Inertiality Implies the Lorentz Group

In his seminal paper of 1905, Einstein derives the Lorentz group as being the coordinate transformations of Special Relativity, under the main assumption that all inertial frames are equivalent. In that paper, Einstein also assumes the coordinate transformations are linear. Since then, other investigators have weakened and varied the linearity assumption. In the present paper, we retain only the inertiality assumption, and do not even assume that the coordinate transformations are continuous. Linearity is deduced. Our result is described in the affine space, Rn+1, with coordinates x0,x1, . . . ,xn . Using the notation t = x0 and y = (x1, . . . ,xn), the slope of a line in Rn+1 is defined to be |∆y/∆t|, computed from any two points on the line. The slope is non-negative and possibly infinite. A line in Rn+1 is said to be time-like if the slope of the line is strictly less than 1. Since inertial frames agree on who is inertial, coordinate transformations must carry time-like lines to time-like lines. A bijection from Rn+1 to Rn+1 is said to be time-like if it maps any time-like line onto another time-like line. The bijection is not assumed to be continuous. This paper proves that a time-like bijection is continuous (in fact, affine linear). The bijection is said to be strictly time-like if both it and its inverse are time-like. It is elementary to deduce that the strictly time-like bijections form the group generated by the extended Poincaré group and the dilations. Subject Classification: Primary 51N10, 51P05, 83A05