Let us now state the precise nature of those modifications in lengths and durations which will arise as a result of relative motion. The Einstein-Lorentz equations show that a rigid material object, say a yardstick, which, when at rest in our frame, would measure out as one yard in length, would appear to contract more and more in value in the direction of its relative motion (FitzGerald contraction) as this relative velocity increased. Could the relative velocity reach that of light, the rod would become completely flattened down to nothing. [Pg 224] Likewise, an observer fixed to a clock would note that the durations between the successive second-beats of the clock were truly seconds. But with an increase in the relative motion existing between clock and observer, these beats would measure out as lengthened; and for a relative speed equal to that of light an infinite duration would separate two successive beats. It must be noted that this lengthening in the durations defined by the tickings of the clock must not be attributed to the fact that the observer is moving farther and farther from the clock and therefore that longer and longer periods of time are necessary for him to take cognisance of these tickings. The results mentioned are those which would be found to hold after all the necessary corrections had been performed. It is solely the relative motion of the observer and the clock which is involved, and not the variations in their mutual distance. Our results would be exactly the same whether we were approaching or receding from the clock, so long as the same magnitude of our relative motion was maintained. It is customary to designate by the names proper length and proper duration the length and duration obtained by an observer at rest with respect to the rod or clock. Beginners appear to have considerable difficulty in convincing themselves that all these durations, whether proper or elongated, are equally real. We have already explained in what measure they are so, but it may be helpful to give the following additional illustration: Suppose we are suffering from a headache and that this headache lasts an hour when timed by our watch. This will be the proper duration of the headache. Now we might be inclined to claim that inasmuch as we alone are suffering from the headache, we alone are in a position to determine its duration; and when some rapidly moving observer tells us that we have held our watch in our hand for one week and that our headache has therefore lasted a week, we might be tempted to tell him to mind his own business. But it must be realised that although the headache is ours and ours alone, and nobody else can feel it for us, yet the duration occupied by the headache is common property, and any other observer is just as much entitled to live it and to measure it as we ourselves are. [Pg 225] CHAPTER XXII AND THE TRIP TO THE STAR