The Railway Problem
By Tom Brown
When doing a post-graduate Topology course I asked another student for advice, “I don't understand this what does it say?” that was a seemingly obscure especially strange definition. He had a good look and said “I think it says what it says” and I thought “Yes must be”. I think this is what relativity theory is, you never understand it you just get used to it.
It would be a good thing in anycase to first give a discussion and solution to my exercise in ordinary experience not in special relativity. I do believe the problem as given is well-posed, and it can be solved in both cases.
The first is easy given one very simple observation.
The description of a physical model and mathematical abstraction follows, the station is the origin of our inertial reference frame, distance is measured on the train tracks' counting “rungs” like a ladder, there is a clock on the train and another “on” or “at” the sound signal and a large bell and clock at the station, the locomotive has a very loud whistle. See the sketch on my scans.
Given is the constant u the speed of the locomotive on the track and s the distance to the train measured at the station.
As in ordinary everyday experience, at low speeds firstly we consider the simple classical description, i.e. Galilean transformations. One can add up the infinite geometric series' but it could be complicated.
Our solution is elegant the sum of distances is found indirectly. The problem in the end turns out very simple provided noticing one elementary fact.
A signal travels with the speed of sound v, much faster than the train. Suppose we have the train nearing the railway station from a initial distance S , taken where the clock measures zero. The locomotive whistle's scream travels and a gong is struck when the signal reaches the station the bell replying again with a loud sound signal reaching the train. Subsequently a whistle back to the station and then back again and thus back and forth.
The question is, how far does the signal travel altogether? Or what is the total sum of distances travelled, inventing notation, Sum[s].
For the case of relativity the model is identical just very large speeds and different transformations instead. Trains generally don't travel that fast.
We now realize that if the time the train itself takes to arrive is T , the signal must also travel for exactly the same time duration since it also has to stop once the locomotive reaches the station and that we don't have to add individually all the distances s.
For the whole time duration the signal moves at its speed v and total distance travelled added (zig-zag) at the same speed until stopping is the same total time T, which we may calculate very simply. Note that for this “everyday problem” the times are all the same i.e. T
It is to say S = uT and our answer, the total distance of the (sound) signal travelled is now,
Sum[s] = vT = v.(S/u) = v/u S. This is definitely correct it was in great detail too but I think it's Ok. My diagram actually already makes clear most of it.
It has always been my dream to drive a steam locomotive, head out the window in the wind all I want is the throttle and the whistle! And do I love the song of a big steam engine.