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).

Photons are particles that travel in straight lines at the speed of light; if a photon passes through the origin, then its position in three-dimensional space is (x, y, z)=(a1t, a2t, a3t), where . Show that a photon lies on a “cone” in ℝ4 (i.e., that x2 +y2+ z2= kt2 for some constant k). Show that any particle traveling at a constant speed v less than c through the origin lies “inside” the light cone in the sense that . Why would the expression in (2.1.12) prohibit particles moving with a speed greater than that of light?