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Physics
Special and General Relativity
Clock Thought Exp: A Light-Year Time Paradox
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[QUOTE="George Jones, post: 5464580, member: 31570"] The alien sees the Earth's clock almost infinitely fast. Because of Lorentz contraction, the alien takes an almost infinitely short time (according to the alien) to get to the Earth. In terms of mathematics, the alien sees the Earth's clock ticking faster (setting ##c = 1##) by the Doppler shift fact of $$\sqrt{\frac{1+v}{1-v}}.$$ The time taken by the alien (according top the alien) for the trip is distance/speed, where the distance (according to the alien) is a Lorentz contraction of 1 light-year, i.e., ##\sqrt{1-v^2}## light-years. The time that the alien sees elapse on the Earth's clock is (rate of ticking seen) times time for trip. Putting this together gives $$\sqrt{\frac{1+v}{1-v}} \frac{\sqrt{1-v^2}}{v} = \sqrt{\frac{1+v}{1-v}} \frac{\sqrt{\left(1-v\right) \left(1+v\right)}}{v} = \frac{1}{v} +1.$$ In the limit that ##v## approaches one (light speed), the time that the alien see elapse on the Earth's clock is 2 years. Edit: Note that this agrees with the results according to people on Earth. According to Earth, there is no Lorentz contraction for the alien's trip, i.e., the aliens travel a distance of 1 (light-years), and the time (according Earth) for the alien's trip is distance/speed ##=1/v##. According to Earth, the alien's trip started one year after the clock turned on, so, again, ##1/v +1##. [/QUOTE]
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Physics
Special and General Relativity
Clock Thought Exp: A Light-Year Time Paradox
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