The Physical and Mathematical Basis of the Lorentz Transformations — A Consistency Analysis

This study presents a consistency analysis of the algebraic, geometric and physical relations that form the basis of the Lorentz transformations (LT) between two inertial reference frames in uniform relative motion. The algebraic analysis shows that homogeneity, continuity and the condition of relative motion reduce the general affine transformation to x' = A(x-vt), t' = Cx + At. The geometric analysis of the same configuration (Figure 1) gives d(O',P)_{(F)} = x - vt. With x = ct for the light path from O in F and d(O',P)_{(F)} = (c-v)t for the distance O'–wavefront in F, the identification d(O',P)_{(F)} ≡ x' — i.e. that O'P is the same physical distance in both systems measured with the same units — gives A=1, t'=t, C=0, i.e. x' = x-vt, t' = t (Galilean). As an independent test, Bondi's k-calculus is analyzed. The comparison between the k-calculus determination of the distance OP' and the geometric determination of the same distance leads to the condition 1-v^2 = 1+v^2, which is satisfied only for v=0. The same inconsistency appears in the determination of O'P. Either the Euclidean identity OP = OO' + O'P for the same distance holds, or Lorentz holds, but not both. The paper clarifies what is meant by "distance" and "unit" in two inertial frames, the meaning of relative velocity v = d(O,O')_F / t = d(O,O')_{F'} / t', and why introducing a scale factor λ between F and F' is mathematically identical to postulating γ = 1/λ.