How can the Pole&Barn Paradox be solved?

[Dale]

By using the same math, you can easily prove that:


$$X_{T1,front}=X_{T2,rear}$$

So, the trains line up perfectly as measured from T1. You can repeat the same exact reasoning and you'll get the same result from T2.

Which also doesn't provide enough information to answer the question. Since we need to know if the gun on the T1 rear goes off before or after the intersection events that you have described, and that depends on an unspecified notion of simultaneity with the T1 front event.

 

[Dale]

The two trains match up perfectly in all frames of reference.

What do you mean by this?

 

[GAsahi]

It doesn't show any such thing. It shows merely that the intersection of two worldlines (the rear of T1 and the front of T2) is a frame-independent occurrence. I.e. if they are at the same position at the same time in one frame then they are at the same position at the same time in all frames.

Good, you got this correct.

The scenario had to do not only with what was happening at the rear of T1 and the front of T2, but also what was happening "simultaneously" at the front of T1 and the rear of T2. Your math did not address that at all. And as ghwellsjr correctly pointed out "simultaneously" was not specified sufficiently to determine the outcome of the scenario.

Sure it does, u need to read post 22 from the beginning. It says clearly that the front of T1 lines with the rear of T2 at the SAME location at time $\tau_1$ as measured from the track frame. Please go back and re-read the post.

 

[Dale]

Sure it does, u need to read post 22 from the beginning. It says clearly that the front of T1 lines with the rear of T2 at the SAME location at time $\tau_1$ as measured from the track frame. Please go back and re-read the post.

Where does it say that? I have been through post 22 a half dozen times and cannot find one single mention in either text or math of the front of T1, only the rear of T1. If it indeed says it at all then it certainly doesn't say it clearly.

 

[Dale]

There are two trains T1 and T2 of equal rest length "L" (say) running on two parallel tracks in opposite direction with a relative velocity V such that due to length contraction one appears of length L/2 w.r.t. the other. Train T1 has a gun right at the "back end" which can shoot perpendicularly right towards the track of train T2. Suppose, it has been arranged for the the gun to shoot at the same time when the front of T1 passed the rear of T2 according to T1 clocks. Now one can see that in the frame of T1, T2 will appear contracted (to L/2) so that T2's front wouldn't have crossed the back-end of T1 when gun shoots, and thus gunshot will not hit T2

This is a completely specified scenario and a correct analysis. All other frames will necessarily also agree that the shot will not hit.

But in the frame of reference of T2, T1 will appear contracted (to L/2) and thus, the gunshot will hit T2. So is T2 hit or not hit?

T2 is not hit. Although T1 is contracted in T2's frame, the shot and the passing of the ends are not simultaneous. The gun fires too late, thereby missing T2.

I can't check your math right now, but it looks like you got the right conclusion.