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Alice and Bob both work at a factory and are both capable of making 10 widgets per hour. If Alice decided to work remotely from her “centrifuge office” that spins her at a tangential velocity close to the speed of light, I understand that at the end of the work day (i.e. 5pm, Bob’s time), after she powers down her centrifuge and returns to the factory, she will have produced less widgets than Bob, who stayed in the factory, right? (Please tell me if this is right! I may have just confused myself.)

Is there anything that Alice could do involving time dilation to increase production rate relative to Bob?

Am I missing something entirely with this example?

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In space there's a shortest distance between two points (a straight line). You can't find any shorter path. A person who deviates from that path will always travel a longer distance.

In spacetime there's a longest time between events (in some sense time and space have opposite signs). That longest time is achieved by a person who is at rest between the events, i.e. never accelerates. Anyone who accelerates gets a shorter elapsed time between the events.

In your example, Bob is unaccelerated, and so follows the path of longest time. There is no longer time available to Alice; the best she could do is to match Bob's time.

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  • $\begingroup$ nice way to think of it $\endgroup$
    – chaimp
    Jul 8, 2022 at 17:20
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after she powers down her centrifuge and returns to the factory, she will have produced less widgets than Bob, who stayed in the factory, right?

Right.

Is there anything that Alice could do involving time dilation to increase production rate relative to Bob?

Sure. Put Bob in the centrifuge.

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Your first assumption is correct, that at 5pm Bob's time it will only be 4pm Anna's time, so she will have produced fewer widgets. In order to use this idea to produce more widgets, it would be necessary to speed up time relative to Earth time - not slow it down. But if you look at the universe, you'll see that Earth time is very very close to the fastest possible time. In fact the difference between Earth time and time at 10,000km altitude is only about 1 second per century. So there is no way that you could reasonably speed up time for Anna to make her more productive. (And I'd hate to consider the Union Rules regarding having someone die earlier.)

I should add that if you were to build your factory on the moon, then time would pass 0.66 parts per billion faster than on Earth. So you might get an extra widget in 10,000 years with 1,000,000 workers.

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Ignoring the acceleration of hugely accelerated frames will always predict unphysical results. You've duplicated the Twin Paradox here, which is asked and answered extensively on this site and has a good treatment on Wikipedia. Your understanding of the scenario is incorrect. It happens to be true (once you include frame acceleration) that Alice is in fact younger than Bob when she returns to Bob's frame, but your reasoning got you no better odds of reaching the correct answer than a coin flip would.

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