Is the Killer Crate Paradox Resolved?

[Gamma Anon]

TL;DR Summary

Length contraction paradox that I created but cannot solve, please help (Not homework)

Hi Physics Forums,

I've devised a thought experiment called the "Killer Crate Paradox" to put a spotlight on an issue I'm having, with regards to understanding length contraction, specifically in instances where multiple objects are observable and they have different velocities and directions of motion.

Please read it (see below) and provide your enlightened thoughts in the comments.

The Killer Crate Paradox

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 0.70711c (or exactly (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 some 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 train's 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.

Diagram of the crate and window* (Ground frame of reference)

*Dimesions not exact and are for illustrative purposes only

Distance between SW and NE will be 500mm as a result of length contraction

Distance between SE and NW as it is at rest (~707.1mm) due to being perpedicular to the direction of motion

Distance between South and North ~433.9mm as a result of length contraction

If the Assassins manage to line up the South Eastern edge of the crate with the Southern edge of the window frame, by timing the launch correctly, the North Western corner will be further North than the Northern edge of the window frame. As a result, the crate will collide with the Northern edge of the window frame.

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 or carraige no matter the timing.

Thanks for reading

γ

 

[Orodruin]

They are describing two different situations. Bob is describing a situation where the box is fired in such a configuration that its side is parallel to the window in the train rest frame. Alice is not.

 

[PeroK]

Thanks for reading

I didn't get through your post, but my guess is you are forgetting about the relativity of simultaneity. Especially when you say things like:

If the Assassins manage to ... by timing the launch correctly

 

[Orodruin]

I didn't get through your post, but my guess is you are forgetting about the relativity of simultaneity. Especially when you say things like:

This would be one of the 1% of cases where relativity of simultaneity is not directly the culprit. Instead, the culprit is that the composition of Lorentz transformations in two directions is not a pure boost.

 

[Dale]

Summary: Length contraction paradox that I created but cannot solve, please help (Not homework)

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.

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.

Distance between SW and NE will be 500mm as a result of length contraction

Distance between SE and NW as it is at rest (~707.1mm) due to being perpedicular to the direction of motion

Alice’s incorrect reasoning is highlighted here. The SE-NW and SW-NE distances are irrelevant. What matters is the distance between the North and South faces in the N-S direction.

 

[Orodruin]

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.

Alice’s incorrect reasoning is highlighted here. The SE-NW and SW-NE distances are irrelevant. What matters is the distance between the North and South faces in the N-S direction.

I do not agree with this. Who is correct will depend on the orientation in which the box is launched. Alice and Bob are considering different scenarios.

Both will agree that contraction does indeed happen along the diagonal. The point is that Alice’s box is launched with a Lorentz boost such that the norrh/south sides were perpendicular to the north/south direction before the boost. Bob’s box is instead launched from an orientation such that those sides are perpendicular to the north/south orientation after the boost.

 

[Dale]

Alice and Bob are considering different scenarios.

I don't think that is what is under consideration. I think that both Alice and Bob are considering the situation where in the train's frame the box is coming straight at the window at $0.5 \ c$, but Alice is just considering it from the ground frame instead of the train frame.

For sure there is a mistake in the description, so I think that the OP intends that they are looking at the same scenario from different frames. With that assumption then the OP's description of the geometry in Alice's frame is incorrect. I can see how you would also be able to read it as the OP's description of the geometry is correct but they are describing different scenarios.

Perhaps @Gamma Anon can clarify. Do you intend this to be one single scenario described from two frames or do you intend them to represent two separate scenarios?

 

[Orodruin]

I don't think that is what is under consideration. I think that both Alice and Bob are considering the situation where in the train's frame the box is coming straight at the window at $0.5 \ c$, but Alice is just considering it from the ground frame instead of the train frame.

I guess that is two different ways to look at the same issue. My point is that this:

Summary: Length contraction paradox that I created but cannot solve, please help (Not homework)

Diagram of the crate and window* (Ground frame of reference)

*Dimesions not exact and are for illustrative purposes only

results from considering a diagonal boost of an object that was originally aligned to the north/south so that seems to be OP’s consideration.

 

[Nugatory]

I think that both Alice and Bob are considering the situation where in the train's frame the box is coming straight at the window at 0.5 c, but Alice is just considering it from the ground frame instead of the train frame.

The catch is that if that's what Alice is doing she has done the transformation from the train frame incorrectly, or at least drawn a picture that does not properly represent that transformation.

I found it easier to not think about length contraction at all - intuition about the behavior of extended objects just gets in the way here - and write down the equations of the trajectories of the NW and SE corners in the coordinates of the train where they are trivial, then transform these to the frame in which Alice is at rest, compare with the trajectories of the top and bottom edges of the window.

 

[Dale]

The catch is that if that's what Alice is doing she has done the transformation from the train frame incorrectly, or at least drawn a picture that does not properly represent that transformation.

Exactly.

I found it easier to not think about length contraction at all - intuition about the behavior of extended objects just gets in the way here - and write down the equations of the trajectories of the NW and SE corners in the coordinates of the train where they are trivial, then transform these to the frame in which Alice is at rest, compare with the trajectories of the top and bottom edges of the window.

Yes, this is precisely why Alice’s reasoning is incorrect. She took the lazy way out and made a mistake in her transformations. Bob shouldn’t have been convinced by such lazy reasoning in Alice’s frame. If you want to do lazy reasoning then you need to use a frame that allows the lazy reasoning.

 

[jartsa]

They are describing two different situations. Bob is describing a situation where the box is fired in such a configuration that its side is parallel to the window in the train rest frame. Alice is not.

If OP does not know about Wigner-rotation, then Bob does not know about it either.

Alice does not need to know about it.

Bob wrong, Alice right.

 

[Orodruin]

If OP does not know about Wigner-rotation, then Bob does not know about it either.

Alice does not need to know about it.

Bob wrong, Alice right.

The question is not whether Alice and Bob know about Wigner rotation. The question is if the assassins do. If they do then they can make the plot work by launching the box in the appropriate orientation. If they don’t, they will fail. So whether Alice or Bob will be right will depend on the assassins’ knowledge of relativity.

 

[Gamma Anon]

I didn't get through your post, but my guess is you are forgetting about the relativity of simultaneity. Especially when you say things like:

I've thought about this, but I'm not sure how it will come into play and allow the box to pass through the window. I'd really appreciate you read the whole thing and provide a clear resolution if you can. Thanks

 

[Gamma Anon]

To clarify, each perspective is the same scenario, just viewed from a different frame of reference.

As stated at the end of the paradox "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"

The implied question is 'Do they understand how Special Relativity works in this instance?'

Is Alice's prediction wrong or is Bob's?

 

[Dale]

I found it easier to not think about length contraction at all - intuition about the behavior of extended objects just gets in the way here - and write down the equations of the trajectories of the NW and SE corners in the coordinates of the train where they are trivial, then transform these to the frame in which Alice is at rest, compare with the trajectories of the top and bottom edges of the window.

When you do this (assuming knowledgeable assassins) a cubic box of width 2L can fit through a window of width 2L. The corners of the box in the ground frame are:
$$NE = \left(t_g,L \left(\sqrt{1-v^2}-v^2\right)-v \sqrt{1-v^2} t_g,L \sqrt{1-v^2}-v t_g,0\right) $$ $$ NW = \left(t_g,-v \sqrt{1-v^2} t_g-L \left(v^2+\sqrt{1-v^2}\right),L \sqrt{1-v^2}-v t_g,0\right) $$ $$ SE = \left(t_g,L \left(v^2+\sqrt{1-v^2}\right)-v \sqrt{1-v^2} t_g,L \left(-\sqrt{1-v^2}\right)-v t_g,0\right) $$ $$ SW = \left(t_g,L \left(v^2-\sqrt{1-v^2}\right)-v \sqrt{1-v^2} t_g,L \left(-\sqrt{1-v^2}\right)-v t_g,0\right) $$
And the edges of the window are:
$$N = \left(t_g,0,L \sqrt{1-v^2}-v t_g,0\right) $$ $$ S = \left(t_g,0,L \left(-\sqrt{1-v^2}\right)-v t_g,0\right) $$

So the assassins do have a "shot"

 

[Nugatory]

Is Alice's prediction wrong or is Bob's?

Have you read through all the responses above? If the crate is moving as you are specified (using the frame in which the train is at rest it is moving straight towards the window, untilted so that its eastern face is parallel to the side of the train) the crate goes through the window.

However, figuring out how to launch the crate on that trajectory is not as obvious as it seems, and the procedure you described in your initial post turns out not to do it. So either we’ve launched the crate on a trajectory which makes Alice right and the crate tilted according to Bob, or we’ve launched it on a different trajectory that makes Bob’s description right and Alice’s wrong. But whichever it is, they will both get the same answer if they correctly analyze the same trajectory.

The easiest and least error prone way of finding how to launch the crate to get through the window is suggested in #9 (and while I was writing this post @Dale carried through and posted the calculations in #15).

To clarify, each perspective is the same scenario, just viewed from a different frame of reference.

They are not. They are two different scenarios resulting from launching the crate in different ways. Are we supposed to analyze the problem with the crate launched in a way that let's it through the window, or with the crate launched in a different way?

 

[Nugatory]

I've thought about this, but I'm not sure how it will come into play and allow the box to pass through the window.

Relativity of simultaneity comes into play because whether the east face of the cube is parallel to the y-axis or not depends on where the points on the east face of the cube are at the same time. It is the underlying reason that the launch procedure you described in your original post doesn’t line the crate up in the train frame the way you were thinking.

 

[Gamma Anon]

Relativity of simultaneity comes into play because whether the east face of the cube is parallel to the y-axis or not depends on where the points on the east face of the cube are at the same time. It is the underlying reason that the launch procedure you described in your original post doesn’t line the crate up in the train frame the way you were thinking.

If you read the paradox, it presumes that the assassins will be able to get the timing right.

In the diagram of Alice's perspective, she assumes that the assassins are able to time the launch so that the South Eastern edge of the crate will be minutely North of the Southern edge of the window frame allowing that South Eastern Edge to enter the window, while leaving as much space as possible for the rest of the crate to enter also.

The problem is that due to length contraction of both the crate and the train in their respective directions of motion, the North Western edge of the crate is much further North than the Northern edge of the window frame. Hence, the crate won't just snuggly side in, which is what Bob expects, based on the train's frame of reference.

 

[Gamma Anon]

When you do this (assuming knowledgeable assassins) a cubic box of width 2L can fit through a window of width 2L. The corners of the box in the ground frame are:
$$NE = \left(t_g,L \left(\sqrt{1-v^2}-v^2\right)-v \sqrt{1-v^2} t_g,L \sqrt{1-v^2}-v t_g,0\right) $$ $$ NW = \left(t_g,-v \sqrt{1-v^2} t_g-L \left(v^2+\sqrt{1-v^2}\right),L \sqrt{1-v^2}-v t_g,0\right) $$ $$ SE = \left(t_g,L \left(v^2+\sqrt{1-v^2}\right)-v \sqrt{1-v^2} t_g,L \left(-\sqrt{1-v^2}\right)-v t_g,0\right) $$ $$ SW = \left(t_g,L \left(v^2-\sqrt{1-v^2}\right)-v \sqrt{1-v^2} t_g,L \left(-\sqrt{1-v^2}\right)-v t_g,0\right) $$
And the edges of the window are:
$$N = \left(t_g,0,L \sqrt{1-v^2}-v t_g,0\right) $$ $$ S = \left(t_g,0,L \left(-\sqrt{1-v^2}\right)-v t_g,0\right) $$

So the assassins do have a "shot"
View attachment 301846

Thanks for the animation.

However, I don't understand how the shape would shift to be in this orientation.

For instance, I can swap the Northbound train out for an Eastbound train going 0.5c with South facing windows of the same dimensions. The velocity and the direction of motion don't need to change for the crate, for Bob to conclude it can enter a window, as based on the Train's frame of reference.

However, based on the orientation of the crate in this animation, it won't enter the window from the ground's frame of reference, the South Eastern edge of the crate will be too far East of the Eastern edge of the window frame.

 

[PeterDonis]

each perspective is the same scenario, just viewed from a different frame of reference.

If that is the case, then one of the descriptions in your OP must be wrong: either Alice's or Bob's. That is the first point.

The second point is: the only way to answer the question of which description in the OP is correct and which is wrong, Alice's or Bob's, is for you to stipulate it. There is no other way because the two descriptions you give in the OP are inconsistent, and no information in the OP tells us which one you intended to be the correct one. So you, the person posing the problem, have to choose which description, Alice's or Bob's, is stipulated to be correct. (The statement that others have made, that you are talking about two different scenarios, not just one, is the same issue stated in different words: the scenario in which Alice's description, as you give it in the OP, is correct, is a different scenario from the scenario in which Bob's description, as you give it in the OP, is correct. So you need to choose which of those two different scenarios you want to discuss.)

Once you do that, then we can help you to understand what error the person whose description in the OP is not correct is making. But we can't do that until you make the choice described above, since that choice is not a matter of physics, it's a matter of the problem specification. In other words, the problem specification you give in the OP is, as it stands, inconsistent and therefore unanalyzable. You need to fix that before discussing anything else.

 

[Gamma Anon]

Once you do that, then we can help you to understand what error the person whose description in the OP is not correct is making. But we can't do that until you make the choice described above, since that choice is not a matter of physics, it's a matter of the problem specification. In other words, the problem specification you give in the OP is, as it stands, inconsistent and therefore unanalyzable. You need to fix that before discussing anything else.

I apologise if it isn't clear what I'm looking for from this post.

Bob forms an opinion based on one frame of reference and Alice forms an opinion based on another. These opinions are conflicting, hence its a paradox. The resolution will be one of the below:

(a) Alice's opinion is wrong
(b) Bob's opinion is wrong
(c) Both their opinions are wrong
(d) There is a flaw in Special Relativity

I don't know the answer, so for anyone reading this, please choose the one you think is correct and explain your reasoning.

 

[Dale]

However, I don't understand how the shape would shift to be in this orientation.

That is why I didn’t just wave my hands and guess. I actually worked through the math so that I could get the answer without knowing it in advance. Sometimes it is hard to see in advance how this stuff works out.

I can swap the Northbound train out for an Eastbound train going 0.5c with South facing windows of the same dimensions. The velocity and the direction of motion don't need to change for the crate, for Bob to conclude it can enter a window, as based on the Train's frame of reference.

However, based on the orientation of the crate in this animation, it won't enter the window from the ground's frame of reference, the South Eastern edge of the crate will be too far East of the Eastern edge of the window frame

I challenge you to actually work out the math to show this. It cannot happen as you say. Your analysis is incorrect because you are just guessing instead of doing the work.

The issue is that non-colinear boosts do not form a group. A pair of non-colinear boosts is not itself a boost. It is a boost and a rotation. So you cannot just swap the Northbound train for an Eastbound train. That would be a different rotation. If the assassins are attacking an Eastbound train then they have a shot, but it is not the same shot as for a Northbound train.

The resolution will be one of the below:

(a) Alice's opinion is wrong
(b) Bob's opinion is wrong
(c) Both their opinions are wrong
(d) There is a flaw in Special Relativity

I don't know the answer, so for anyone reading this, please choose the one you think is correct and explain your reasoning.

(a) as shown. The assassins have a shot, as Bob stated. Alice got the shape wrong.

However, please be aware that (d) is not actually an option. It is possible for special relativity to be contradicted by experimental evidence, but it is self consistent so there is never any thought experiment or theoretical scenario that will produce an actual self contradiction. This is for the same reason that Euclidean geometry is self consistent but may not represent physical reality.

 

[Orodruin]

Bob forms an opinion based on one frame of reference and Alice forms an opinion based on another. These opinions are conflicting, hence its a paradox. The resolution will be one of the below:

(a) Alice's opinion is wrong
(b) Bob's opinion is wrong
(c) Both their opinions are wrong
(d) There is a flaw in Special Relativity

I suggest you reread Peter’s last post again.

 

[Dale]

the only way to answer the question of which description in the OP is correct and which is wrong, Alice's or Bob's, is for you to stipulate it.

No, it is unambiguously Alice’s description which is wrong. The problem is to identify if there is any possible way the assassins could shoot a 500 mm cube box through a 501 mm square window at 0.5 c relative velocity (ideal everything). There is a possible way, which Bob correctly identified and Alice incorrectly said was not possible.

 

[Orodruin]

No, it is unambiguously Alice’s description which is wrong. The problem is to identify if there is any possible way the assassins could shoot a 500 mm cube box through a 501 mm square window at 0.5 c relative velocity. There is a possible way, which Bob correctly identified and Alice incorrectly said was not possible.

Let’s hope Bob wins the argument or that the assassins are unaware of Wigner rotation!

 

[Nugatory]

Bob forms an opinion based on one frame of reference and Alice forms an opinion based on another. These opinions are conflicting, hence its a paradox. The resolution will be one of the below:
(a) Alice's opinion is wrong
(b) Bob's opinion is wrong
(c) Both their opinions are wrong
(d) There is a flaw in Special Relativity

We do have to clearly specify the setup before we can analyze it to see which if either of them is right.

Going back to your original post, you have specified that - working in the frame in which the train is moving - the crate starts at rest with its east face parallel to the y-axis and is accelerated to velocity $\vec{v}=\hat{v}\sqrt{v_x^2+v_y^2}$. Is that what you intended?

In this case Bob is wrong. He has correctly analyzed a different situation in which the crate was launched in a different way - he has the right answer to a different problem but the wrong answer for this one.
Alice has lucked out. She has analyzed the problem incorrectly but has come up with the right answer - crate doesn't make it through the window - for the wrong reasons.

However this problem is remarkably hard to analyze for reasons that are unrelated to the perceived paradox: to see the difficulties google for "Born rigid acceleration", look at the comments upthread about relativistic velocity addition and the composition of Lorentz boosts, and remember the relativity of simultaneity.
So I'm going to propose a simplification: Instead of of Alice and Bob having to launch the crate towards the train, they have friend somewhere behind them, and they can tell the friend to launch the crate so that it passes them at whatever moment they want, moving at any constant speed and in any direction that they want, with the east face of the crate making any angle with respect to x-axis that they want. With this simplification everything is moving a constant speeds relative to everything else.

Now it should be clear that whether the crate makes it through the window or not depends on the instructions Alice and Bob give their friend - they can tell him to aim it to go through or aim it to miss, but there's no paradox either way.

 

[PeterDonis]

Bob forms an opinion based on one frame of reference and Alice forms an opinion based on another. These opinions are conflicting, hence its a paradox.

No, it's not, it's a failure on your part to specify a unique scenario. None of your options are correct. The correct option is:

(e) The scenario as you give it in the OP is not uniquely specified. To uniquely specify it, you need to tell us which description in the OP, Bob's or Alice's, is stipulated to be correct. Either of them could be correct; both of them are potentially correct descriptions of a physically possible scenario. They're just descriptions of different physically possible scenarios. And you need to pick which one of those scenarios is the scenario that we are going to talk about.

 

[PeterDonis]

No, it is unambiguously Alice’s description which is wrong.

If the OP stipulates that the assassins' plan is one that can succeed, then yes, Alice's description would be the one which was wrong (since her description is of a physically possible scenario in which the plan fails because its initial conditions are not set up correctly), and Bob's description would be the one which was correct (since his description is of a physically possible scenario in which the plan succeeds because its initial conditions are set up correctly).

But the OP has not stipulated that. He thinks there's some sort of paradox involved, because he thinks that his OP is describing a single scenario. But it isn't, and there isn't any paradox here at all. There is just a failure on his part to stipulate which of two distinct scenarios (the one that Bob's description in the OP is a valid description of, and the one that Alice's description in the OP is a valid description of) he wants to discuss.

 

[Gamma Anon]

I challenge you to actually work out the math to show this. It cannot happen as you say. Your analysis is incorrect because you are just guessing instead of doing the work.

This is not a paradox that I've written up after 2 minutes of thought, based on a simple hunch, I've done the math.

The paradox is designed and written in a way to make it as accessible as possible, in that the math is minimal within the paradox writeup itself, but is also pretty straight forward for anyone wanting to examine it closer.

For instance, the velocity and dimensions of the crate and train have been chosen so that math is easier than if they were some arbitrary numbers.

The crate is to be launched at speed of ~0.70711c at an azimuth angle of 45° relative to the ground.

Prior to launch, if someone draws a line from the South Western edge of the crate, to the North Eastern edge of the crate, this line will also be drawn at an azimuth angle of 45° and it will have length of ~707.11mm

Because this line matches the direction of motion, in will contract to exactly 500mm (~707.11mm x ~0.70711)

While this line will be shortened, once launched, it will not shift away from its azimuth angle of 45° according ground frame of reference.

Draw a line between the South Western edge and the North Eastern edge of the crate in your animation and measure its azimuth angle. I've got a hunch its less than 45°, but happy to be proven wrong.

Also, put that crate in your animation through a South facing window on an Easterly bound train. I've got a hunch it won't just slide through, but happy to be proven wrong.

 

[Orodruin]

This is not a paradox that I've written up after 2 minutes of thought, based on a simple hunch, I've done the math.

The paradox is designed and written in a way to make it as accessible as possible, in that the math is minimal within the paradox writeup itself, but is also pretty straight forward for anyone wanting to examine it closer.

For instance, the velocity and dimensions of the crate and train have been chosen so that math is easier than if they were some arbitrary numbers.

The crate is to be launched at speed of ~0.70711c at an azimuth angle of 45° relative to the ground.

Prior to launch, if someone draws a line from the South Western edge of the crate, to the North Eastern edge of the crate, this line will also be drawn at an azimuth angle of 45° and it will have length of ~707.11mm

Because this line matches the direction of motion, in will contract to exactly 500mm (~707.11mm x ~0.70711)

While this line will be shortened, once launched, it will not shift away from its azimuth angle of 45° according ground frame of reference.

Draw a line between the South Western edge and the North Eastern edge of the crate in your animation and measure its azimuth angle. I've got a hunch its less than 45°, but happy to be proven wrong.

Also, put that crate in your animation through a South facing window on an Easterly bound train. I've got a hunch it won't just slide through, but happy to be proven wrong.

Please reread all of the replies you have obtained. The scenario you are describing here is not the same scenario that Bob is describing.

 

[PeterDonis]

The crate is to be launched at speed of ~0.70711c at an azimuth angle of 45° relative to the ground.

In other words, you are stipulating that Alice's description in the OP is correct?

 

[Gamma Anon]

But the OP has not stipulated that. He thinks there's some sort of paradox involved, because he thinks that his OP is describing a single scenario. But it isn't, and there isn't any paradox here at all. There is just a failure on his part to stipulate which of two distinct scenarios (the one that Bob's description in the OP is a valid description of, and the one that Alice's description in the OP is a valid description of) he wants to discuss.

Alice and Bob form their respective opinion based on different frames of reference (Alice uses the ground and Bob uses the train), they do not know which description is valid description.

All they know is that the Assassins intend to launch the crate (500mm x 500mm x 500mm) in the North East direction (Azimuth angle 45°) at a speed of 0.70711c (or exactly (0.5c)^0.5 ).

In each of their opinions they have made assumptions about the orientation of the crate relative to the window. Bob has assumed that due to the dimensions of the window matching that of the crate's faces, that the assassins intend to launch the crate face first at the window and not at angle (it won't go in otherwise).

Alice has assumed the crate prior to launch has each of its vertical faces orientated to exactly face North, South, East and West. Once launched however, orientation of these changes due to length contraction.

Are they distinct scenarios and thus Alice and Bob shouldn't be comparing them?

What would crate orientation need to be for Alice's perspective to match Bob's?

 

[Orodruin]

All they know is that the Assassins intend to launch the crate (500mm x 500mm x 500mm) in the North East direction (Azimuth angle 45°) at a speed of 0.70711c (or exactly (0.5c)^0.5 ).

As specified above, this is insufficient information. Even without special relativity, whether it goes in or not depends on orientation. The new thing with special relativity is Wigner rotation.

If they are distinct scenarios and thus Alice and Bob shouldn't be comparing them. How should the assassins orientate the the crate prior to launch so the plot will be successful?

In such a way that their setup matches Bob’s setup would be the obvious answer.

 

[PeterDonis]

Alice and Bob form their respective opinion based on different frames of reference

It's not a matter of opinions or frames of reference. The initial conditions for how the crate is launched need to be properly specified. Right now, as your OP stands, they are not. "Properly specified" means "a specification that can be translated into invariant terms, i.e., terms that do not depend on any choice of frame of reference". There is no such specification in your OP as it stands now.

All they know is that the Assassins intend to launch the crate (500mm x 500mm x 500mm) in the North East direction (Azimuth angle 45°) at a speed of 0.70711c (or exactly (0.5c)^0.5 ).

Angle 45 degrees relative to what? Speed of 0.70711c relative to what? Ideally these things would be specified relative to some actual object, not just a "frame", and the object's own state of motion would be specified as well.

Also, since the crate is not a point object (if it were this scenario would be a non-problem), it's not enough just to specify an initial velocity of one point. You need to specify enough to determine a unique state of motion for the crate as a whole. You haven't.

 

[PeterDonis]

In each of their opinions they have made assumptions about the orientation of the crate relative to the window.

And their assumptions are inconsistent with each other, so they do not describe the same scenario, they describe different scenarios. That is why they are coming up with different answers.