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I'm a bit rusty on my special relativity and have been thinking about this problem recently:

Suppose that a massive projectile (i.e. something large enough that we could see with a telescope) is ejected from the Sun towards the Earth at 0.9c at $t=0$, for example. Let's approximate that the distance between the Sun and Earth is such that light takes 10 minutes to reach the Earth from the Sun.

After 10 minutes have passed on the Earth, the light from the ejection event reaches the Earth at $t=10$. However, during these 10 minutes, the projectile has traveled 90% of the distance from the Sun to the Earth (i.e. if $d=0$ at Sun and $d=10$ on Earth, then $d=9$ for the projectile at $t=10$ in the Earth's frame). It seems to me that in another 1/0.9 minutes, the projectile will hit the Earth. However, at $t=10$, we will have just seen the projectile leave the Sun. Thus, will it appear as if the projectile travels from the Sun to the Earth in only 1/0.9 minutes? This seems to violate the principle that nothing travels faster than the speed of light. I know that this must be incorrect, but I can't seem to figure it out without dredging up years-forgotten SR. Could anyone point me in the right direction for rectifying this seemingly paradoxical situation?

WillO
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Sheridan Beckwith Green
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  • Nothing did travel faster than light. If you measure the velocity of any object that moves towards you, you have to take this effect into account. – thi Jul 12 '18 at 16:54
  • The search term is (no surprise here) "apparent velocity", and the situation is discussed in any reasonably complete introduction to special relativity. Alas, many introductions (even some in texts and many on the internet) simple aren't not in anyway complete. Local posts on the matter. – dmckee --- ex-moderator kitten Jul 12 '18 at 17:16
  • Special relativity doesn't tell you what an observer "sees." It tells you what an observer would "calculate," after taking the finite speed of light into account. – D. Halsey Jul 12 '18 at 21:27
  • Note that this question has absolutely nothing to do with relativity. It is about calculating velocity in a single reference frame, not about comparing two different frames. I have therefore changed the tag. – WillO Jul 12 '18 at 23:28

1 Answers1

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I believe

Thus, will it appear as if the projectile travels from the Sun to the Earth in only 1/0.9 minutes

is a correct statement. (Appear being the key word here). Nothing in this situation actually traveled faster than the speed of light, and the appearance of the object traveling so quickly is an illusion.

A similar illusion and thought-experiment is flicking a laser beam across the surface of the moon, as described and explained here: https://www.universetoday.com/109147/how-a-laser-appears-to-move-faster-than-light-and-why-it-really-isnt/

Mxracer424
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    Indeed, after I posed the question I also found this https://www.newscientist.com/article/2131889-weird-energy-beam-seems-to-travel-five-times-the-speed-of-light/ so you may be right that the appearance of FTL travel doesn't actually violate SR – Sheridan Beckwith Green Jul 12 '18 at 17:01