Solving the Mystery of Relativistic Length

[Aeneas]

I have been trying to read Einstein's small book on relativity and to work out the results for myself, but I seem to be getting to the wrong conclusions. In Chapter 12 he says "The rigid rod is thus shorter when in motion than when at rest, and the more quickly it is moving, the shorter the rod." For a train moving away from the observer on the emankment, that seems to be right, but is the train is coming towards you, it seems to me it should be longer. Imagine you are standing by the track as the front of a train, say one light year long relative to a man on the train measuring it, and traveling at 1/3 the speed of light reaches you. The point on the embankment where the back of the train would be "simultaneously", relative to you, would be much further than one light year away from you because the light would take a while to arrive.

In fact I make it that the length would be 1 + 1/3 + 1/9 + 1/27...= 1.5 light years.
If the speed of the train was 1/N the speed of light the length would be N/N-1 times its length at rest.

If the train was going away from you it would be shorter, N/N+1 times its length at rest, in this particular case, 1 - 1/3 + 1/9 - 1/27...

Obviously these are the wrong results. Have I 1. miscalculated something or 2. misunderstood what is meant by relativistic length or 3. is this result in fact directional, putting -v for v if it is coming towards you?

Can someone please sort me out? Simple explanation please! Many thanks in anticipation.

 

[JesseM]

You're misunderstanding the nature of length contraction, it's not based on what you see using your eyes, which is affected by different light-signal delays from sections at different distances from you; rather, length contraction is based on the difference between the position of the front and back of the object at simultaneous moments in your frame. One way to find this would be to have an array of clocks which were at rest and synchronized in your frame, so that if the back of the object was passing next to a clock 12 meters from you when that clock read 49 seconds, and the front of the object was passing next to a clock 6 meters from you when that clock read 49 seconds, you could conclude the object was 6 meters in length. Another way of doing it would be to note when the light from the front and back passing by markers at different distances from you reached your eyes, and then factor out the light signal delays--if I see the front of the object passing a marker 5 light years away in 2020, and then I see the back of the object passing a marker 10 light years away in 2025, by subtracting the light-delays I can conclude that in my frame both these events "really" happened simultaneously in 2015, and thus the object must be 5 light-years long in my frame.

 

[Aeneas]

Many thanks JesseM