MPEJ Volume 13, No. 2, 8 pp.
Received: Jul 11, 2006. Revised: Jan 28, 2007. Accepted: Mar 28, 2007.
N.Goldstein
Inertiality Implies the Lorentz Group
ABSTRACT: 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, R(n+1), with coordinates
x0,x1,...,xn. Using the notation t = x0 and y = (x1,...,xn), the slope of a
line in R(n+1) is defined to be |\delta y/\delta t|, computed from any two
points on the line. The slope is non-negative and possibly infinite. A line in
R(n+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 R(n+1) to
R(n+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 Poincare group and the dilations.
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