A heuristic derivation of Minkowski distance and Lorentz transformation

Abstract

Students learn new abstract concepts best when these concepts are connected through a well-designed analogy, to familiar ideas. Since the concept of the relativistic spacetime distance is highly abstract, it would be desirable to connect it to the familiar Euclidean distance, but present the latter in such a way that it makes a transparent contact with the former. Starting with some intuitive and ‘obvious’ assumptions concerning distance in one dimension, we ‘derive’ the two-dimensional Euclidean distance between two points in terms of their coordinates. Then, assuming the invariance of this distance, we deduce the (familiar) two-dimensional orthogonal coordinate transformation. We present the derivation in such a way that the transition to spacetime becomes ‘self-evident.’ Thus, following exactly the same procedure, we derive the Minkowskian distance and the corresponding transformation that respects the invariance of that distance, i.e., the Lorentz transformation.