AlMetis said:
I want to construct an experiment similar to that described by Einstein in his thought experiment:
https://en.wikisource.org/wiki/Rela..._I#Section_9_-_The_Relativity_of_Simultaneity
How would I design it to ensure the light from the flashes at the back (A) and front (B) of the platform arrive at the center of the platform simultaneously?
Here are my thoughts.
I set a mirror at A and B and flash a light between them. I note which reflection arrives back at the source first (let’s say it is A) and continue this process while moving the source toward the other (B) until the reflections return to the source simultaneously.
Have I found the center of the platform?
It'd be helpful if you could draw a space-time diagram, but of course it'd take some work.
Also, you used a different reference than the one used. The reference I used is
https://www.bartleby.com/lit-hub/re...ral-theory/ix-the-relativity-of-simultaneity/. Yours doesn't seem to have a diagram - mine does.
To clarify a few things - I'm assuming your setup is described in the platform frame. When you say A and B are "mirrors" , I'm assuming that A and B are to be be regarded as worldlines, at rest with respect to the platform. However, in Einstein's thought experiment, A and B are specific events on these worldlines.
Give this, you've (almost) defined three worldlines, A, B, and C in the frame of the platform such that worldline C is at the midpoint of wordlines A and B in the frame of rerference of the embankment. This is a good start but not sufficient yet to replicate Einstien's thought experiment.
I say "almost", because you haven't addressed the spacing of the wordlines. That'd be the next task. You'd need to arrange the proper separation between worldlines A and B.
This shouldn't be too hard - it'll be the Lorentz contracted length of the train, but of course you'll need to know the proper length of the train (in it's own frame), and the velocity of the train relative to the station, so that you can compute the proper Lorentz contraction. You're also assuming that relativity is correct and setting up the experiment accorfdingly, so you'll know if it fails that relativity is false (or you made some error in the setup).
Your final task will be to work out some arrangment for the specific events along worldlines A and B which generate the lightning flashes, which involves not only their location, but their timing, which I assume you are all doing with devices conceptually in the embankment frame. The timing is, in fact, the main point of the experiment.
There are a number of ways you might attempt to do this with sensors and cables, and a lot of attention to delays. Conceptually, I'd suggest some high-speed sensor at the "right place" along the track, something on the train to activate the sensor (possibly a magnet at the center of the train running over a current loop or something similar), and a cable or signaling system that transmits the signal from the sensor first to C, then splits it and retransmits it to A and B. Then the events A and B are generated on worldlines A and B when the signal from the sensor reaches them, and you carefully arrange things so that the total delays in the sensor path match the time it takes the train to move into position.
[add-afterthought]
I'll attempt to find the position of the sensor, making some idealizations as to lack of any delay in the sensor or flash generators, and that cables transmit signals at the speed of light (there would probably be some dielectric effects that would slow the signals slightly in reality, but I don't want to be bothered with this level of detail). There's some chance I've made an error, this is my own work, not from a textbook.
The equation for the location of the sensor, assuming there aren't any other systematic delays in the sensors or the flash generators, would be as follows. Let X be the distance from the sensor to the midpoint of the platform, C. Here A, B, and C are conceptually locations in space in the frame of reference of the embankment, i.e. A, B, and C are worldlines on a space-time diagram. Let L be the distance / spacing between A and B. The center of the train must reach C at the same instant as the two flashes of light. The time it takes the midpooint of the train to move to the embankment is ##\frac{X}{\beta c}##. This must equal the time it takes for the signal to propagate from the sensor, to C, to to A, and then back to C, equal to the time it takes for the signal to go from the train, to C, to B, and then back to C. This is equal to ##\frac{X+L}{c}##. We then can solve for X, finding that that X = ##L \frac{\beta}{1-\beta}##.
Note that in this formulation, L is not the proper length of the train, due to length contraction, but is rather the contracted length of the train as seen in the embankment frame.