I might have this wrong but I remember reading that when a rocket ship is moving away from the earth at some significant fraction of the speed of light, time will pass more slowly on the moving ship relative to time passing back on earth. Now, the puzzling thing to me is that we are told that from the point of view of the rocket ship time would also seem to be passing more slowly on earth, which seems counterintuitive to me. I understand that from both perspectives, each one is moving relative to each other but since it is the rocket ship that is moving very fast and not the earth how can this be? Sorry if I have this completely wrong but that’s why I’m asking. 
What does “moving very fast” mean?
Well, what I meant was before the rocket ship begins travelling, it is moving at the same speed as the earth and then begins to move away from it at say, near the speed of light, so it is the rocket ship that is moving away from the earth so shouldn’t it be the on-board time that slows and not vice-versa?
It’s not the speed that causes time to slow down on the rocket ship. It’s acceleration. There is no absolute frame of reference with respect to speed, so you’re right that from the point of view of someone on the rocket, the earth could be seen as moving near the speed of light. But the acceleration that causes the rocket to approach the speed of light is something that isn’t felt on earth.
By the way, acceleration and gravitational attraction are the same thing in Einsteinian physics. If you’re on a big planet with a lot of gravity time will run slower for you than if you’re on a small planet with less gravity.
Is the rocket moving away from the earth, or is the earth moving away from the rocket?
That’s not correct. Time dilation is a function of relative velocity, given by the Lorentz transformation.
I think you’ve got it right. For an explanation, check out Wiki on the Twin Paradox; specifically this section: Twin paradox - Wikipedia
The question and answers are hinting at thetwin paradox.
Acceleration is how you move from one inertial reference frame to another. Critically for the twin paradox you need to turn around and come home, and it is this change of reference frames that is needed. The rocket’s deceleration, turn around, and acceleration to effect the jump from one frame to another is what allows the paradox.
Otherwise, it is just about the Lorentz transform, and that gets you the time dilation once you are comparing inertial frames. Which means both the earth bound and rocket clocks see the other as slow.
Ninga’d 
Ummmm…both, I guess. :o
Thanks, Ignotus. 
Okay, thanks for the explanation, although it’s still hard to grasp it.
Abashed, I think you would learn considerably from a classic book for non-scientists by George Gamow, One, Two, Three, Infinity. Gamow was a famous theoretical physicist; the Tyson of his Time or the Sagan of the Century. He tackles many of the subjects you have been asking about here, like relativity, motion, and the square root of -1. Pick up a cheap used copy and enjoy!
That’s the idea. The whole point of relativity is that you have to stop thinking in terms of one thing (like you, or the earth) being “still” while the other thing is in motion. That’s called a “privileged” (i.e. special) reference frame; relativity tells us that no reference frames are privileged. If a rocket is moving relative to the earth, it is exactly the same as the earth moving relative to the rocket.
The Lorentz transformation is a formula which takes relative velocity (how fast the rocket is moving relative to the earth) and tells you how much time dilation there will be. (It also will tell you the length contraction.)
It’s not important to grok all the math, but the formula tells us a few important things:
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As your rocket’s relative velocity approaches the speed of light, the speed of a clock you observe on earth approaches zero.
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If you plug into the equation a velocity faster than the speed of light, the result is a complex number, which is nonsensical in our sadly physical universe. So that’s why you can’t go faster than the speed of light. So we say that the Lorentz transformation is only defined for absolute velocities less than c.
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If you plug into the equation a velocity of exactly the speed of light, you get 1/0. So don’t do that. You’ll break the universe.
Thanks for the tip, Musiccat, I’ll take a look at that.
Yeah, I’m coming round to the idea although it still seems a bit crazy. So the earth and the rocket do not have any special status in terms of movement because both are moving relative to one another, yes? And since both are moving near the speed of light in relation to one another, from both viewpoints the other’s time seems to slow down, okay?
But I’m still not quite sure why, when the rocket decelerates, turns round and re-accelerates, more time has passed back on earth than on the rocket. How does changing time frames cause this?
Also, it seems to me this is why the speed of light is always the same whatever speed you happen to be travelling at because it is as if the rest of the universe has alter its frame of reference to accommodate this, which seems extremely weird.
Right.
The reason is because relativistic velocities cause both time dilation and length contraction. Let’s say you go on a mission to a place 10 light-years away, traveling at just under the speed of light. (We’ll ignore the time it takes to accelerate.) From Earth’s point of view, you are traveling a total distance of 20 light-years (there and back) which should take just over 20 years to do. From your point of view, you are actually traveling less distance - call it 16 light-years there and back - and so it actually takes you less time to complete the mission than Earth would observe.
It is weird, and counterintuitive to our everyday experience. I think it helps to think in terms of “the speed of light is constant” as an axiom, and then progress through thought experiments to see what the consequences of that are. That’s how Einstein did it.
As long as they’re just moving away from each other, in each’s reference frame, the other’s time is moving slower. This only becomes a problem if they meet up again, but if they both stay in their respective reference frames, that’ll never happen. The only way they’ll ever meet up again is if (at least) one of them changes reference frames. If that happens, then there will be an asymmetry in which one changed. You can then pick any reference frame you like to do the calculations in, and in any of them you’ll get the same answer for how their clocks compare when they meet back up.
Length contraction…so are you saying because going near the speed of light ‘shrinks’ things it reduces the actual distance you have to travel?
Yes, I think once you understand that the speed of light is always constant, no matter what your speed is, you can see that something has to be happening to your local time to allow light to maintain its speed otherwise it would not be able to ‘catch up’ with you. I can see the logic of it but must have taken some working out at the time as it seems so ridiculous. I guess this is why Einstein is so admired - he was able to think right out-of the-box.
Okay, thank you, but I’m still having trouble with this. What is it about returning that makes the time on earth pass so quickly? Is it because the light has less and less distance to travel to you and, therefore, the time on earth does not have to pass as slowly as it did? So although your ‘on-board’ time is travelling very slowly, the time elapsing on earth has speeded up greatly? Is it like in the first case you are really ‘stretching’ time, while in the latter case you are ‘contracting’ time?
Right, but although Jeff Lichtman is wrong to say that time dilation is [only] due to acceleration, it is of course due to both relative velocity AND to acceleration (e.g.- the presence of a gravitational field). This is famously seen in the adjustments that have to made to the atomic clocks in GPS satellites, which run slightly slower than clocks on earth due to their relative velocity, but faster due to being away from earth’s gravity – the latter effect turns out to be dominant.
Think of it this way. Imagine a pool where a cork is bobbing in the water creating one wave per second and the waves radiate outward at 1 inch per second (the analog to the speed of light). If I am standing still in the lake I see a wave once every second thus 1 second to the cork equals 1 second to me. So I can measure time passage at the cork. Every time a wave hits me, one second has passed at the cork. To help the analogy, imagine every second I take a picture from my spaceship. In this scenario, a picture will show Earth one second after the previous picture i.e. a picture is taken as each wave reaches me.
Now I move away from the cork at 0.5 inch per second starting when wave #1 hits me. One second later I will be halfway between wave #1 and wave #2. To go to the spaceship analogy, my photo will show the Earth 1/2 second after the previous one*.
Now when wave #2 hits me I instantaneously increase my speed to 1 inch per second. This means I will always be on top of wave number #2 thus according to my data, time never passes at the cork. On the spaceship this would be the photographs showing the same moment (Second #2) every picture.
Now imagine that in reverse. I travel back to the cork at 1 inch per second. This means that every second of my time I pass two waves meaning two seconds has passed at the cork. A picture from my spaceship shows the Earth two seconds after the previous one. Is that the effect you are talking about?
*I know time dilation is not linear like this.
Question for Chronos: I travel along a geodesic on Earth at close to the speed of light or for the sake of something bigger imagine the universe as the surface of a sphere. Doesn’t that contradict “but if they both stay in their respective reference frames, that’ll never happen.”?