Mass, Energy, Momentum

Scientific Paper
Title	Mass, Energy, Momentum
Read in full	Link to paper
Author(s)	Diego Jos? Arturo Sa
Keywords	Mass, Energy, Momentum
Published	2008
No. of pages	15

Read the full paper here

Abstract

The author offers the interpretation that the mass is a relativistic invariant. Next it is evidenced that the formula for the momentum four-vector cannot be applied, given the current interpretations of Physics, neither to particles nor to photons, and that the photons should have mass. Then it is proposed a postulate about the electromagnetic constitution of matter, which gets rid of these problems. Finally, the validity and applicability of the equivalence of mass and energy is evaluated and is confirmed the requirement that the constitution of mass must be electromagnetic.

Overview

Diego Saá, of the Escuela Politécnica Nacional in Quito, Ecuador, offers a short critical essay on three linked concepts: mass, energy and momentum. The paper proceeds in four moves. It argues that mass is a relativistic invariant and that "relativistic mass" is a mistake; it argues that the momentum four-vector as usually written cannot be applied to a photon without absurdity, and that photons must therefore carry mass; it argues that a particle at rest has a non-zero momentum component that current physics tacitly ignores, implying an internal velocity equal to c; and it concludes that matter must be electromagnetically constituted, elementary particles being made of circulating photons.

Saá is careful to limit his target. He explicitly leaves the constancy of the speed of light untouched, calling it "a fact well supported by several experiments", and changes nothing not named in the paper. The disagreement is thus with the interpretive apparatus that has grown around special relativity rather than with its kinematics.

The argument

Mass as an invariant

Saá sets out the standard construction. The Lorentz factor γ = 1/√(1 − v²/c²) is also expressible as dt/dτ, the ratio of coordinate to proper time. The coordinate velocity u^a = dx^a/dt is not a tensor; the proper velocity U^a = dx^a/dτ is, and the two are related by U^a = γu^a, with U^aU_a = c². The four-momentum is then P^a = mU^a = γm{c, v_x, v_y, v_z}.

His point is that the γ in this expression demonstrably came from the four-velocity, not from the mass, so attaching it to m to make a velocity-dependent "relativistic mass" misassigns it. He adds a mathematical argument — an invariant times a tensor gives a tensor, whereas "relativistic mass" times coordinate velocity has no clear tensorial standing — and a physical one: if the particle is left untouched and an observer begins to move, the particle cannot have acquired mass, and a second observer at a different speed cannot give it a third value. Mass, he concludes, is invariant like electric charge, while energy and momentum are the frame-dependent quantities. He cites Taylor and Wheeler, Okun's 1989 Physics Today article and Einstein's 1948 letter to Lincoln Barnett in support, and notes that Einstein's own 1905 "longitudinal" and "transverse" masses, γ³m₀ and γ²m₀, are now regarded as incorrect.

The photon

Setting v = c makes γ infinite, so every component of U^a and P^a diverges. Physicists escape, Saá says, only by setting m = 0 to obtain an indeterminacy. He proposes instead that γ = 1 for photons, that the temporal component of the four-momentum vanishes, and that the spatial speed is c, giving P^a = {0, mc, 0, 0}. This, he argues, is automatically Lorentz invariant since both m and c are invariant, and it assigns the photon the momentum mc. Since photons demonstrably carry energy and momentum, and both would vanish if m were zero, photons must have mass. He quotes Einstein's remark in The Evolution of Physics that "a beam of light carries energy and energy has mass".

The particle at rest

Letting v → 0 in the same expression gives P^a = {mc, 0, 0, 0}. Saá contends that physics "forgets" this non-zero first component, reinterpreting it as energy divided by light speed "with the secret intention of avoiding the evident need of the velocity of light in the expression for momentum". His reading is that the c is a real internal velocity, and that this vindicates a postulate he had advanced earlier, in an unpublished paper submitted to Galilean Electrodynamics: that elementary particles are constituted of photons in closed circulation. He acknowledges the referee's objection, and the editor Cynthia Whitney's, that no mechanism is offered for curving the photon path, replying that light is bent by gravity and can be looped tightly in optical fibre, and pointing to the spiral tracks left in bubble chambers when particles "unwind".

Mass-energy equivalence

Saá argues that the usual treatment breaks E = mc² in both directions. If a photon's energy corresponds to no mass, then energy does not universally have a mass equivalent; and if mass converts to radiation of zero mass, the original mass has vanished rather than been converted. The substitution of E = hν merely "conceals the mass within the Planck's constant", since h equals the electron mass times its Compton wavelength times c. Giving photons mass, he says, restores the equivalence and makes sense of the Compton effect and of Anderson's 1932 result that a 1.02 MeV gamma yields an electron and a positron of about 0.511 MeV each. Descriptions of pair production as conjuring particles from empty space or from vacuum virtual particles he likens to the magician's account of where the rabbit came from.

Assessment

The first section is simply right, and it is worth saying so plainly. Mass as a Lorentz invariant, with γ belonging to the four-velocity rather than to m, is not a dissident position but the settled view among relativists; Okun's article, which Saá cites accurately, is the canonical statement of it, and Taylor and Wheeler's textbook teaches it. Saá's tensorial argument is correct, and his two-observer argument is a clean way of putting the physical point. That the majority of introductory textbooks still teach "relativistic mass" is a real pedagogical complaint, not an invented one. Where the paper is on this ground it is careful, well referenced and modest in scope.

The trouble begins when the same formula is pushed onto light. P^a = mdx^a/dτ is defined only along timelike worldlines, because only those possess a proper time; for a null worldline dτ = 0 identically and the expression is not ill-behaved so much as inapplicable, in the way that dividing by the arc-length parameter of a zero-length curve is inapplicable. The standard treatment does not evade this by a trick: it parameterises the null geodesic by an affine parameter and writes P^a = (E/c, p) with P^aP_a = 0, which is finite, well defined, and reproduces E = pc. Saá's indeterminacy is an artifact of the parameterisation he has chosen, not a defect in the physics.

His replacement is worse than the problem it is meant to solve, and by the paper's own criteria. P^a = {0, mc, 0, 0} has zero time component, hence zero energy — the very thing Saá insists photons must have, and the reason he rejected m = 0 two paragraphs earlier. Its invariant square is P^aP_a = −m²c², a spacelike momentum, which describes a tachyon rather than light. And the claim that it is "automatically Lorentz invariant" is unsupported: a four-vector's components are not invariant, only its square is, so writing a vector with constant entries in one frame does not make it frame-independent. The construction is offered with the honest caveat that it "still needs to be confirmed with further studies", but as it stands it is internally inconsistent.

The photon mass is also not a free parameter. A massive photon obeys the Proca rather than the Maxwell equations, which turns Coulomb's law into a Yukawa form with an exponential cut-off; laboratory tests of the inverse-square law and, more stringently, measurements of the solar-wind magnetic field and of Jupiter's magnetosphere bound the photon rest mass below roughly 10⁻¹⁸ eV, some twenty-four orders of magnitude under the electron mass. Vacuum dispersion is bounded independently: gamma-ray burst photons of widely different energy arrive together to within seconds after travelling billions of years. Any mass Saá's scheme requires to give the photon its momentum mc would be enormously larger than these limits allow.

The rest-particle argument mistakes a known result for a suppressed one. The first component of {mc, 0, 0, 0} is not forgotten; multiplied by c it is precisely the rest energy mc², which is the content of the equation the paper is defending. The appearance of c there is a unit conversion between the time and space entries of a four-vector, not evidence of an internal circulation, and no step in the paper derives such a circulation — the photon-constituted-matter postulate is introduced as already held and then said to be confirmed. Bubble-chamber spirals, offered as unexplained, have a quantitative standard explanation: a charged track in a uniform magnetic field has radius proportional to its momentum, so a particle losing energy by ionisation spirals inward, and the pitch of that spiral is used routinely to measure momentum.

Similarly, the criticism of pair production attacks a popular-science phrasing rather than the physics. The textbook account of Anderson's result is exactly the one Saá says it should be — photon energy converting into rest energy — with the threshold at 1.022 MeV being 2m_ec² to the digit, and with a nearby nucleus required to absorb recoil momentum. Nor does h conceal the electron's mass: m_eλ_Cc = h holds because the Compton wavelength is defined as h/m_ec, and the same identity holds for the proton with a different mass and a different wavelength, so no particular mass is hidden in a universal constant.

The paper is best read, then, as two unequal halves: a sound and well-sourced argument for invariant mass, and a set of inferences from it that do not follow and that conflict with measured bounds. Saá's own framing is more assertive than his results warrant — "it is evidenced", "we have proved" — for conclusions that rest on a formula applied outside its domain of definition.