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- TL;DR Summary
- The twin paradox is resolved by claiming if you instantaneously switch frames, your new frame's measured earth clock instantaneously changes. Try to compute that
Spacetime physics chapter 4 describes this wonderfully well chosen set of speeds/distances for the twin paradox.
A traveler departs from Earth at a speed of 99/101 (1=speed of light), traveling to a star that is 99light years away. From Earth's perspective, the traveler takes 101 years to go to the star, and 101 years to come back. Because of time dilation, Earth thinks the travelers' clock advances by ##101\sqrt{1-(99/101)^2} = 20## years outbound, and another 20 inbound. From the rocket's perspective, because of length contraction it thinks the Earth and the star are ##99\sqrt{1-(99/101)^2}\approx 19.6## light years away, and since the earth/star are moving at a speed of 99/101 times the speed of light, the time between the Earth passing and the star passsing takes ##19.*101/99=20## years each way on the rocket traveler's clock. Thanks to time contraction, from the rocket traveler's perspective, the Earth's clock only appears to advance ##20\sqrt{1-(99/101)^2}\approx 3.96## years each trip.
So the twin paradox is worse than expected. The traveler goes out to the star and back, and when they arrive, they should only expect the Earth to have aged about 8 years, when actually it has aged 202. The book explains the discrepancy by noting that for the traveler to go to the star and back, it must switch frames from outward to inward. Therefore, all is consistent as long as that frame switch instantaneously moves the traveler's perspective of the clock time on Earth by about 194 years.
I wanted to try to compute this (no hint as to how you would derive this other than 'it must be so for consistency!' is given). My thought was to imagine a third rocket traveler, who starts on the far side of the star from earth, and travels towards Earth at a speed of 99/101. Then the inbound rocket traveler can compute how much time it thinks has passed on Earth between the two events of the outbound rocket leaving, and the the outbound rocket reaching the star. I guess the claim here is that the always-inbound rocket traveler should measure the Earth clock as advancing by should advance by about 198 years.
So I guess here's my weak attempt at it. From the always-inbound rocket perspective, the Earth is moving at a speed of 99/101. The outbound rocket is moving at a speed of 99/101 in the Earth frame, so the velocity addition formula says that from the perspective of the always-inbound rocket, the outbound rocket moves at a speed of ##\frac{2 \frac{99}{101}}{1+(\frac{99}{101})^2}\approx 0.9998##. That's pretty fast! From here, I get pretty confused. Because of length contraction, the always-inbound rocket should think the outbound rocket needs to travel 19.6 light years at a speed of 0.9998, which takes, well, about 19.6 years to do. Because of time contraction, it should think the Earth clock only moves ##19.6\sqrt{1-(99/101)^2} \approx 3.88## years between these two events. This seems to be moving in the wrong direction, since 3.88 is smaller than 3.96, and it's certainly not anywhere close to 198 years.
I'm not sure if I'm missing the point of the claim that the apparent Earth clock jumps instantaneously when the frame switches, or if I'm just doing a bad job of attempting to compute the Earth clock from the point of view of the inbound rocket. Any advice or insight into this is appreciated.
A traveler departs from Earth at a speed of 99/101 (1=speed of light), traveling to a star that is 99light years away. From Earth's perspective, the traveler takes 101 years to go to the star, and 101 years to come back. Because of time dilation, Earth thinks the travelers' clock advances by ##101\sqrt{1-(99/101)^2} = 20## years outbound, and another 20 inbound. From the rocket's perspective, because of length contraction it thinks the Earth and the star are ##99\sqrt{1-(99/101)^2}\approx 19.6## light years away, and since the earth/star are moving at a speed of 99/101 times the speed of light, the time between the Earth passing and the star passsing takes ##19.*101/99=20## years each way on the rocket traveler's clock. Thanks to time contraction, from the rocket traveler's perspective, the Earth's clock only appears to advance ##20\sqrt{1-(99/101)^2}\approx 3.96## years each trip.
So the twin paradox is worse than expected. The traveler goes out to the star and back, and when they arrive, they should only expect the Earth to have aged about 8 years, when actually it has aged 202. The book explains the discrepancy by noting that for the traveler to go to the star and back, it must switch frames from outward to inward. Therefore, all is consistent as long as that frame switch instantaneously moves the traveler's perspective of the clock time on Earth by about 194 years.
I wanted to try to compute this (no hint as to how you would derive this other than 'it must be so for consistency!' is given). My thought was to imagine a third rocket traveler, who starts on the far side of the star from earth, and travels towards Earth at a speed of 99/101. Then the inbound rocket traveler can compute how much time it thinks has passed on Earth between the two events of the outbound rocket leaving, and the the outbound rocket reaching the star. I guess the claim here is that the always-inbound rocket traveler should measure the Earth clock as advancing by should advance by about 198 years.
So I guess here's my weak attempt at it. From the always-inbound rocket perspective, the Earth is moving at a speed of 99/101. The outbound rocket is moving at a speed of 99/101 in the Earth frame, so the velocity addition formula says that from the perspective of the always-inbound rocket, the outbound rocket moves at a speed of ##\frac{2 \frac{99}{101}}{1+(\frac{99}{101})^2}\approx 0.9998##. That's pretty fast! From here, I get pretty confused. Because of length contraction, the always-inbound rocket should think the outbound rocket needs to travel 19.6 light years at a speed of 0.9998, which takes, well, about 19.6 years to do. Because of time contraction, it should think the Earth clock only moves ##19.6\sqrt{1-(99/101)^2} \approx 3.88## years between these two events. This seems to be moving in the wrong direction, since 3.88 is smaller than 3.96, and it's certainly not anywhere close to 198 years.
I'm not sure if I'm missing the point of the claim that the apparent Earth clock jumps instantaneously when the frame switches, or if I'm just doing a bad job of attempting to compute the Earth clock from the point of view of the inbound rocket. Any advice or insight into this is appreciated.