If I was running a mile. Even if I could run at the speed of light. The distance I would run would still be 1 mile, regardless of how fast I ran. I admit it would seem like it took almost no time to run it, since I just blazed thru it. But it would have still been a mile.
But when you start getting the distances out there to 30 million light years, It would take me a little while to run that.
When you say that you ran a mile, you measured that mile in some reference frame.
So, suppose you measured out a mile on the Earth (in the reference frame at rest on the Earth).
Now, suppose you run this race at 86% of c with respect to the Earth. In the earth's reference frame, you would take 1 mile/.86c = .000006242 seconds. This is using the speed of light as 186282 miles per second.
Now, in *your* frame of reference, the earth is going past *you* at 86% of c. Are you OK with my saying this? if not, then we have to discuss this. If you are going past the Earth at 86% of c, then the Earth is going past you at 86% of c.
Now, the reason I use 86% of c so often is that the dilation factor is simple: it is .5.
So, in *your* frame, that mile on the Earth is only 1/2 a mile. This is the precise, accurate measurement. It is going past you at 86% of c, and so it takes .000003121 seconds for *you* to go from the start of the race to the end.
Next, when you say that star is 30 million light year away, you measure it in some reference frame, probably that of the Earth. If you are in a spaceship going at 86% of the speed of light, it would take you 30 million/.86 = 34.88 million years to get there *from the reference frame of the Earth*.
But, in *your* reference frame, that star is moving towards you at 86% of the speed of light. And in *your* reference frame, that star was originally not 30 million light years, but was 15 million light years away. So it only takes 15 million/.86 = 17.44 million years until the star reaches you.
Both the time of travel and the distance of travel depend on the specific frame used to measure them. No frame is more or less correct than any other frame.
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