Minkowski Geometry

Abstract

In this chapter we introduce Minkowski geometry, a geometric formulation of the special theory of relativity. The starting point is the representation of events as points on the plane by means of their space and time coordinates as measured in a particular inertial frame. This is akin to identifying points on the Euclidean plane with pairs of real numbers by means of their Cartesian coordinates relative to a particular system of orthogonal axes. We show that it is possible to define a formula for the distance between two events, called the interval, which is preserved under a change of inertial frame, just as the usual formula for the Euclidean distance between two points is preserved under a change of the system of orthogonal axes. The interval, which physically is just the time measured by a free particle travelling between the two events, is very different from the Euclidean distance: the length of one side of a triangle is always larger than the sum of the lengths of the other two (twin paradox), and lines are the curves with maximum length (generalized twin paradox).