|
In Ladder Paradox (ii) we viewed events from the perspective of the box. When the events are viewed from the arrow's inertial frame, figure 4 of the previous post is essentially just rotated so that the arrow is horizontal and the box is tilted the other way, as shown in figure 7. The box now moves at -0.8c relative to the arrow.
Figure 7 
The 10ft box is projected onto the arrow's coordinates as a 6ft long box, with the rear end clock desynchronized by 8ns, as indicated. Events at the same special location again are on the same vertical lines and simultaneous events are on the same horizontal lines. Space-time movements are as before, illustrated in figure 8 for event C.
Figure 8 
The change in reference frame and the inescapable difference in "what is vertical", translating to "what is viewed as being in the same spatial location at a given time" causes the perception that the box is far too short to hold the arrow. It's also the cause of the effect that events B and C are swapped in temporal order (compared to the box view).
When we carry the argument to its conclusion, the near side of the box will take 18.75ns to reach the tail of the arrow, as shown in figure 9.
Figure 9 
We now have a complete rationale for the timings of events that were shown in Ladder Paradox (i) and it is clear that the timing is a matter of how clocks are synchronized in inertial frames. In a way, it is the different clock synchronizations that cause the apparent length contraction.
To measure the length of a moving object, one must read the x-coordinate of both ends simultaneously and do a subtraction of x-coordinates. Surely, if the definition of simultaneity varies with relative velocity, inertial observers in relative motion will not agree on length measurements.
And that's all there is to it! A more technical treatment of the relativity of simultaneity is available on the website Relativity 4 Engineers. In part 4 of "Paradoxes of Relativity" we will have a quick look at some other interesting ones.
| 
