B3 Tertio: Moving Yardsticks Are Shorter
The inertial frame B (prime, red) moves with constant speed of v along the x axis of the inertial frame A (non-prime, black):
The STR should be consistent in the following sense: both A and B make the same statements about which time intervals or lengths in A and B are measured. They will not measure the same values, but they both can figure out, what the other one measured, and agree about these values. We draw from this fact the following important conclusion: If B moves for A with velocity v in the positive x-direction, then A moves for B with the velocity -v in the x'-direction! In addition to the speed of light c both also have the amount of their relative velocity in common. Most authors assume that this is self-evident. Is it really?
We consider what alternatives might be possible: Assume that B measures a relative velocity u of the two systems where |u| < |v|. In this case A also knows that B measures a smaller relative velocity. If space is isotropic (looks the same in all directions) and the STR is consistent as described above, then the situation is perfectly symmetrical. In this case B will correspondingly state that A measures a smaller relative velocity. Thus we have a contradiction: it follows for the magnitudes of the relative velocities that v < u < v, which is impossible. Therefore B can measure neither a smaller nor larger relative speed of the two systems than A, it must be that u = - v and |u| = |v|. Here again (remember B1 !) we need the postulate of isotropy!
System A has 2 synchronized clocks at a distance ∆x from each other. B moves with relative velocity v over this distance and measures with its clock the time ∆t', which elapses between the meetings with the two clocks of A. B uses ∆t' and v to determine the distance between the two clocks in system A:
∆x' = v·∆t'.
But what does A observe? A measures ∆t between the two clock meetings and the distance ∆x of its clocks and determines the speed v of B: v = ∆x/∆t.
The value of v is the same for A and B, and so we have the following equation
using the result of the last section. The distance ∆x which is at rest in the system A appears in system B to be shortened by the same factor we have already encountered.
We have preferentially treated the x-direction (which corresponds to the x'-direction and the direction of the relative velocity); actually we know only that lengths of moving objects in the direction of relative motion are shortened. What is the behavior in perpendicular directions?
Consider Einstein’s train, moving with speed v = 0.6·c in the x-direction on a long straight railroad line. If the train has the length 300 m in its own reference system, then we will measure it at a shortened length of 240 m (do the math!). Did the train also become narrower? If so then it would have fallen between the rails (which are at rest) when it reached a certain speed. That would, in principle, be possible. However, if moving objects would contract themselves perpendicular to the direction of motion, then from the point of view of the travelers in the train it would mean that the separation between the moving rail tracks became smaller! And the theory would require the track width to be too large and too small at the same time - impossible! Therefore we conclude that there is no ‘lateral contraction’.
Moving objects appear shortened in their direction of motion, that is, the length of an object measured at rest is always the longest. We call this the principle of maximum proper length. (Note: curiously google comes up empty in a search for this expression whereas the principle of maximum proper time is well known.) Perpendicular to the direction of the relative motion the measured values of all observers agree. The following formulas apply:
In addition the value of the radical agrees for both reference systems, since in both the square of the relative velocity has the same value.
By the way, the absence of lateral contraction is quite significant for our argument in B2! Otherwise the path of the light perpendicular to v would not be of equal length in both systems, and we would not have a unique length of the vertical leg of the triangle. The perpendicularly standing light clock only becomes more narrow and not shorter or longer. We were indeed fortunate…
Thus Epstein's small fleet, if at first it is at rest and then speeds past an observer at very high speed, appears so (Illustrations [15-39f]):