- The paper derives a general one-parameter family of inertial-frame transformations from homogeneity, isotropy, relativity, and consistency, without initially assuming light-speed invariance.
- It shows algebraically that transverse cross-terms vanish, the transverse scale equals one, and the velocity-composition law is w = (u + v)/(1 − uv/R), while continuity in velocity follows from the resulting formulas.
- The light postulate eliminates the Galilean branch and selects R = −c², yielding the Lorentz transformation, invariant light speed, and preservation of subluminal motion under the stated assumptions.
Overview and motivation
This paper presents an elementary, expository derivation of the Lorentz transformation organized around symmetry principles, with the light postulate deferred to the final step. The work situates itself squarely in the Ignatowski tradition [2], in which the general inertial-frame transformation family is derived from the Principle of Relativity, spacetime homogeneity, and isotropy alone, with the invariant speed entering only at the end as an empirical input. The stated aims are modest: not a new kinematics, but a derivation that (i) remains at an elementary algebraic level, (ii) makes explicit steps that are usually compressed—linearity, the elimination of transverse cross-terms, and the algebraic determination of the velocity-composition law—and (iii) clarifies a methodological point about the equivalence of regularity assumptions used to pass from additivity to linearity.
The paper's central structural innovation is a staged use of transformation consistency. The identity-and-inverse part of consistency constrains the transformation coefficients, while the composition part yields a one-parameter family of transformations characterized by a universal constant R with dimensions of velocity squared. The light postulate then selects the physically relevant branch by fixing R=−c2, simultaneously recovering the Lorentz transformation and identifying c as the maximal speed.
Physical assumptions and the linearity step
The derivation rests on five assumptions: spacetime homogeneity, isotropy, and continuity; the Principle of Relativity; the absence of a distinguished inertial frame; transformation consistency (identity, inverses, closure under composition); and the constancy of the speed of light, used only in Section 2.4.
The linearity argument follows Berzi and Gorini [3]. Homogeneity implies that the image increment T(e+h,v)−T(e,v) is independent of the base event e, defining a function G(h,v) satisfying the additive Cauchy equation. Rational homogeneity follows by induction; continuity of T in the event coordinates then extends rational to real homogeneity, excluding pathological additive solutions. In the standard configuration the transformation is therefore purely linear: e′=F(v)e.
A methodological point the paper emphasizes is that, once homogeneity has reduced the problem to an additive functional equation, the standard regularity routes—continuity at a point, differentiability, boundedness on an interval—are equivalent paths to the same linear class. The paper accordingly assumes only continuity in the event coordinates; notably, continuity in the velocity parameter is not assumed but recovered a posteriori from the explicit coefficient formulas, which turn out to be elementary functions of v. This is a genuine, if modest, sharpening of the usual hypothesis structure.
Symmetry constraints and the one-parameter family
The paper takes a definite physical position on a contested issue: the inverse transformation is written with parameter −v not as a technical theorem derived from a velocity-labeling map, but as the direct kinematical expression of inertial-frame equivalence. The author explicitly rejects the view (associated with Moylan [7]) that reciprocal descriptions of the same relative motion could carry unequal speed magnitudes, arguing that any such discrepancy would itself furnish a criterion for distinguishing frames, contradicting the no-distinguished-frame assumption.
An explicit rotational comparison—rotating both frames by R=−c20 about the common R=−c21-axis and equating coefficients—eliminates all cross-terms between the longitudinal and transverse sectors, yielding R=−c22, R=−c23, and R=−c24. A geometric argument then fixes the transverse scale factor completely: R=−c25 (hence R=−c26), derived without any continuity assumption on R=−c27. This is a notable expository improvement, since many treatments either assume transverse unity or invoke smoothness in R=−c28.
In the longitudinal sector, isotropy under reversal of the R=−c29-axis gives c0 even and c1 odd in c2; the requirement that c3's origin map to c4 gives c5; and inverse consistency gives c6 and c7, with c8 from axis alignment. The remaining inverse-consistency constraint is c9.
Imposing composition closure across three frames produces the cross-relation T(e+h,v)−T(e,v)0, which splits the analysis into two branches. If T(e+h,v)−T(e,v)1, one obtains T(e+h,v)−T(e,v)2—the Galilean transformation, retained as a live possibility until the light postulate is imposed. On the non-Galilean branch, the ratio T(e+h,v)−T(e,v)3 must be a universal constant T(e+h,v)−T(e,v)4, giving
T(e+h,v)−T(e,v)5
with reality of the coefficients requiring T(e+h,v)−T(e,v)6. The composition law follows purely algebraically:
T(e+h,v)−T(e,v)7
and the generalized transformation is T(e+h,v)−T(e,v)8, T(e+h,v)−T(e,v)9 with e0. Continuity of the composition law in e1 is again a consequence rather than an assumption.
Selection by the light postulate and maximality of e2
The light postulate first eliminates the Galilean branch: under e3, e4, invariance of a signal speed e5 would require e6, impossible for e7. On the non-Galilean branch, applying the transformation to a collinear worldline e8 gives the transformed speed e9; setting G(h,v)0 forces G(h,v)1.
The paper is explicit that G(h,v)2 is a selection, not a uniqueness result. The case G(h,v)3 is a singular degeneration rather than a distinct branch. The case G(h,v)4 yields, via the parametrization G(h,v)5, a trigonometric angle-addition composition law G(h,v)6, which the paper dismisses on the grounds that the tangent's poles preclude a natural global real-velocity domain and that it is not selected by the light postulate. This is a defensible but somewhat brief treatment; a reader might ask whether the G(h,v)7 branch can be excluded on more fundamental grounds than domain convenience. The Galilean branch is retrospectively recovered as the limit G(h,v)8.
With G(h,v)9, the standard Lorentz transformation follows, real and finite precisely for T0. Two further results are established. First, the one-dimensional light postulate is extended to three dimensions: substituting the full velocity-transformation formulas shows that a light ray with T1 in T2 satisfies the same relation in T3 for arbitrary propagation direction. Second, maximality follows from the identity T4: subluminal speeds remain subluminal in every inertial frame, and T5 implies T6. The paper notes the caveat that this argument presupposes T7, i.e., that the relative velocity of the frames is itself subluminal—an assumption consistent with, but not derived within, the derivation.
The paper closes the derivation by recording the group structure: closure of T8 under the relativistic composition law, together with identity, inverses, and associativity, establishes a one-parameter Lie group acting linearly on spacetime.
Limitations and open questions
The paper is explicitly expository, and its contributions are organizational and pedagogical rather than substantive; the mathematical content is close to Berzi–Gorini [3] and Pal [5]. Several limitations bear on the results. The physical reading of velocity reciprocity—writing the inverse with T9 as a direct expression of frame equivalence—is asserted on physical grounds and contrasted with the technical treatment of Moylan [7], but the paper does not fully engage with the possibility that reciprocity requires independent derivation; it adopts a position rather than settling the dispute. The exclusion of the e′=F(v)e0 branch rests on domain considerations (poles of the tangent) and non-selection by the light postulate, leaving open whether a sharper physical argument exists. The maximality proof assumes frame velocities are subluminal, and the derivation is confined to the standard configuration, with the general case (arbitrary orientations and offsets) only implicitly covered. No new experimental content is claimed.
Conclusion
The paper delivers a clean, staged, symmetry-first derivation in which the Lorentz transformation emerges as one member of a one-parameter family labeled by e′=F(v)e1, with the light postulate doing only the final work of selecting e′=F(v)e2. Its clearest contributions are pedagogical and methodological: the demonstration that transverse unity and the velocity-composition law follow without continuity assumptions in e′=F(v)e3; the clarification that continuity, differentiability, and boundedness are interchangeable regularity routes once homogeneity yields additivity; and the explicit two-branch treatment that keeps the Galilean case in view until the empirical input arrives. For researchers concerned with the logical economy of special relativity's postulates, the paper offers a careful, self-contained account of exactly which assumptions carry which load.