. It is not that the relativistic clocks differ from the classical ones; we are always considering our ordinary chronometers. The novelty is solely due to the fact that, thanks to ultra-precise experiment, our understanding of the behaviour of clocks and rods is more accurate than it used to be.
From all this, we may infer that the space-time distances from
to any of the points on the hyperbola represent congruent space-time distances in the space-time geometry. We may also note that, were space-time Euclidean instead of semi-Euclidean, the locus of points defined by the hyperbola would be given by a circle with
as its centre.
And now let us consider the world-line of a light ray leaving
at a time zero. We see that this world-line (