Paradox: Rocket ship moving in a circle

[andrewpareles]

Say there is a circular fence that has a diameter of 10 meters, and a rocket ship that is normally 20 meters goes very quickly so that its relativistic length is 1m from the position of an observer standing at rest with relation to the fence.

The rocket ship starts to go in a circle inside the fenced area. From the perspective of the person at the fence, this is perfectly fine and normal since the rocket is 1m long.

However, from the perspective of the people in the rocket, the fence is contracted in the direction of movement and the rocket couldn't possibly fit in the fenced area.

Who is right and how do you resolve the paradox?

I'm thinking it has to do with acceleration but have no idea what is wrong with it! Normally questions like this are resolved with time and lack of simultaneity, but the fence is always closed

 

[PeterDonis]

from the perspective of the people in the rocket, the fence is contracted in the direction of movement

No, it isn't, because to go in a circle inside the fenced area, the rocket needs to be moving, at any given instant, perpendicular to the diameter of the fence, and there is no length contraction perpendicular to the direction of motion.

There are more complications involved in this scenario, but the above is the gist of the answer.

 

[Ibix]

It's simultaneity again.

If you look at the worldlines of the rocket's nose and tail, what the rocket calls its length is the perpendicular distance between those lines. However the fence frame does not use the same definition of space or time, and the length it measures is not the perpendicular distance between the worldlines but some diagonal distance. Diagonal distances between parallel lines are shorter than the perpendicular distance in Minkowski spacetime.

What the rocket sees of the fence is complicated. It's not using an inertial frame, so naive SR results don't apply. And clock synchronisation is problematic. You can work something out, but it's not unique. I'd suggest Googling the Ehrenfest Paradox, since this is a variant.

 

[jartsa]

Say there is a circular fence that has a diameter of 10 meters, and a rocket ship that is normally 20 meters goes very quickly so that its relativistic length is 1m from the position of an observer standing at rest with relation to the fence.

The rocket ship starts to go in a circle inside the fenced area. From the perspective of the person at the fence, this is perfectly fine and normal since the rocket is 1m long.

However, from the perspective of the people in the rocket, the fence is contracted in the direction of movement and the rocket couldn't possibly fit in the fenced area.

A person in the rocket: "The fence is contracted in the direction of the movement of of the fence. The ship is mostly located next to that part of the fence that is not very much contracted length wise. And most of the ship is contracted - those parts that are moving relative to me are contracted"(I think this scenario might actually be unphysical. The direction of the motion might be changing too fast for the contraction to keep up) (Arbitrary Born-rigid acceleration of arbitrarily long rod is not possible)

 

[Ibix]

Say there is a circular fence that has a diameter of 10 meters, and a rocket ship that is normally 20 meters goes very quickly so that its relativistic length is 1m from the position of an observer standing at rest with relation to the fence.

The rocket ship starts to go in a circle inside the fenced area. From the perspective of the person at the fence, this is perfectly fine and normal since the rocket is 1m long.

To understand the difference between @PeterDonis' answer and mine you need to say how the rocket is oriented. Peter has - I think - taken it that the rocket is aligned with its nose pointing at the centre of the circle. I've taken it that the rocket is aligned parallel to the fence. So we're answering different questions. I can read either into your post, @andrewpareles. Which one did you intend?

 

[A.T.]

The rocket ship starts to go in a circle inside the fenced area

To make the scenario less ambiguous, I suggest you replace the fence with a circular train track of circumference C, and the rocket with a train of rest length L > C, that goes in circles on the track fast enough that its contracted length is L' < C, so it fits completely onto the circular train track (all frames must agree on that).

This was discussed here:
https://www.physicsforums.com/threa...elativistic-speeds.960744/page-6#post-6096648

 

[PeterDonis]

Peter has - I think - taken it that the rocket is aligned with its nose pointing at the centre of the circle.

No, I assumed that the rocket's nose was pointed perpendicular to the diameter of the circle. That means it's pointing 90 degrees away from the direction of the center of the circle.

If the rocket's nose were pointed towards the center of the circle, then I'm not sure how to interpret "the rocket ship goes in a circle inside the fenced area".

 

[PeterDonis]

The direction of the motion might be changing too fast for the contraction to keep up

It's not a matter of the contraction "keeping up". It's a matter of, as the rest of your post suggests, it being impossible for all of the parts of the rocket to be at rest relative to one another in the scenario as given. This is one of the complications I hinted at in post #2 of the thread, and you're right that it makes the analysis much more complicated if you actually try to work through the details. But the length contraction doesn't have to "keep up" with anything, since it's not something that's happening in the first place, it's just a coordinate effect. The physical thing that is happening is that the rocket's parts will have different directions of both velocity and acceleration.

 

[PeterDonis]

To make the scenario less ambiguous, I suggest you replace the fence with a circular train track of circumference C, and the rocket with a train of rest length L > C, that goes in circles on the track fast enough that its contracted length is L' < C, so it fits completely onto the circular train track

Note that in this scenario, the train is allowed to bend. I'm not sure the OP intended for his rocket ship to bend.

 

[Dale]

Also, the rocket frame is non inertial so the usual formulas do not apply at all.

 

[PAllen]

A note on this is that for the rocket in this example to behave approximately as expected (e.g. its front and back not have high velocity in the momentary rest frame of its middle), it is necessary that the angle its subtends in the fence frame is less than around π / 16γ². However, the case desired by the OP is for the subtended angle to be greater than 2π/γ, so this will never be remotely an object roughly at rest in the momentary rest frame of its middle. In fact, its front and back will be moving at close to c in opposite directions compared to the middle, and the shape in this frame will be more like a U shape rather than a straight line (bending around the squashed circle that is the fence in this frame). Note, this is the case where we allow the rocket to bend to stay close to the fence, or better, the train on a track example. However, if the approximately linear requirement is met, it doesn't really matter whether you consider the rocket to be a chord versus an arc. In the quasi-linear regime, the rocket is longer in its rest frame by the expected factor, and the fraction of fence adjacent to it is greater by the expected factor (because the fence contracts, the fraction of it spanned by the rocket in its rest frame is larger), compared to the fence frame.

 

[sweet springs]

But what makes you think it is valid the way you used it?

Oh! it is not valid? I appreciate your teaching.

[EDIT]Now I do not see my preveous posts in the line. I am afraid this post has no reference. I would just advise OP to see Landau Lifshitz classical theory of field section 89. [Link removed - we're not sure about the copyright status]
rocket length : fence periphery
= 1 meter : 62.8 meter in the fence system
= 20 meter : 1256 meter in rotating system, i.e. the rocket system

[EDIT2]
Rotating system needs 20 times more materials in making its own fence adjacent inside. President of that system would love the idea that 20 times cheaper boundary wall will be built only if he allows it to rotate (reversely). 1 meter brick material would inflate to 20 meter one under (reverse) rotation in rotating system.

 

[PeterDonis]

Oh! it is not valid?

Not for what you're trying to use it for, no.