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khashayar

I'm kinda new to relativity. I've worked realy hard to understand the idea but there are still some thought going in my head trying to convince me that it doesn't make sense or I don't understand it. I was wondering if you can help me.

So here is my first problem regarding special relativity. Sorry that it's long but i had to explain the thought experiment.

In the twin paradox the problem is solved by invoving accelaration needed for the traveler to come back and arguing that the traveler is not in one inertia frame of reference at all time. Now let’s consider the following scenario. Suppose we have three identical stopwatches, and a car that can travel at speeds close to speed of light. One of the stop watches is placed at the start point of the track and another at the finish line. The third stop watch is installed on the car. Let’s assume that the road is perfectly straight and flat and that the stop watches are designed such that as soon as they meet they turn each other on if they were off and off if they were on. Also let all the stopwatches to be initially off.

The driver starts at some point far from the start point accelerating and reaches a speed close to that of light before he reaches the start line. As soon as he passes the start line both stopwatches start counting. When he reaches the finish line the stopwatch at the finish line starts counting and the stopwatch on the car stops. Driver slows down and brings the stop watch back to us. Meanwhile we have picked the other two stop watches from the start line and finish line. We stop both of them at the same time. Subtracting the time on the first stopwatch from the second one we have the time that took the car to travel from start to finish line from our point of view. The stopwatch on the car on the other hand should have the time measured from the driver’s point of view for completing the drive.

Since the stop watch installed on the car was not working while the driver was accelerating or slowing down it could not be affected by acceleration, and also we assume that the other two stop watches are brought together at really low speeds with almost zero acceleration. Also there was no communication between the driver and the observer involved to obtain the results.

Now the question is, “Which of the values is greater? The one measured by the stopwatch on the car, or the one obtained from subtracting the values shown by the other two stopwatches.” Suppose that we say since the car was in motion with respect to the observer the value shown by the stopwatch on the car is smaller, but then the driver can claim that he was not in motion and it was the observer and everything else which were moving at that speed in the oposit direction.

I used Lorentz_Einstein trasformations to find the time needed for the trip from two different points of view, but my results were crazy. They were telling me no matter what the time frrom observers poit of view is longer than the time recorded by the car. Did I make a mistake somewhere? Or is it simply because the action of stop watches being turned on and off happenes in the same space coordinate with respect to the car frame of reference?

Can someone help me please?

Thanks