+PK+ wrote:I've been studying it a bit and I think I get it, but I'm still not very satisfied with the explanation of the twin paradox. I understand that length and time are contracted for the traveling twin,
"length and time are contracted for the traveling twin" isn't a very clear explanation--it's not as if you are required to analyze the problem in the inertial rest frame of the Earth twin, in relativity you're always free to use whatever frame you like to analyze a problem. For example, you could choose to analyze things from the perspective of an inertial frame where the Earth twin is moving at a constant velocity, whereas the traveling twin is at rest for the first phase of the journey, then after the turnaround is traveling towards the Earth twin at an even faster velocity so he can catch up. In this frame the Earth twin's clock is running slower during the first phase, but then in the second phase the traveling twin's clock is running even slower than that since his velocity is higher.
+PK+ wrote:but the SR explanation is kinda weak. The doppler shift explains why they see each other's clocks moving at various speeds but it doesn't explain the time dilation. Even though the traveling twin see's the earth twin's clock moving more slowly the traveller would notice the time dilation as soon as he slows down. Which would appear to him as if light is catching up to him.
If by "catching up" you mean he'd see the Earth twin's clock suddenly jumping forward that's not correct. If the traveling twin travels away from Earth at 0.8c, then while he's traveling he sees the Earth clock visually slowed by a factor of 3 due to the Doppler effect, so if he travels for 25 years in the Earth frame = 15 years according to his own clock (using the time dilation equation), he'll visually only see the Earth clock showing that 15/3 = 5 years have passed. Then if he "slows down" relative to Earth (there is no non-relative notion of 'slowing down' in relativity), and comes to rest relative to Earth at a distance of 0.8c*25 = 20 light-years away, from then on he'll see the Earth clock ticking at the same rate as his own since he's at rest relative to it, but visually it'll always be 10 years behind his own, just like it was when he first came to rest relative to Earth and saw his clock reading 15 years and the Earth clock reading 5 (so if he stays at that position for another 40 years, then when his own clock shows 55 years have passed since his original departure, he'll see the Earth clock reading only 45 years).
+PK+ wrote:Do you guys see where I'm getting confused? If inertial reference frames are all we can use for SR I don't think it properly explains the time dilation. I mean if you had to travel ten light-years to deliver a message, and you were moving so close to the speed of light it only seemed like five light-years to you, it doesn't mean you traveled five light-years. The people on the new planet will say that the radio broadcast of you leaving earth arrived just moments ago, but that both planets had agreed, through a very long series of communications of course, that you would leave ten years ago. Yet you saw earth's clock moving slower than your own the whole time while traveling. So at some point light is going to appear to you to have gone faster than it should, as you're view of earth's clock while you sere traveling catches up to the earth clock you now see at rest.
No, the key to a problem like this (and many others, like the one I hinted you should think about with the train on the straight track) is the relativity of simultaneity
, which says different inertial frames disagree about whether two events happened at the "same time", which means they also disagree about whether a pair of clocks at rest relative to one another are "synchronized" or not. If two clocks are at rest relative to each other and a distance d apart in their own inertial rest frame, and they are synchronized in that frame, then in a different inertial frame where the two clocks are both moving at speed v along a straight line, at any given moment in this new frame the rear clock's time is ahead of the front clock's time by an amount vd/c^2.
So if this other planet is at rest relative to the Earth, and in their mutual rest frame their clocks are synchronized (and the distance d between them is 10 light-years in this frame), then in the inertial rest frame of a traveler moving relative to them at 0.99999c, they are out-of-sync by 9.9999 years. Thus if the Earth clock shows a reading of 0 years when this traveler is leaving it, in the traveler's frame the other planet's clock already
reads 9.9999 years "simultaneously" with the event of the traveler leaving Earth. So there is nothing inconsistent from the traveler's point of view about the fact that the other planet's clock reads just over 10 years when he reaches that planet, in spite of the fact that the other planet's clock was ticking very slowly in his frame, since it already had a "head start" of just under 10 years.