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The math is quite simple; the problem is that you are using bits and pieces from one theory and another bits from the reality which the same theory is explicitly excluding. The length contraction was a postulate in SR (never confirmed experimentally; some people still argue that it is not physical phenomenon) which is applicable to a moving body relative to another body which is "at rest". It is not possible to determine which body is in motion and which one is at rest.
And there is no possibility of discovering if the whole system is in motion, so the only possibility to determine length contraction is to make arbitrary one frame at rest relative to another which is constant motion.
Your idea that both the rod and the lasers may travel at some constant speed (0.86c) should not be detectable by any theoretical or practical means. However the fact the rod traveling in one direction is shortened and lengthened in opposite direction can be detected by measuring by one clock the time it takes for both edges of the rod to pass the laser .
Let's say the rod is traveling in the direction of the system (which is moving with 0.86c). First front edge is passing A, then trailing edge. Because the rod is lengthened, it should take a bit more time in comparison to the rod of original length. Now we transport the rod in opposite direction. The rod is shortened, so the time it takes for both ends to pass point A will be a bit shorter. Carefully measuring the time difference (while monitoring rigorously the speed of the rod) we could calculate the absolute speed of the system.
This is definitely invalidated by the theory which introduced length contraction in first place.
If the rod has less contraction on it than the clocks, the leading end will start clock B before the trailing end starts clock A. If the rod has more contraction on it than the clocks, the trailing end will start clock A before the leading end starts clock B. In both cases, the trailing clock is started first, giving it extra time to tick up a high score before light from the other clock can reach it. The clocks will count up the same number of ticks as if the system was stationary and you will generate a null result from the experiment just like the MMX.
That is not what I have said. Please read again.Let's say the rod is traveling in the direction of the system (which is moving with 0.86c). First front edge is passing A, then trailing edge. Because the rod is lengthened, it should take a bit more time in comparison to the rod of original length. Now we transport the rod in opposite direction. The rod is shortened, so the time it takes for both ends to pass point A will be a bit shorter. Carefully measuring the time difference (while monitoring rigorously the speed of the rod) we could calculate the absolute speed of the system. This is definitely invalidated by the theory which introduced length contraction in first place.
Or look at this example:Let's have a pipe 1m diameter at 45 degrees to your 0.68c moving frame. The ball of 1m diameter is just moving through it . Any increase in physical dimension of the ball would stop the movement of the ball. If the ball is moving at 10(sqrt2)m/s, there will be 10m/s speed component parallel (or antiparallel) to your 0.68c frame. Following your reasoning the movement of the ball will be possible in only one direction through the pipe (with the component 10m/s antiparallel to the speed of your 0.68c frame)