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I have started reading about relativity and am grappling with the following issue.

If I shine a light beam from a device no matter whether I'm standing still or moving at e.g. 50km/h in the direction of the beam, the light beam will move away from me at the constant speed of light. I understand that the speed of light relative to the observer is always the same no matter the speed of motion of the observer.

However, if I shine a strong enough beam of light towards a planet exactly 1 light year away, it will take that light 1 year to reach the planet. Let's say I then within 10 seconds travel behind the beam at a fraction below the speed of light. If the beam I shone from my device is always moving away from me at the speed of light despite my speed, when do we both reach this planet? That is from the planets perspective, (assuming there are people thereon to observe the approaching beam and me).

Do we not both reach the planet within 10 seconds of each other, yet from my perspective if the light is travelling at a difference being the speed of light ahead of me, it will reach the planet that much quicker than me and not 10 seconds? What would the difference in time be from my perspective and the people on the planet's perspective and why? I'm confused.

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And with that thought experiment, you rediscovered Einstein's theory of relativity. You should be proud to come close to the main questions that lead to this theory on your own.

Time depends on the observer. When you said you sent light to 1 light year away, it will take 1 year for the observer you leave behind on the earth. For light, no time at all passes (technically, there is no inertial observer that can vouch for what light would feel, but in the limit, we can sort of say that). If you travel very close to light speed, the observer on the earth will see you arrive a bit later than the light. Your travel time will be longer, consistent with definition of speed: $t=x/v$. However, for you, much less time will pass (the ratio is the infamous $\gamma$ factor from special relativity). So if the earth observer looks at your watch, he will see your time pass very slowly. That actually means that with relativity, you get there faster (in your own time) than the time suggested by the distance, when calculated by the outside observer. But that's ok - in your perspective, the observed distance to the planet shrinks by the same factor so you agree about the outcome with the outside observer. You just need to know what velocity is: it's distance measured by person A divided by travel time measured by person A. If you mix observers, you get nonsense.

This whole thought experiment leads to the twin paradox, which then argues who's younger when you turn around and come back to earth. The paradox is caused just by sloppy treatment of acceleration times, in fact, the traveller always experiences less time.

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