The Accelerating Relativistic Rockets Paradox (Bell's Spaceship Paradox)
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- Опубліковано 4 жов 2024
- The Accelerating Relativistic Rockets Paradox (Bell's Spaceship Paradox) is a special relativity thought experiment concerning two accelerating spaceships attached by a rope or string. It was first thought of in the 1900s, and is a great, tricky, paradox for those new to relativity. Also, the video's pretty fast, so feel free to comment any questions down below.
The "obvious" answer by special relativity is that the rope as seen in the ground frame is length contracted, and thus the rope's proper length is longer and so it breaks. However, a second argument considering the frame of spaceship A makes it no longer obvious. The resolution is the loss of simultaneity, as is for nearly all relativity "paradoxes". Once the spaceships pick up some speed, they still accelerate simultaneously in the ground frame, but if we consider the frame of either ship, due to the loss of simultaneity, one accelerates before the other, and so the distance between the ships increase in their frames as well, resolving the paradox.
A spacetime diagram can also be drawn of the situation, which some might prefer to this explanation.
If you would like a slower more careful introduction to relativity than a one minute video, I recommend chapter 11 of David Morin's Introduction to Classical Mechanics which is available as a sample chapter here: scholar.harvar....
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This is a great video, being concise and to the point while still allowing me to gain a deeper understanding of this interesting situation. Thank you Mr. Slipping Hexagons!
You're welcome! :D
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If two equal distances are in relative motion to each other at speed v, the distances overlap. (the distance between the two spaceships tied by a string and the distance between two points on the frame of the Earth) In my opinion, if we denote with t the elapsed time in the frame of the Earth and if we denote with t_1 the elapsed time in the frame of the two spaceships (tied by a string), it is t = t_1. (and two equal distances overlap, in this case no problem and the string does not break )
The length of the string contracts in the Earth's frame and the length of the Earth contracts in the frame of the two spaceships, the two lengths are equal! (suppose they are equal to d, even if in two different frames)
d = gamma * v * t and d = gamma * v * t_1.
When two distances overlap it is t = t_1 = d / (gamma * v), for each value of d.
If a spaceship is moving at speed v not tied to a second spaceship, then the situation is different. (there are no two distances in relative motion between them)
The light pulse hits both spaceships at the same time if it starts from the center of the rope. Remember, the spaceships aren't moving if there's nothing to move relative to!
What if instead of a light pulse telling the ships to accelerate, they had a synchronised clock onboard that told them when to accelerate?
Great vid
Thank you!
Physics Wallah OP 🔥🔥🔥🔥🔥🔥
Alakh pandey ka beta aagya😂
ur channel banner is serially capping
With school right now, I'm having trouble finding time to create videos--please accept my apologies.
@@SlippingHexagons all g brotha man. in murdock we trust
Wow
Special relativity has nothing to do with it. The only problem is the lag caused by the perception between the two spaceships. If the front one moves first, the rope is pulled, then the second ship is towed. If the back one moves first, the rope slackens. That is it, it happens with or without special relativity. If the ships are in the same rest frame, simultaneity is not an issue, the rope and the ships move together, no length contraction happens.
Consider this: the ships both accelerate at the same instant, using constantly series of light pings to make the difference and start at the same time, then adjust their accelerations so the pings remain constant.. there would always be the same distance between them regardless of their speed relative to outside frames, and the rope just gets dragged along as usual.
Maybe I wasn't clear due to the 1 minute time limit. Special relativity has everything to do with this problem because it's an artificial special relativity thought experiment. The setup is such that there is no "lag" in the ground frame between when they accelerate (I should have said this more clearly, sorry).
Sure, simultaneity is not an issue when they're both at rest in the ground frame, but once they both begin accelerating, there is no inertial frame in which both ship are at rest. This may be counterintuitive due to the fact that they are always moving at the same speed in the ground frame, but because they are accelerating and the loss of simultaneity, this is in fact the case. This might be easier to see in a space time diagram: math.ucr.edu/home/baez/physics/Relativity/SR/BellSpaceships/spaceship_puzzle.html (see the first figure).
I hope this helps clear some things up! :)
@@SlippingHexagons so you are forcing them to not be at rest with respect to each other? If so, then don't say they accelerate together.
No, they are accelerating together; this is the condition that is forced. There isn't an inertial frame in which they are both at rest even when they're accelerating at the same time with the same magnitude in the ground frame because of the loss of simultaneity. Check out the link I sent previously; I think it will be helpful.
@@SlippingHexagons If they start at rest with respect to each other, they accelerate at even remotely close to the same time, at the same rate, and are trying to maintain that state, they are so close to at rest to respect to each other at all times. There is no length contraction, that only happens at relative high-wrt-c velocities.
That is the implied state. They might as well be two engines on different parts of a the same long spaceship.