Clock Thought Exp: A Light-Year Time Paradox

In summary, 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
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ZawL
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This is a thought experiment I came up with this evening. It must surely have been considered before but I couldn't find any reference and I've come up with it independently.

NASA has positioned a clock in Earth's thermosphere that can be viewed from outer space with a powerful enough telescope and activates it just as the time hits New Year's on 1 Jan 2020. Aliens live on planet XYZ one light year away from earth. They have been monitoring Earth for a while and notice the clock as soon as it is activated. They send a being in a spaceship towards Earth at close to the speed of light. The being constantly watches the clock as he approaches Earth (except when rotation of Earth means the clock isn't visible to him). Since the light from the clock takes one year to reach planet XYZ, the aliens first notice the clock when it is 1 Jan 2021 on Earth, although they will see the time on the clock as 1 Jan 2020 on XYZ. The alien traveling towards the Earth will take 1 year to reach it, so when he arrives to Earth, the clock will show 1 Jan 2022. He has been watching the clock go from 1 Jan 2020 to 1 Jan 2022 in the year it takes him to reach Earth from the moment he first saw the time on the clock. My questions are, how does he observe two years' worth of clock ticks in the space of 1 year, and does he perceive the clock ticking as we do on Earth, ie one second in the space of one second?
 
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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##.
 
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Thank you! Your answer is explained really well and makes complete sense to me. I've only read about relativity in popular science media but maybe I should pick up a textbook and study the details.
 

1. What is the "Clock Thought Exp" and how does it relate to the concept of a light-year time paradox?

The "Clock Thought Exp" is a thought experiment that explores the concept of time dilation and its implications on our perception of time. It involves two clocks, one stationary and one in motion, and shows how time can pass differently for each clock depending on their relative speeds. This thought experiment is related to the light-year time paradox because it demonstrates how time can appear to pass at different rates depending on the observer's frame of reference, leading to a paradoxical situation where two observers may perceive different amounts of time passing for the same event.

2. Can you explain the concept of time dilation in more detail?

Time dilation is a phenomenon in which time appears to pass at different rates for objects moving at different speeds. This is one of the key principles of Einstein's theory of relativity. According to this theory, time slows down for objects in motion relative to a stationary observer. This means that the faster an object moves, the slower time appears to pass for it. This effect becomes more pronounced as the speed of the object approaches the speed of light, resulting in a time dilation factor known as the Lorentz factor.

3. How does the "Clock Thought Exp" illustrate the concept of time dilation?

The "Clock Thought Exp" uses the example of two clocks, one stationary and one in motion, to demonstrate how time dilation works. In this experiment, the clock in motion is moving at a high speed relative to the stationary clock. As a result, time passes slower for the moving clock, leading to a time dilation factor. This means that when the moving clock returns to the stationary clock, it will show a later time, even though both clocks started at the same time. This illustrates how time can pass differently for two observers depending on their relative speeds.

4. Is the "Clock Thought Exp" purely hypothetical or has it been observed in real-life scenarios?

The "Clock Thought Exp" is a thought experiment and has not been observed in real-life scenarios. However, the concept of time dilation has been confirmed through experiments such as the Hafele-Keating experiment, which showed a small but measurable time dilation effect for atomic clocks traveling on commercial airplanes. This effect has also been observed in particle accelerators and is taken into account in GPS technology.

5. What are the potential implications of the "Clock Thought Exp" and time dilation?

The implications of the "Clock Thought Exp" and time dilation are far-reaching and have been studied extensively in the fields of physics and cosmology. One of the main implications is that time is not absolute, but rather relative to the observer's frame of reference. This can have implications on our understanding of the universe, including concepts such as the age of the universe and the possibility of time travel. Time dilation also plays a crucial role in modern technologies, such as GPS, that rely on precise time measurements to function accurately.

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