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I constructed a thought experiment, can't figure the answer.

Two long (say $1\ \rm{km}$ long) railway cars are on parallel east-west tracks in a zero gravity environment. Each car has a light source at one end (say the west end) and a device to measure the speed of light at the other end (the east end). The cars are started a long way apart, accelerated toward each other then allowed to coast at the same speed but opposite directions until they are next to each other. Mechanisms on each car touch the other and trigger the light sources. [edit - the idea here is that by mechanisms touching the light sources are triggered at the same location at the same time in the stationary observer's frame of reference] The beams travel to the detectors and Einstein says each will register the speed of light as the constant c. [edit - and here is my error. As pointed out in one answer the one way trip can't be measured, so the detectors on the cars are useless]

I'm OK with that [edit - I was OK with that, not any more]. Here is the part I don't understand.

There is a stationary observer positioned further east. Does he see both beams arrive at the same time? Since both cars are travelling at the same speed relative to the observer but in opposite directions I would expect one to be delayed a bit. If they do arrive at the same time how does one reconcile this with the detectors on the cars saying the speed is constant but they would register when the two cars have moved slightly apart so the beams are not at the same place as they pass the detectors on the cars, so one must 'speed up' to catch up with the other when they reach the stationary observer.

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    $\begingroup$ Your setup is unclear. Where are the sources relative to each other and relative to the observer in the north train when the light source on the north train fires? You are attempting to analyze a "simultaneous events" problem, but you must define reference frames and determine in which frame the events (two light sources firing) are simultaneous. You must also define/determine the positions of the events. Your thought experiment is not fully specified. $\endgroup$ – Bill N Apr 8 at 1:25
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Your question can be reduced to simple “how can one way speed of light can be c in the “stationary observer’s frame” and in the frame of the each “moving traincar””. Please note that one way speed of light cannot be measured even in principle (see article in Wikipedia).

One can only measure speed of light from a light source and back and this speed of light appears to be equal to c.

It is allowed to assume, that One way speed of light can be different from c. Special Relativity employs Einstein synchrony convention that keeps one way speed of light to be equal to the speed of light back and forth.

In a moving frame effects of time dilation and lenght contraction make roundtrip speed of light isotropic (equal to c in all directions). It is experementally confirmed fact.

So, an observer on a moving traincar can only measure that speed of light from a ligtsource an back is c.

He is free to assume that speed of light from a lightsource to opposite side of the traincar is c, or for example c/2 or even infinitely great (see Reichenbach’s synchrony convention in the Stanford encyclopedia of phylosophy)

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Yes he would see both beams at the same time and so would an observer positioned further west if the sources pointed both ways. The propagating light and the observer are completely separate from the light sources, detectors or anything on the trains. Unless the observers were moving at relative speeds compared to the sources they wouldn't notice anything.

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