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I am trying to do a thought experiment to determine the longitudinal length contraction (length along the direction of motion). Experiment is something like this:

Here I'll be using $left$ and $right$ w.r.t. rod $B$.

Suppose two rods which have same proper length $L$ are moving parallel and towards each other. Rod $A$ is moving towards right with respect to rod $B$. On the right ends of both rods (w.r.t. $B$), there is a sword attached to both of them. There is an observer sitting on both the rods. The job of the observers is to cut the other rod when they see that $left$ ends of both rods are coinciding (w.r.t. $B$).

Here is the problem. My understanding is that both observers will agree when the left ends coincide. Since both of them agree, they will try to cut the other rod. But since each one of them sees the other rod as shortened, so, his rod will be cut as his rod is longer and other rod should remain intact. Both of them would have to agree at the end of the result and would check that whose rod survived. But this is the problem, both can't be true as according to both of them one rod survives and other remains intact.

Where am I wrong?

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  • $\begingroup$ "both can't be true as according to both of them one rod survives and other remains intact." There is a unstated assumption about the nature of time in the logic leading up to that. You need to spend some more time thinking about the "relativity of simultaneity" (or use a tool —such as a Minkowski diagram—which deals with that concept automatically) before the resolution is clear. $\endgroup$ – dmckee Feb 26 '18 at 21:28
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    $\begingroup$ BTW, having drawn the diagram I get both attacks missing. Rod A misses by chapping too late, and rod B by chopping too soon. $\endgroup$ – dmckee Feb 26 '18 at 21:35
  • $\begingroup$ There are some subtleties and tacit assumptions regarding the concept of simultaneity in your question. Observer B may think that he is chopping his sword at one end of his rod simultaneously with the instant that the other end of his rod is coincident with the end of Observer A's rod. But in special relativity, two events which are simultaneous according to one observer are not necessarily simultaneous according to another observer. In the case of your problem, while Observer B may think that his two events were simultaneous, Observer A will see them occurring at different times. $\endgroup$ – Samuel Weir Feb 27 '18 at 0:13
  • $\begingroup$ Based on above comments I think I have got the answer. Please correct me if I am wrong. In the frame of rod B, the events that A sees the left ends coincide and he immediately cuts the rod are not simultaneous. So he cuts the rod at a later instant and within this instant his left end moves a bit to right. So even if he is able to cut the rod (I am still confused how both will miss), the instant he does so, his left end is not coincident with rod B's left. So B won't accept it as a legitimate means of measuring length. $\endgroup$ – Ankur Singh Feb 27 '18 at 3:15
  • $\begingroup$ @a.b - I think that you've got the right idea. Things that are simultaneous to Observer A are not necessarily simultaneous for Observer B and vice versa. $\endgroup$ – Samuel Weir Feb 27 '18 at 19:59
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The thing with length contraction is that it's only an apparent effect, so it makes no real sense to think of cutting the rod like this.

If a cheese wire were to be lashed so as to slice through a moving rod, it would cause a skew cut in the rod.

If a sword were lashed at the rod, either the sword or the rod, or both, would be shattered in the attempt, because there is no way to strike a plane surface against a moving rod and maintain the mechanical integrity of both. You can prove that if you imagine a handsaw trying to cut a moving piece of wood. There is no orientation of the blade that will allow you to cut through (other than making a purely longitudinal cut), if the blade does not move with the wood. And of course, if your blade moves with the wood, then your length contraction problems are solved.

Two co-moving cheesewire-men each riding a rod, and lashing the counterpart rod, will not produce rods of different lengths. They will produce rods with identical skew-cuts of equal overall length. They cannot produce straight-edge cuts, because that would require the cheese wire to move through the rod at infinite speed.

A more relevant thought experiment is to note that spherical objects do not contract into an ellipsoid shape. Their outline remains fully circular, which of course it could not if there was any real contraction of length.

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