Two Relativistic Interpretations

Abstract

Aphysical theory is comprised of two components: a mathematical formalism (a set of equations and a set of calculational rules for making predictions that can be compared with experiment) and a physical interpretation (what the theory tells us about the underlying structure of phenomena, that is to say, an ontology).1 Thus, a single formalism with two different interpretations counts as two theories. Einstein’s original formulation of his Special Theory was mathematically algebraic in nature and metaphysically a space and time theory. By this latter characterization, I mean that Einstein presupposed an ontology of spatial objects which endure through time, howbeit that no single, universal time exists. But in 1908 the German mathematician Hermann Minkowski proposed a formulation of SR which was strikingly different mathematically and metaphysically. Minkowski proposed that space and time be united in a four-dimensional mathematical space, three of whose dimensions represent physical space and the fourth time.2 In this manifold, relativity theory and the Lorentz transformations can be exhibited with great clarity. Events in spacetime are specified by giving their four coordinates, and although the temporal and spatial distances between two specified events will differ from one coordinate system to another (relativity of simultaneity and length), nevertheless the composite spacetime interval between events is absolute. Letting cis represent the spacetime interval, and dt represent the temporal distance, and dx, dy, dz the spatial distance between the events, Minkowski spacetime has a metric of the form ds 2 = dx 2 + dy 2 + dz 2 − dt 2,or alternatively expressed, ds 2 = dt 2 − dx 2 − dy 2 − dz 2.The notion of spacetime interval between two events can be understood as an extension of spatial interval. In two dimensions x and y, we can calculate the spatial interval between two points by means of the Pythagorean Theorem (Figure 5.1).