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I've devised a thought experiment called the "Killer Crate Paradox" to further improve my understanding of Lorentz Transformations. This is not a paradox I am able to resolve, so please read it (see below) and provide a resolution if you are able to determine one. Thanks γ

Two police officers, Bob and Alice, receive intel about an assassination plot against prominent exiled physicist, Dr Katze, that they protect from the vindictive regime she has fled. The assassins plan to attack the physicist while she is on a train she regularly catches, and want to make it look like an accident at first glance, to provide them time to escape.

While the train is heading due North along its route and Dr Katze sits by her usual Western facing window, the assassins intend to launch a cargo crate at a high velocity that will enter the carriage via the window and strike her. Because the train will be traveling at 0.5c, the assassins intend to launch the crate in the North East direction, at a speed of (0.5c)^0.5, so that the Northbound velocity vector of the crate is the same as the Northbound velocity of the train.

In order to ensure that it kills Dr Katze, they have designed the crate to be as large as it possibly can and still pass through the window. The crate they have made is cube shaped, with dimensions (500mm x 500m x 500mm) minutely shorter than the square window frame (501mm x 501mm), when both are at rest, which should allow it to completely slide through the window frame and strike the physicist, despite its large size.

Bob ponders the feasibility of the plot, using the knowledge of special relativity he has picked up from Dr Katze, and decides that the plot has decent potential of being successful, and thus it is far too dangerous for the physicist to continue catching this train.

As the crate is traveling at the same speed in the Northbound direction, Bob reasons, it would appear to anyone on the train that the crate is traveling in an eastwards direction relative to the carriages. While the crate would no longer look like a cube due to length contraction between the eastern and western faces of the crate, the distance between the northern and southern faces of the crate would appear the same as it is at rest, allowing the crate to snuggly slide through the window frame, assuming the assassins get the launch timing right.

Alice is more sceptical of the likelihood that such a plot would be successful, having taken the perspective of someone stationary relative to the ground. In that frame of reference, the distance between the North Eastern edge and the South Western edge of the crate will have contracted, as this is the direction of motion, while the distance between the North Western edge and the South Eastern edge will be the same as when the crate is at rest.

As the shape of the crate would be rhombohedron once launched, and the trains windows are narrowed as a result of length contraction in the direction of motion, Alice reasons that it will be impossible for the crate launch to be timed in such a way, that the crate will snuggly slide through the window, without one of the crate's faces colliding with the window frame.

What the crate and window would look like using the ground as a frame of reference

Bob and Alice see the merits of each other's arguments and come to the realization that either their understanding of Special Relativity is wrong or they have just picked up a flaw. They reason that the outcome of the two perspectives must be the same; either the crate can slide snugly through the window if timing is right or it will collide with the window frame no matter the timing.

Thanks for reading γ

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    $\begingroup$ This is not a puzzle site... You may have better luck on puzzling. SE. Please bol down your question to the pure physics if you want to keep it here $\endgroup$
    – Brian
    Commented May 22, 2022 at 11:29
  • $\begingroup$ It appears that Alice is wildly guessing rather than reasoning; either that, or she has forgotten that in the Earth frame the crate's speed is considerably higher than the train's speed, and hence it will be more length contracted than the window. In any case, if she actually does the math, she will find that it will fit, just as Bob predicted. $\endgroup$
    – Eric Smith
    Commented May 22, 2022 at 15:11
  • $\begingroup$ Thought experiments such Schrödinger's Cat and the Twins Paradox have been used by physicists in the past to comprehend physics theories. This is no different. $\endgroup$
    – γ Anon
    Commented May 23, 2022 at 1:07
  • $\begingroup$ Yes, length contraction with be greater for the crate, but not in same direction. In Alice's perspective, the length between the North Western edge and and the South Eastern edge will be unchanged, at ~707.11mm and the distance between the North Eastern edge and the South Western edge will now be 500mm. This shape now has to fit through a gap that is now ~433.88mm wide, due to the trains length contraction. Try and do that while making sure that each of the crate's corners face their original orientation. $\endgroup$
    – γ Anon
    Commented May 23, 2022 at 1:33

3 Answers 3

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I think the question is: "Is it possible for the crate to slide through the window?" The answer is yes. The key is that the crate is not a cube, but a rhomboid. The length of the diagonal of a cube is sqrt(3) times the length of an edge. The length of the diagonal of a rhomboid is sqrt(2) times the length of an edge. So the length of the diagonal of the crate is sqrt(2) times the length of an edge. The length of the diagonal of the window is sqrt(2) times the length of an edge. So the crate can fit through.

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  • $\begingroup$ I've added an image to the post that will give you a better understanding of why Alice doesn't think the crate will pass through in one clean motion. If the South Eastern edge of the crate is due west of the southern edge of the window frame, then the North Eastern edge of the crate will be ever so slightly South of the Northern edge of the window. The problem is the North Western edge of the crate is more North then the Northern edge of the window. $\endgroup$
    – γ Anon
    Commented May 23, 2022 at 5:02
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Alice reasons that it will be impossible for the crate launch to be timed in such a way, that the crate will snuggly slide through the window, without one of the crate's faces colliding with the window frame

Alice reasons incorrectly. The East-West contraction is a red herring, it exists but is irrelevant to the problem. Only the North-South contraction matters, and as the crate and the window have the same Northward speed the North-South contraction is the same for both. Thus the crate fits through the window in Alice’s frame just fine.

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  • $\begingroup$ The contraction between Northern and Southern faces is not the same in both frames. In Alice's frame, the distance between the North Western and South Eastern edge are unchanged, as they are perpendicular the crate's direction of motion. $\endgroup$
    – γ Anon
    Commented May 23, 2022 at 1:52
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Since SR is entirely symmetrical, you can imagine the train and crate are initially at rest, and the crate is then thrown through the window at some speed. Since in that frame, in which the train and crate were both at rest before the crate was thrown, the crate is narrower than the window, it must pass through. Therefore, in any other frame it must also pass through, so Alice's reasoning must be false. I suspect the fault is that in Alice's frame the distorted crate is oriented to the window at an angle, which allows it to pass through notwithstanding the fact that the northwest to southeast dimension is unchanged.

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  • $\begingroup$ I've added a picture to the post to clarify Alice's perspective. As you will see, the North Western edge of the crate is too North, even with the Easterly edges of the crate lined up with the window's Northern and Southern edges. $\endgroup$
    – γ Anon
    Commented May 23, 2022 at 5:10
  • $\begingroup$ It is only too North if it approaches the window at a right angle in the frame in which it is distorted. Have you considered that? $\endgroup$
    – Professor Sushing
    Commented May 23, 2022 at 7:42
  • $\begingroup$ Yes I have Marco. There is no phenomenon that im aware of that will result in the crate yawing counterclockwise from its launch angle, to an angle that would allow it to pass through, and if there was I would have quite a few issues. The biggest is that the crate would require advanced knowledge of the orientation of the window. For instance, the same crate launch velocity and direction could be used on an Eastbound train going 0.5c with South facing windows. This yaw phenomenon would have to now occur in a clockwise direction for the crate to pass through. $\endgroup$
    – γ Anon
    Commented May 23, 2022 at 8:52
  • $\begingroup$ I think you are missing the point, or I am missing a different one, which is that if you can pass the crate through the window at 90 degrees in the train's rest frame, then it must pass through when viewed from any moving frame. If the crate has to yaw in another frame for that to happen, it is not the case that the crate decides how to yaw- it is that the yawing effect arises in the moving frame according to the speed and direction of motion of that frame, so the change of angle would take care of itself. $\endgroup$
    – Professor Sushing
    Commented May 23, 2022 at 9:27
  • $\begingroup$ I agree that if the crate can pass through in Bob's perspective then it should in Alice's also. I just dont know by which mechanism it can happen. I considered yawing as a potential solution at first, but as said, if the train is Nothbound the crate must yaw counter-clockwise so that the North Western corner isn't more North than Northern edge of the window frame. If the train was going East instead, the crate would have to yaw clockwise so that the South Eastern edge isnt more East that the eastern edge of the window frame. There are also some other problems with yawing. $\endgroup$
    – γ Anon
    Commented May 23, 2022 at 10:38

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