to indicate the difference between the x-coordinates of two points, we employ the more compact expression dx; if for the benefit of readers who have a little algebra but no analysis we state explicitly that this expression is a single symbol for a single quantity, and has nothing to do with any product of two quantities d and x; and if we extend this notation to all our coordinates: then it is clear from previous essays that the distance S between two points in a plane referred to a rectilinear system OX, OY, is given by the simple equation
. Einstein and Minkowski show that the value for the separation interval
, the analogue of S, referred to an inertial system is given by the equation
which is seen to be a modified extension to four dimensions of the equation for S. We must measure t in the same units as x, y, z. By taking the constant velocity of light (300,000 kilometres per second) as unit velocity, we can measure in length or time indiscriminately.[2]
We will analyse briefly this equation as it epitomizes the Special Theory of Relativity. If the World-Frame had been Euclidean the equation would have been
but this would not satisfy the “transformation equations” which resulted from the Special Theory. These transformation equations arose directly from a reconciliation between two observed facts; (a) the observed agreement of all natural phenomena with the “Restricted Principle of Relativity”—a principle which shows that absolute rectilinear motion cannot be established—(as regards mechanics this was recognized by Newton; the Michelson-Morley and other experiments showed this principle also applied to optical and electro-dynamical phenomena); and (b) the observed disagreement of optical and electro-dynamical phenomena (notably the constancy of light velocity) with the laws of dynamics as given by classical mechanics, e.g., in regard to the compounding of relative velocities. Einstein effected this reconciliation by detecting a flaw in classical mechanics. He showed that by regarding space and time measurements as relative to the observer—not absolute as Newton defined them—there was nothing incompatible between the Principle of Relativity and the laws of dynamics so modified. Newton’s definitions were founded on conception. Einstein’s recognition of the relativity of space and time is based on observation.