Addition of Velocities and Electromagnetic Interaction: Geometrical Derivations Using 3D Minkowski Diagrams

Scientific Paper
Title	Addition of Velocities and Electromagnetic Interaction: Geometrical Derivations Using 3D Minkowski Diagrams
Read in full	Link to paper
Author(s)	Calin Galeriu
Keywords	classical electrodynamics, Minkowski space
Published	2003
Journal	Apeiron
Volume	10
Number	1
No. of pages	14
Pages	1-14

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Abstract

This article presents intuitive, geometrical derivations of the relativistic addition of velocities, and of the electromagnetic interaction between two uniformly moving charged particles, based on 2 spatial + 1 temporal dimensional Minkowski diagrams. We calculate the relativistic addition of velocities by projecting the world-line of the particle on the spatio-temporal planes of the reference frames considered. We calculate the real component of the electromagnetic 4-force, in the proper reference frame of the source particle, from the Coulomb force generated by a charged particle at rest. We then obtain the imaginary component of the 4-force, in the same reference frame, from the requirement that the 4-force be orthogonal to the 4-velocity. The 4-force is then projected on a real 3 dimensional space to give the Lorentz force.

Overview

Galeriu's paper, published in Apeiron while he was in the physics department at Worcester Polytechnic Institute, is a piece of pedagogical geometry rather than a challenge to established physics. Its complaint is about how special relativity is taught: the 1+1-dimensional Minkowski diagram is used to introduce the Lorentz transformation, time dilation and length contraction, and is then abandoned for differential calculus and linear algebra, leaving the student with "little intuitive understanding" of the velocity-addition law or of Purcell's dictum that "magnetism is a kind of 'second-order' effect arising from relativistic changes in the electric fields of moving charges".

The remedy proposed is to extend the diagram by one spatial dimension — 2 spatial + 1 temporal — and then to obtain two standard results purely by projection and elementary trigonometry. Nothing in the derivations departs from the accepted content of special relativity; both results are the textbook ones. Where the paper points outward is in its closing motivation: the geometrical treatment of interactions between finite world-line segments, developed in Galeriu's companion work on time-symmetric action-at-a-distance electrodynamics, implies that the electron's rest mass is not constant under acceleration, and this, he argues, gives "strong motivation to investigate alternative theories of the electromagnetic interaction, which allow for the variation of the electron's rest mass."

The derivations

Setting up: imaginary time and angles

Galeriu uses the older imaginary-time convention, with the time axis carrying ict rather than ct, so that a Lorentz boost appears as a rotation through an imaginary angle. A world-line drawn at angle θ to the time axis in one of the spatio-temporal planes then satisfies tan(iθ) = iv/c for the corresponding velocity component. This is what allows the entire argument to be conducted with plane trigonometry: the addition of velocities becomes the addition of angles.

Relativistic addition of velocities

Take frame K′ moving with velocity V along x relative to K, with the origins and a moving particle all coinciding at O. The particle's world-line OP is projected onto the six planes (x, O, ict), (y, O, ict), (z, O, ict) and their primed counterparts; the (x, O, ict) and (x′, O, ict′) planes coincide, since the boost is along x. The angles of these projections encode the velocity components in the two frames.

Within the boost plane the angles simply add, iθ = iα + iβ, and the tangent addition formula gives immediately

v_x = (V + v′_x) / (1 + Vv′_x/c²)

The transverse component requires one more step. The projection process yields two rectangles APCD and BPCE in the diagram, from which two independent expressions for the ratio CP/OC can be read off, one via EB and one via DA. Equating them and substituting the tangents gives

v_y = v′_y(1 − V²/c²)^1/2 / (1 + Vv′_x/c²)

with the same form for v_z. Both are the standard results, but obtained here by reading lengths off a figure rather than by manipulating the transformation matrix.

The electromagnetic interaction: the target expressions

For the second derivation, a source charge Q₁ moves uniformly along Ox with velocity V, and a test charge Q₂ sits at separation R = R[cos(φ)x̂ + sin(φ)ŷ], moving with arbitrary uniform velocity v. Galeriu first records the answer that conventional electrodynamics gives. In Gaussian units the field of the uniformly moving charge is

E = (Q₁/R²)(1 − V²/c²)[1 − (V²/c²)sin²(φ)]^−3/2 R̂

with H = (1/c)V × E, and the Lorentz force F = Q₂E + (Q₂/c)v × H then has the Cartesian components given in his Eqs. (16)–(18), with F_z = 0.

The geometrical route to the same force

The alternative derivation starts from Coulomb's law alone, in the rest frame K′ of the source. The key input, which Galeriu labels explicitly as "one key assumption or experimental fact", is that in a frame where all source charges are at rest the force on a charge q is F = qE, independent of that charge's velocity in that frame. In K′ the interaction is with the source at the retarded-construction point B rather than at O, and the Coulomb force is F′ = Q₁Q₂R̂′/R′².

The 4-force is then assembled from two pieces. Its real part follows from the Coulomb force times the Lorentz factor, F′_real = γ(v′)Q₁Q₂R̂′/R′². Its imaginary part is fixed not by any further physical assumption but by the orthogonality of the 4-force and the 4-velocity, F · V = 0, which gives

F′_imag = γ(v′)(Q₁Q₂/R′²)(v′_rad/c) î′

where v′_rad = v′ · R̂′ is the radial component of the test particle's velocity in K′. This is the paper's most elegant move: the "magnetic" content of the interaction is not postulated but is the timelike component that orthogonality forces on a 4-vector whose spacelike part is Coulombic.

Projecting the same 4-force back onto the real 3-space of K is then pure geometry. The segments BA and BO are decomposed as BA = BD + DE + EA and BO = BD + DO; since neither expansion has a component along Oz, F_z = 0 follows at once. The remaining components come from the ratios of segment lengths, all of which Galeriu computes explicitly (EA = R sin φ, OE = R cos φ, BE = R cos φ cos(iα), DE = R cos φ cos²(iα), and AB = R cos(iα)[1 + tan²(iα)sin²(φ)]^1/2 = R′). Substituting the transformed velocity components and the identities sin(iα) = i(V/c)γ(V), cos(iα) = γ(V) and tan(iα) = iV/c reproduces Eqs. (16)–(17) exactly.

The closing motivation

Galeriu then raises a conceptual objection to his own expressions. The forces (22)–(23) depend explicitly on the test particle's velocity and implicitly on the source's, "but, from a geometrical point of view, a point in Minkowski space is just a fixed point, it does not have a velocity". The consistent geometrical object is therefore an interaction between segments of finite length along the two world-lines. He reports the result from his companion paper that once accelerated motion is admitted on this footing, the electron's rest mass is no longer constant, though "its averaged variation is still null" — and cites a body of theoretical and experimental work on alternative expressions for the electromagnetic force as the context for pursuing this.

Assessment

Judged as what it is, this is a clean and successful piece of work. The two derivations are correct, they reproduce the standard results exactly rather than approximately, and the 3D Minkowski diagram genuinely does what the paper claims for it: the transverse velocity-addition formula, which in the usual treatment appears as an unmemorable consequence of dividing one transformation equation by another, here falls out of the geometry of two rectangles. The device of fixing the timelike component of the 4-force by orthogonality, rather than importing the magnetic field, is pedagogically the paper's best idea — it makes concrete Purcell's remark that magnetism is a relativistic aspect of electricity, since the magnetic term is literally what the geometry of Minkowski space adds to a Coulomb interaction. This is the sort of derivation that belongs in a teaching supplement, and the author's own stated aim is exactly that.

Its limitations are equally clear and mostly acknowledged. Both derivations are restricted to uniform motion; the moment acceleration enters, the projection construction that produces point B from A no longer holds and the method gives nothing. The paper is also not self-contained on its most interesting claim. The assertion that a segment-based geometry forces a variable electron rest mass with null average variation is stated, not derived — it is carried entirely by reference to the companion paper on time-symmetric action-at-a-distance electrodynamics, so a reader of this article alone cannot assess it. Since that claim is the whole justification offered for "investigating alternative theories of the electromagnetic interaction", the conclusion is more of a signpost than a result.

The use of the imaginary time coordinate ict is a defensible choice here — it is what turns boosts into angle additions and makes the trigonometry work — but it is worth noting that the convention has been abandoned in modern treatments precisely because it obscures the difference between the Lorentzian and Euclidean signatures, and it does not generalise to curved spacetime. A reader taking the diagrams as a picture of the geometry, rather than as a calculational device, may take away a misleading impression on that point.

Finally, on the physics rather than the pedagogy: a variable electron rest mass is a strong claim, and the constancy of the electron mass is among the better-constrained quantities in physics through atomic spectroscopy and the agreement of the measured electron magnetic moment anomaly with quantum electrodynamics to better than a part in 10¹². Galeriu's own qualification — that the averaged variation vanishes — is what would have to be shown to survive those constraints, and this paper does not attempt it. But it does not claim to; within its stated scope, the work does what it sets out to do and does it neatly.

See also

• Calin Galeriu
• Hermann Minkowski
• Lorentz Force
• Coulomb's Law
• Length Contraction
• Time Dilation
• Relativity
• Apeiron