial reference frame S moving with respect to S<sub>0</sub> with (absolute) velocity V. Using the Maxwell equations in the form suitable to S, we show that the e.m. plane waves in S have the same properties as in S<sub>0</sub>: the fields are perpendicular to one another and perpendicular to the propagation direction of the field energy. The latter direction, however, does not coincide, in S, with the propagation direction of the e.m. plane wave. From the corpuscular point of view, we show that in the framework of the IT the usual equations relating the photon energy and momentum to frequency also hold. The result of this research on the Compton effect is a complete empirical equivalence between the TSR and the IT approach. | The Inertial Transformations (IT) are a new set of transformations of the space and time variables providing an alterative (but empirically equivalent) approach to the Theory of Special Relativity (TSR). With the IT, the one way velocity of light is isotropic only in a privileged reference frame, S0. In this new theory only a weak form of relativity principle holds. We apply the IT to the collision of an energetic photon with an electron (Compton effect). A theoretical description of the dual quantum mechanical photon having both wave and particle properties is required. From the undulatory point of view we use the IT to deduce the e.m. wave equations in an inertial reference frame S moving with respect to S<sub>0</sub> with (absolute) velocity V. Using the Maxwell equations in the form suitable to S, we show that the e.m. plane waves in S have the same properties as in S<sub>0</sub>: the fields are perpendicular to one another and perpendicular to the propagation direction of the field energy. The latter direction, however, does not coincide, in S, with the propagation direction of the e.m. plane wave. From the corpuscular point of view, we show that in the framework of the IT the usual equations relating the photon energy and momentum to frequency also hold. The result of this research on the Compton effect is a complete empirical equivalence between the TSR and the IT approach. | ||
==Overview== | |||
Buonaura's paper belongs to the research programme founded by [[Franco Selleri]], which holds that the [[Lorentz Transformation]]s owe part of their content not to nature but to a ''convention'' — Einstein's rule for synchronizing distant clocks. Following Mansouri and Sexl, Selleri showed that a whole one-parameter family of space and time transformations, differing only in the coefficient ''e''<sub>1</sub> of ''x'' in the time transformation, reproduces every experimental result of [[Special Relativity]]. Setting ''e''<sub>1</sub> = 0 gives the '''inertial transformations''' (IT), in which clocks are synchronized absolutely: two events simultaneous in the privileged frame ''S''<sub>0</sub> are simultaneous in every frame, and the one-way [[Speed of Light|speed of light]] is isotropic only in ''S''<sub>0</sub>. What survives of Einstein's postulate is the invariance of the ''two-way'' speed, and what survives of the relativity principle is only a weak form (the "Weak Relativity Principle"): absolute motion cannot be detected experimentally. | |||
Until this paper, Buonaura notes, the published applications of the IT "mainly had a kinematical nature". His purpose is to push them into dynamics, choosing the case usually taken to be the most relativistic and most quantum-mechanical at once: the [[Compton Effect|Compton scattering]] of an energetic [[photon]] by a free [[electron]]. He treats the photon in both of its aspects — as a wave, by deriving the electromagnetic wave equation in the moving frame from [[Maxwell's Equations]] in the IT form, and as a corpuscle, by transforming energy and momentum — and concludes that the IT reproduce Compton's 1923 formula exactly. The result is offered not as a new prediction but as a demonstration of ''complete empirical equivalence'': relativity's successes here are shown to be successes of the two-way light postulate, not of Einstein synchronization. | |||
==The argument== | |||
===The equivalent and inertial transformations=== | |||
Buonaura begins from Selleri's equivalent transformations (ET) from ''S''<sub>0</sub> to a frame ''S'' moving with absolute velocity ''V'': | |||
: ''x'' = (''x''<sub>0</sub> − ''Vt''<sub>0</sub>)/''R'', ''y'' = ''y''<sub>0</sub>, ''z'' = ''z''<sub>0</sub>, ''t'' = ''Rt''<sub>0</sub> + ''e''<sub>1</sub>(''x''<sub>0</sub> − ''Vt''<sub>0</sub>) | |||
with ''R'' = (1 − β<sup>2</sup>)<sup>1/2</sup> and β = ''V''/''c''. These follow from three assumptions: an isotropic frame ''S''<sub>0</sub> exists; the two-way light speed is the same in all frames and directions; and clocks moving with respect to ''S''<sub>0</sub> retard by the factor ''R''. Special relativity is recovered at ''e''<sub>1</sub> = −β/(''Rc''); the inertial transformations are ''e''<sub>1</sub> = 0, leaving simply ''t'' = ''Rt''<sub>0</sub>. | |||
The one-way light speed in ''S'' becomes direction-dependent: 1/''c''(θ) = [1 + Γ(β) cos θ]/''c'' with Γ(β) = β + ''e''<sub>1</sub>''Rc''. [[Length Contraction|Lorentz–FitzGerald contraction]] and Larmor retardation follow as before, and Michelson- and Fizeau-type experiments, stellar [[aberration]], Römer's occultations of Jupiter's satellites, radar ranging of planets and the International Atomic Time are all shown to be independent of ''e''<sub>1</sub>. Buonaura notes the theorem of [[Guido Rizzi]] and collaborators that Selleri's assumptions are equivalent to Einstein's, whose corollaries are that no experiment discriminates between values of ''e''<sub>1</sub> and that ''S''<sub>0</sub> is undetectable. He dissents on the first point: the indifference of physical reality to the synchronization choice "holds only so far as Weakly Accelerated Reference Frames are excluded", and physical continuity with such frames, he argues, singles out ''e''<sub>1</sub> = 0. Under the IT the origin of ''S''<sub>0</sub> seen from ''S'' recedes at β''c''/''R''<sup>2</sup>, which may exceed ''c'' without being superluminal, since a pulse travelling in the −''x'' direction moves at ''c''/(1 − β), always faster still. Absolute velocities, by contrast, never exceed ''c''. | |||
===Energy propagation and aberration=== | |||
Tracking a small light pulse, Buonaura substitutes the inverse IT into the ''S''<sub>0</sub> description and obtains for the direction of energy transport in ''S'': | |||
: tan θ<sub>E</sub> = ''R'' sin θ<sub>0</sub> / (cos θ<sub>0</sub> − β) | |||
which he emphasises "coincides with the well known relativistic aberration formula", with every quantity on the right belonging to ''S''<sub>0</sub>, where the Lorentz and inertial theories agree numerically. The aberration angle is therefore identical in the two theories for arbitrary ''S''. The energy speed comes out as ''c''<sub>E</sub> = ''c''/(1 + β cos θ<sub>E</sub>), reproducing the known IT one-way light speed. | |||
===Wave equations in the moving frame=== | |||
Transforming the derivative operators, he finds ∇<sup>2</sup><sub>0</sub> = ∇<sup>2</sup> + [β<sup>2</sup>/(1 − β<sup>2</sup>)] ∂<sup>2</sup>/∂''x''<sup>2</sup>, and hence, from the ''S''<sub>0</sub> wave equations, coupled equations for the field components in ''S''. Using the IT field transformations established in his earlier ''Foundations of Physics Letters'' paper, the couplings collapse — the structure ''F'' = −β''G'', ''G'' = −β''F'' forces ''F'' = ''G'' = 0 — and each Cartesian component satisfies the same equation, giving in vector form | |||
: ∇<sup>2</sup>'''E''' = [(1 − β<sup>2</sup>)/''c''<sup>2</sup>] ∂<sup>2</sup>'''E'''/∂''t''<sup>2</sup> − (2β/''c'') ∂<sup>2</sup>'''E'''/∂''x''∂''t'' | |||
and identically for '''H'''. Because the strong relativity principle does not hold under the IT, these differ in form from the ''S''<sub>0</sub> equations — the frame's absolute motion appears explicitly in the cross term. | |||
===Plane waves: a non-transverse wave with a transverse energy flow=== | |||
Inserting a plane wave into this equation yields a quadratic in ''k'' whose positive root is ''k'' = (ν/''c'')[β cos θ + (1 − β<sup>2</sup> sin<sup>2</sup>θ)<sup>1/2</sup>]. The apparent phase velocity is thus independent of frequency (no vacuum dispersion) but depends on direction. Phase invariance gives the transformations of wave vector and frequency, including the [[Doppler Effect]] formula ν = (ν<sub>0</sub>/''R'')(1 − '''V'''·'''n'''<sub>0</sub>/''c''), and Buonaura notes that his expressions for '''n''' and for ''c''<sub>φ</sub> coincide with the Puccini–Selleri formulae obtained from absolute motion. | |||
The section's central result is a genuine departure from the familiar picture. Imposing the IT form of Maxwell's equations, he obtains '''ε'''·'''h''' = 0 and |'''ε'''| = |'''h'''| — the electric and magnetic fields remain mutually perpendicular and of equal modulus — but also '''k'''·'''ε''' = β'''k'''·('''k'''×'''h''')/''k'' and its counterpart, which show that "the plane wave solutions of the wave equations, differently than in ''S''<sub>0</sub>, in general are '''not transverse''' in ''S''". The fields are not orthogonal to the wave normal '''k'''. Buonaura resolves this by defining an '''effective wave vector''' '''k'''<sub>E</sub> ≡ ρ('''k''' − βν/''c''), which he proves is perpendicular to both fields and parallel to the energy direction '''n'''<sub>E</sub> found earlier. The transversality that fails for the phase normal thus holds for the energy flow, and the effective wavelength ''L'' = ''c''<sub>E</sub>/ν = ''c''/(''R''ν<sub>0</sub>) is elongated by 1/''R'' relative to λ<sub>0</sub>, as expected. | |||
===Photon energy, momentum, and the Compton formula=== | |||
Applying Selleri's IT transformations for particle energy and momentum to a photon with ''E''<sub>0</sub> = ''h''ν<sub>0</sub>, Buonaura finds ''E'' = ''h''ν and |'''p'''| = ''h''ν/''c'' in the moving frame, with components ''p''<sub>x</sub> = (''h''ν/''c''<sub>E</sub>) cos θ<sub>E</sub> and ''p''<sub>y</sub> = (''h''ν/''c''<sub>E</sub>) sin θ<sub>E</sub>. The Planck relations therefore hold in formally identical fashion in ''S'' and ''S''<sub>0</sub> — "of course the standard result in the TSR, but it needed to be proven anew". With these in hand, energy and momentum conservation for a photon striking an electron at rest in ''S'' gives, after eliminating the electron recoil angle, | |||
: 1/ν′ − 1/ν = (''h''/''mc''<sup>2</sup>)(1 − cos θ) | |||
"the famous Compton formula expressed in terms of the synchronization independent frequencies". | |||
===Why the measured wavelength is also unchanged=== | |||
Rewriting this in wavelengths is not trivial, since the light speed in ''S'' is direction-dependent. Buonaura therefore analyses Compton's actual apparatus: x-rays of 0.0709 nm on a carbon target, scattered radiation selected by slits at angle θ, and the wavelength read off from the Bragg angle at a calcite crystal. Since two photons are incoherent, the interference is that of a single photon with itself, represented over the [[Richard Feynman|Feynman]] path sum. The interference at the detector depends on the time delay Δ''T'' between two paths. Computing Δ''T'' with the direction-dependent light speed, the Γ-dependent contribution reduces to a difference of two line integrals of d'''l''', each of which equals the vector from source to detector; the terms cancel exactly and Γ — and with it ''e''<sub>1</sub> — disappears. Every theory of the family therefore predicts the same Bragg angle, so one may use whichever member is convenient, and with λν = ''c'' the standard result λ − λ<sub>0</sub> = λ<sub>C</sub>(1 − cos θ) follows. An appendix derives, from a closed-path time measurable with a single clock, the general relation 1/''c''<sub>XY</sub> = 1/''c''<sub>TSR</sub> + Γcos θ/''c'' — with Γ independent of the medium — which is what makes the cancellation work. | |||
==Assessment== | |||
The paper is careful, internally consistent, and modest about what it claims. Buonaura does not assert that relativity is wrong or that an [[aether]] has been detected; his stated conclusion is equivalence, and he reaches it by honest calculation rather than by assertion. The technical core is real work: deriving the modified wave equation from the IT-form Maxwell equations, showing that the coupled component equations decouple, and then discovering — rather than assuming — that plane waves in a moving frame are not transverse with respect to '''k''' is a nontrivial and somewhat surprising result. His response to it is the paper's best move: instead of treating non-transversality as a defect he isolates the effective wave vector '''k'''<sub>E</sub> and shows that it, not '''k''', is both perpendicular to the fields and aligned with the Poynting direction found independently from the pulse kinematics. That two separate routes converge on the same direction is a genuine internal check. | |||
The treatment of the Compton experiment is the most instructive part, because it is where a competent objection would naturally arise. A direction-dependent light speed threatens to shift the diffraction condition in the calcite crystal, and Buonaura confronts this directly rather than assuming λν = ''c''. His demonstration that the Γ-terms cancel because the two path integrals of d'''l''' share the same endpoints is clean, and it explains ''why'' the synchronization convention is invisible here rather than merely asserting that it is. Anyone tempted to look for a Selleri-type anisotropy in Bragg-diffraction data should read section 8 first. | |||
The difficulties are those of the programme rather than of the execution. The first is that the paper's positive content is, by construction, empirically empty: a result that is "completely empirically equivalent" to special relativity adds no testable consequence. What is gained is interpretive — absolute simultaneity, a privileged frame, and a physically real [[Length Contraction|length contraction]] — and whether that is a gain depends on prior commitments the paper does not argue for. Second, the one step where Buonaura ''does'' break equivalence is the weakest link. He rejects the Rizzi–Ruggiero–Serafini corollary that no experiment can distinguish values of ''e''<sub>1</sub>, asserting that "physical continuity" with weakly accelerated reference frames selects ''e''<sub>1</sub> = 0, but the assertion is referred to Selleri's earlier work and not defended here; no derivation, and no proposed measurement, appears in this paper. Since ''e''<sub>1</sub> = 0 is what makes the whole construction distinctive, that gap sits at the foundation. Third, the identity of the privileged frame is never fixed. It is not tied to the [[Cosmic Microwave Background|cosmic microwave background]] rest frame or to any other physical reference, so β is a free parameter that never has to take a value — which is convenient, but also means no number in the paper is ever confronted with a measurement. The [[Michelson-Morley Experiment|Michelson–Morley]]-type null results are accommodated by construction rather than explained. | |||
Finally, the derivation is entirely first-order in the sense that it treats the electron and photon as free particles in flat space and stops at the two-body kinematics. That is fair for Compton scattering, but it is worth noting that the harder tests of the relativistic framework — the (1+''z'') time dilation of Type Ia supernova light curves, or the precision of [[Quantum Electrodynamics|QED]] radiative corrections to Compton scattering — lie outside what an equivalence proof at this level can address. Within the boundaries Buonaura sets himself, however, the paper does what it says it does. | |||
==See also== | |||
* [[Biagio Buonaura]] | |||
* [[Franco Selleri]] | |||
* [[Guido Rizzi]] | |||
* [[Special Relativity]] | |||
* [[Lorentz Transformation]] | |||
* [[Simultaneity]] | |||
* [[Length Contraction]] | |||
* [[Speed of Light]] | |||
* [[Doppler Effect]] | |||
* [[Maxwell's Equations]] | |||
* [[Apeiron]] | |||
[[Category:Scientific Paper|electromagnetic waves inertial transformations compton effect]] | [[Category:Scientific Paper|electromagnetic waves inertial transformations compton effect]] | ||
[[Category:Relativity|electromagnetic waves inertial transformations compton effect]] | [[Category:Relativity|electromagnetic waves inertial transformations compton effect]] | ||
[[Category:Electrodynamics]] | |||
[[Category:Light]] | |||
[[Category:Quantum Theory]] | |||
[[Category:Time]] | |||
Latest revision as of 11:09, 21 July 2026
| Scientific Paper | |
|---|---|
| Title | Electromagnetic Waves, Inertial Transformations and Compton Effect |
| Read in full | Link to paper |
| Author(s) | Biagio Buonaura |
| Keywords | Electromagnetic Waves, Inertial Transformations, Compton effect |
| Published | 2007 |
| Journal | Apeiron |
| Volume | 14 |
| Number | 3 |
| No. of pages | 30 |
| Pages | 184-213 |
Read the full paper here
Abstract
The Inertial Transformations (IT) are a new set of transformations of the space and time variables providing an alterative (but empirically equivalent) approach to the Theory of Special Relativity (TSR). With the IT, the one way velocity of light is isotropic only in a privileged reference frame, S0. In this new theory only a weak form of relativity principle holds. We apply the IT to the collision of an energetic photon with an electron (Compton effect). A theoretical description of the dual quantum mechanical photon having both wave and particle properties is required. From the undulatory point of view we use the IT to deduce the e.m. wave equations in an inertial reference frame S moving with respect to S0 with (absolute) velocity V. Using the Maxwell equations in the form suitable to S, we show that the e.m. plane waves in S have the same properties as in S0: the fields are perpendicular to one another and perpendicular to the propagation direction of the field energy. The latter direction, however, does not coincide, in S, with the propagation direction of the e.m. plane wave. From the corpuscular point of view, we show that in the framework of the IT the usual equations relating the photon energy and momentum to frequency also hold. The result of this research on the Compton effect is a complete empirical equivalence between the TSR and the IT approach.
Overview
Buonaura's paper belongs to the research programme founded by Franco Selleri, which holds that the Lorentz Transformations owe part of their content not to nature but to a convention — Einstein's rule for synchronizing distant clocks. Following Mansouri and Sexl, Selleri showed that a whole one-parameter family of space and time transformations, differing only in the coefficient e1 of x in the time transformation, reproduces every experimental result of Special Relativity. Setting e1 = 0 gives the inertial transformations (IT), in which clocks are synchronized absolutely: two events simultaneous in the privileged frame S0 are simultaneous in every frame, and the one-way speed of light is isotropic only in S0. What survives of Einstein's postulate is the invariance of the two-way speed, and what survives of the relativity principle is only a weak form (the "Weak Relativity Principle"): absolute motion cannot be detected experimentally.
Until this paper, Buonaura notes, the published applications of the IT "mainly had a kinematical nature". His purpose is to push them into dynamics, choosing the case usually taken to be the most relativistic and most quantum-mechanical at once: the Compton scattering of an energetic photon by a free electron. He treats the photon in both of its aspects — as a wave, by deriving the electromagnetic wave equation in the moving frame from Maxwell's Equations in the IT form, and as a corpuscle, by transforming energy and momentum — and concludes that the IT reproduce Compton's 1923 formula exactly. The result is offered not as a new prediction but as a demonstration of complete empirical equivalence: relativity's successes here are shown to be successes of the two-way light postulate, not of Einstein synchronization.
The argument
The equivalent and inertial transformations
Buonaura begins from Selleri's equivalent transformations (ET) from S0 to a frame S moving with absolute velocity V:
- x = (x0 − Vt0)/R, y = y0, z = z0, t = Rt0 + e1(x0 − Vt0)
with R = (1 − β2)1/2 and β = V/c. These follow from three assumptions: an isotropic frame S0 exists; the two-way light speed is the same in all frames and directions; and clocks moving with respect to S0 retard by the factor R. Special relativity is recovered at e1 = −β/(Rc); the inertial transformations are e1 = 0, leaving simply t = Rt0.
The one-way light speed in S becomes direction-dependent: 1/c(θ) = [1 + Γ(β) cos θ]/c with Γ(β) = β + e1Rc. Lorentz–FitzGerald contraction and Larmor retardation follow as before, and Michelson- and Fizeau-type experiments, stellar aberration, Römer's occultations of Jupiter's satellites, radar ranging of planets and the International Atomic Time are all shown to be independent of e1. Buonaura notes the theorem of Guido Rizzi and collaborators that Selleri's assumptions are equivalent to Einstein's, whose corollaries are that no experiment discriminates between values of e1 and that S0 is undetectable. He dissents on the first point: the indifference of physical reality to the synchronization choice "holds only so far as Weakly Accelerated Reference Frames are excluded", and physical continuity with such frames, he argues, singles out e1 = 0. Under the IT the origin of S0 seen from S recedes at βc/R2, which may exceed c without being superluminal, since a pulse travelling in the −x direction moves at c/(1 − β), always faster still. Absolute velocities, by contrast, never exceed c.
Energy propagation and aberration
Tracking a small light pulse, Buonaura substitutes the inverse IT into the S0 description and obtains for the direction of energy transport in S:
- tan θE = R sin θ0 / (cos θ0 − β)
which he emphasises "coincides with the well known relativistic aberration formula", with every quantity on the right belonging to S0, where the Lorentz and inertial theories agree numerically. The aberration angle is therefore identical in the two theories for arbitrary S. The energy speed comes out as cE = c/(1 + β cos θE), reproducing the known IT one-way light speed.
Wave equations in the moving frame
Transforming the derivative operators, he finds ∇20 = ∇2 + [β2/(1 − β2)] ∂2/∂x2, and hence, from the S0 wave equations, coupled equations for the field components in S. Using the IT field transformations established in his earlier Foundations of Physics Letters paper, the couplings collapse — the structure F = −βG, G = −βF forces F = G = 0 — and each Cartesian component satisfies the same equation, giving in vector form
- ∇2E = [(1 − β2)/c2] ∂2E/∂t2 − (2β/c) ∂2E/∂x∂t
and identically for H. Because the strong relativity principle does not hold under the IT, these differ in form from the S0 equations — the frame's absolute motion appears explicitly in the cross term.
Plane waves: a non-transverse wave with a transverse energy flow
Inserting a plane wave into this equation yields a quadratic in k whose positive root is k = (ν/c)[β cos θ + (1 − β2 sin2θ)1/2]. The apparent phase velocity is thus independent of frequency (no vacuum dispersion) but depends on direction. Phase invariance gives the transformations of wave vector and frequency, including the Doppler Effect formula ν = (ν0/R)(1 − V·n0/c), and Buonaura notes that his expressions for n and for cφ coincide with the Puccini–Selleri formulae obtained from absolute motion.
The section's central result is a genuine departure from the familiar picture. Imposing the IT form of Maxwell's equations, he obtains ε·h = 0 and |ε| = |h| — the electric and magnetic fields remain mutually perpendicular and of equal modulus — but also k·ε = βk·(k×h)/k and its counterpart, which show that "the plane wave solutions of the wave equations, differently than in S0, in general are not transverse in S". The fields are not orthogonal to the wave normal k. Buonaura resolves this by defining an effective wave vector kE ≡ ρ(k − βν/c), which he proves is perpendicular to both fields and parallel to the energy direction nE found earlier. The transversality that fails for the phase normal thus holds for the energy flow, and the effective wavelength L = cE/ν = c/(Rν0) is elongated by 1/R relative to λ0, as expected.
Photon energy, momentum, and the Compton formula
Applying Selleri's IT transformations for particle energy and momentum to a photon with E0 = hν0, Buonaura finds E = hν and |p| = hν/c in the moving frame, with components px = (hν/cE) cos θE and py = (hν/cE) sin θE. The Planck relations therefore hold in formally identical fashion in S and S0 — "of course the standard result in the TSR, but it needed to be proven anew". With these in hand, energy and momentum conservation for a photon striking an electron at rest in S gives, after eliminating the electron recoil angle,
- 1/ν′ − 1/ν = (h/mc2)(1 − cos θ)
"the famous Compton formula expressed in terms of the synchronization independent frequencies".
Why the measured wavelength is also unchanged
Rewriting this in wavelengths is not trivial, since the light speed in S is direction-dependent. Buonaura therefore analyses Compton's actual apparatus: x-rays of 0.0709 nm on a carbon target, scattered radiation selected by slits at angle θ, and the wavelength read off from the Bragg angle at a calcite crystal. Since two photons are incoherent, the interference is that of a single photon with itself, represented over the Feynman path sum. The interference at the detector depends on the time delay ΔT between two paths. Computing ΔT with the direction-dependent light speed, the Γ-dependent contribution reduces to a difference of two line integrals of dl, each of which equals the vector from source to detector; the terms cancel exactly and Γ — and with it e1 — disappears. Every theory of the family therefore predicts the same Bragg angle, so one may use whichever member is convenient, and with λν = c the standard result λ − λ0 = λC(1 − cos θ) follows. An appendix derives, from a closed-path time measurable with a single clock, the general relation 1/cXY = 1/cTSR + Γcos θ/c — with Γ independent of the medium — which is what makes the cancellation work.
Assessment
The paper is careful, internally consistent, and modest about what it claims. Buonaura does not assert that relativity is wrong or that an aether has been detected; his stated conclusion is equivalence, and he reaches it by honest calculation rather than by assertion. The technical core is real work: deriving the modified wave equation from the IT-form Maxwell equations, showing that the coupled component equations decouple, and then discovering — rather than assuming — that plane waves in a moving frame are not transverse with respect to k is a nontrivial and somewhat surprising result. His response to it is the paper's best move: instead of treating non-transversality as a defect he isolates the effective wave vector kE and shows that it, not k, is both perpendicular to the fields and aligned with the Poynting direction found independently from the pulse kinematics. That two separate routes converge on the same direction is a genuine internal check.
The treatment of the Compton experiment is the most instructive part, because it is where a competent objection would naturally arise. A direction-dependent light speed threatens to shift the diffraction condition in the calcite crystal, and Buonaura confronts this directly rather than assuming λν = c. His demonstration that the Γ-terms cancel because the two path integrals of dl share the same endpoints is clean, and it explains why the synchronization convention is invisible here rather than merely asserting that it is. Anyone tempted to look for a Selleri-type anisotropy in Bragg-diffraction data should read section 8 first.
The difficulties are those of the programme rather than of the execution. The first is that the paper's positive content is, by construction, empirically empty: a result that is "completely empirically equivalent" to special relativity adds no testable consequence. What is gained is interpretive — absolute simultaneity, a privileged frame, and a physically real length contraction — and whether that is a gain depends on prior commitments the paper does not argue for. Second, the one step where Buonaura does break equivalence is the weakest link. He rejects the Rizzi–Ruggiero–Serafini corollary that no experiment can distinguish values of e1, asserting that "physical continuity" with weakly accelerated reference frames selects e1 = 0, but the assertion is referred to Selleri's earlier work and not defended here; no derivation, and no proposed measurement, appears in this paper. Since e1 = 0 is what makes the whole construction distinctive, that gap sits at the foundation. Third, the identity of the privileged frame is never fixed. It is not tied to the cosmic microwave background rest frame or to any other physical reference, so β is a free parameter that never has to take a value — which is convenient, but also means no number in the paper is ever confronted with a measurement. The Michelson–Morley-type null results are accommodated by construction rather than explained.
Finally, the derivation is entirely first-order in the sense that it treats the electron and photon as free particles in flat space and stops at the two-body kinematics. That is fair for Compton scattering, but it is worth noting that the harder tests of the relativistic framework — the (1+z) time dilation of Type Ia supernova light curves, or the precision of QED radiative corrections to Compton scattering — lie outside what an equivalence proof at this level can address. Within the boundaries Buonaura sets himself, however, the paper does what it says it does.