Spacetime and Euclidean Geometry

Using only the principle of relativity and Euclidean geometry we show in this pedagogical article that the square of proper time or length in a two-dimensional spacetime diagram is proportional to the Euclidean area of the corresponding causal domain. We use this relation to derive the Minkowski line element by two geometric proofs of the spacetime Pythagoras theorem. Introduction Spacetime diagrams are helpful for understanding relativity since they focus attention on the invariant relations between events, light rays, observers, etc. rather than on coordinate dependent quantities. An inherent limitation of such diagrams is that, in general, the Euclidean lengths of lines in the diagram do not correspond to proper time or proper length in spacetime. In this pedagogical article we use the principle of relativity, together with Euclidean geometry, to show that nevertheless the square of proper time or length of a line segment is proportional to the Euclidean area of the corresponding causal domain. This observation allows visual interpretation of relativistic effects, such as time dilation and the twin effect. We use this relation between Minkowski interval and Euclidean area to derive the Minkowski line element by proving the spacetime Pythagoras theorem. Minkowski space and Euclidean space In a two-dimensional spacetime diagram a spacetime, i.e. a Minkowski space, is represented on a Euclidean plane. This is possible since, like the points in the Euclidean plane, the events in spacetime can be labeled by pairs of real numbers, for example time and space coordinates. What other properties of Minkowski space can this mapping faithfully reproduce? This article is dedicated to Michael P. Ryan on the occasion of his sixtieth birthday. Mike’s passion for, and deft practice of, both geometry and pedagogy is legendary at Maryland. We are pleased with this opportunity to present our pedagogical effort to elucidate the geometry of Minkowski spacetime, the most homogeneous of cosmologies.