only provided our methods of measuring a velocity, and those of the observer in the Galilean frame, were identical; that is, provided our measurements of space and time were identical. This was always tacitly assumed to be the case by classical science; but we know that in Einstein’s theory the identity of our standards of measurement no longer holds, since such a state of affairs would be incompatible with the invariance of the velocity of light.
We may therefore assimilate the conditions surrounding the various space and time measurements in Einstein’s theory to those which would endure in classical science if by reason of some law of nature it were impossible for the observer in the train to fire a bullet in the precise direction of the train’s motion. In other words, in Einstein’s theory, velocities lying in the same direction add up like algebraical numbers, provided the velocities we propose to add and subtract are computed under exactly the same conditions, that is, by the same Galilean observer. It is when one observer computes one velocity and a second observer, in relative motion, computes the second velocity that the classical law of addition is at fault.
Very similar to this problem of the addition of velocities is that of simultaneity. Thus, Einstein’s theory implies that if two events
and
occur simultaneously for one observer, John, and if in the opinion of another observer, Peter, in motion with respect to John, the event
and another event
