A Covariant Lorentz Transformation Analysis of Einstein's Train and the Resolution of Relative Simultaneity

This manuscript re‑examines the standard interpretation of relative simultaneity (RS) in Einstein’s train thought experiment by analysing the Lorentz transformation in its fully covariant form. Using the matrix representation of the Lorentz transformation applied to homogeneous coordinates, the study shows that the fully covariant Lorentz transformation (FCLT) yields consistent equations of motion for light and for arbitrary rectilinear motion, and removes the mixed‑coordinate artefacts that generate RS in textbook treatments. When applied to Einstein’s train scenario, the FCLT predicts that simultaneous emissions in the embankment frame remain simultaneous in the train frame, and that the unique point‑coincidence of the two light signals is invariant. The non‑simultaneous passage of the signals through the train midpoint is shown to be a purely kinematic consequence of the midpoint’s motion rather than an RS effect. The analysis is independently cross‑validated using the inverse Lorentz transformation (ILT) and the Tangherlini transformation (TT), a preferred‑frame relativistic framework derived from general‑relativistic considerations. Despite differing synchronisation conventions, all three approaches yield the same invariant point‑coincidence. The results clarify the internal structure of special relativity and identify how the invariant temporal ordering encoded in the Lorentz transformation coexists with the synchronisation offsets introduced by Einstein, revealing a deeper unity between covariant and standard formulations.