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I'm having trouble conciliating Lorentz contraction with a little thought experiment.

Imagine you have a box full of TNT, the length of the box from the base to the handle that triggers the explosion when extended is L.

Imagine also a fixed crate with a single opening of length L' which is slightly smaller than L.

According to special relativity, if we bring the box of TNT at a speed close to the speed of light, from the point of view of the TNT box L' will have contracted, and we can safely ram it into the fixed crate with no explosion. However, from the point of view of the crate, L will have contracted enough to cause the TNT handle to be triggered.

The thought experiment is visualized here with my mspaint skills: enter image description here

This results in an event that happens from the point of view of one frame, but not in the other. Clearly something is wrong, but what?

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  • $\begingroup$ Just another version of the hole/nail paradox (AKA bug/rivet and other names) with exactly the same resolution. $\endgroup$ Jan 17, 2018 at 20:27
  • $\begingroup$ Are you saying from the point of view of the TNT box, the back portion will continue to move to the right even after it made contact with the crate? $\endgroup$ Jan 17, 2018 at 20:37
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    $\begingroup$ As long as you're putting all this effort into drawing pictures, why not draw the spacetime diagram? Unlike all these pictures of boxes, it's pretty much guaranteed to resolve your paradox as soon as you draw it. $\endgroup$
    – WillO
    Jan 18, 2018 at 0:42

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Assume the crate is somehow fixed in position. (if it were not, then the events would depend on the dynamics of the system)

In relativity, there is a principle called "relativity of simultaneity". This can be extended to say that if event A precedes event B in time IN FRAME L, then in another frame L' it's possible for event B to precede A in time IN FRAME L'. (This can actually only happen if A and B are space like separated and hence not causally related, but don't worry about this for now.)

Analyse this step-by-step.

TNT FRAME

The box is length-contracted, so when the TNT comes in contact with the box, there is no explosion. However the TNT has to decelerate and stop moving (the crate is fixed). This means that the box, in the TNT frame, will grow in length, because the relative velocity between the box and the TNT has gone to zero. This will push up the TNT handle, and it will explode. BOOM

BOX FRAME

The TNT is length contracted, and as from your drawing the TNT box does indeed explode. Ignoring all the crazy things that happen from a relativistic object exploding, perhaps the bottom of the TNT flies forward and hits the crate.

ANALYSIS

Event A is the TNT exploding and Event B is the bottom of the TNT hitting the crate. In the TNT frame, B precedes A in time. In the box frame, A precedes B in time. This resolves the paradox.


ADVANCED MATERIAL

If events A and B are causally related (A causes B), it is absurd to think that B can precede A in any other frame. Thus, the "reversal" of A and B in a frame can only occur if they are "spacelike separated" in space-time. What this means, essentially, is that A cannot influence B by sending a signal such that it hits B in the proper spatial location AND "time-location". This depends on the signal speed (usually light, as it is the fastest thing) and the separation between A and B in terms of length and time.

If I throw a ball up in the air and catch it without moving laterally, the events of me throwing it and catching it are not separated in space, but separated in time. (Time-like separated). If you are standing 1 meter away from me (in the frame of the ground), the event of me throwing the ball at time t=0 and the event of you catching the ball at time t=0 are space-like separated. (Thus I cannot take any action to make you catch a ball at time t=0, due to causality and the speed of light.)

It is worth noting that the notions of "timelike" and "spacelike" separated are INDEPENDENT of the frame in which we're working. (The fancy term for this quality "Lorentz Invariance")

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