Physico-Mathematical Errors in Einstein’s 1905 Derivation of the Lorentz Transformation

Abstract

Hyoungseok Koh

The Lorentz transformation is obtained in the same form both in Lorentz’s ether-based framework and in Einstein’s ether- free framework. Starting from this paradox, we rigorously distinguish two paradigms of spacetime measurement: the mechanical rest frame, constructed from rigid rods, atomic clocks, and light, and the optical moving system, constructed from atomic clocks and light alone. A rest frame is defined as an inertial frame in which the round-trip light time is the same at every position and in every direction. We further distinguish measured coordinates, read directly at stationary points, from borrowed coordinates, which a moving point takes over from the stationary points it passes. Within this framework the invariance of the speed of light is asymmetric: the rest frame exhibits round-trip invariance, whereas the moving system exhibits only borrowed-speed invariance. On this basis we reconstruct the Lorentz transformation and its inverse, using the absolute time dilation of moving atomic clocks and the dilation of the longitudinal coordinate measured with a light ruler. The reconstruction shows that the time of the moving system is the sum of the atomic-clock dilation effect and a time-correction effect required to keep the borrowed speed of light constant. We then reconstruct, step by step, the derivation in Secs. 1–3 of Einstein’s 1905 paper and identify six key errors: (i) an operational asymmetry that treats rigid rods as equivalent to light rulers; (ii) a domain collapse in which the Galilean auxiliary variable is combined with a light-ray condition, reducing the domain of analysis to a one-dimensional ray; (iii) an unjustified extension of relations valid only on that ray to all of spacetime; (iv) an event mismatch in which two distinct events are combined in the construction of the transverse coordinates; (v) a circular parameter fixing in which the invariance of rigid rods is presupposed when the undetermined coefficient is fixed; and (vi) a category mistake in which the ellipsoidal contraction of a rigid sphere is interpreted as a physical deformation. These results indicate that relativistic spacetime relativism originates from a confusion of measuring instruments and from optical illusion. The Lorentz transformation is valid not because spacetime itself is distorted, but because it encodes inter-point relativity and the geometry of borrowed coordinates referred to an absolute rest frame. We therefore propose an interpretation of spacetime founded on an absolute rest frame and on a measurement mechanics that keeps rigid rods and light rulers distinct.

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