Special Relativity from Measuring Rods

Abstract

The mathematical structures associated with a space-time theory, such as the special theory of relativity (SRT) — or the general theory (GRT) for that matter — are numerous and interrelated in complex ways.¹ One may start their analysis with the concept of a point set, the elements of which are identified with events in space-time.² Imposing a continuity structure on this set leads to the concept of space-time as a four-dimensional topological manifold. Restriction to a differentiable structure then leads to the concept of space-time as a differentiable manifold. Various additional mathematical structures may now be introduced on this manifold: projective, affine, conformal and pseudo-metrical (a metrical structure with Minkowski signature). Each of these mathematical structures is closely associated with the behavior of some idealized physical entity in space-time. The projective structure is associated with the trajectories of structureless free test particles. If each particle carries some intrinsic measure of duration along its trajectory, it reflects the affine structure. The conformal structure is associated with the wave fronts of massless fields, such as the electromagnetic. A pseudo-metrical structure with Minkowski signature implies the existence of two fundamentally distinct types of interval which cannot be transformed into one another by any operation of the symmetry group defining the geometry (the inhomo- geneous Lorentz group for SRT).³ These two distinct types of interval are called spacelike and timelike, and physically quite distinct entities — measuring rods and clocks — are associated with their respective measurement.