The Russian-German mathematician Hermann Minkowski (1864−1909) developed a four-dimensiona...

The Russian-German mathematician Hermann Minkowski (1864−1909) developed a four-dimensional geometry in which time is treated essentially as a fourth spatial coordinate. (Minkowski was a teacher of Albert Einstein, whose theories of special and general relativity make use of this geometry.) Writing points in ℝ4as (t, x, y, z), define the “distance” (i.e., time experienced in traveling at a uniform speed) between “events” x1 = (t1, x1,y1, z1) and x2 = (t2, x2,y2, z2) to be

where c is the speed of light measured in appropriate units (e.g., c = 3.0 ×108 m/s).

Find the Minkowski distance between the points x1 = (0, 0, 0, 0) and x2 = (2, 0, 0, 0). (The segment between these two points represents a particle that is stationary for two time units.) Now, let x3 = (1, 10−3c, 10−3c, 10−3c) and calculate dM(x1, x3) and dM(x3, x2). How do these results compare with what happens when the Euclidean distance is used? This is a simple instance of what is called time dilation.