@barrycarter yeah, that's generally a good goal to aim for. Like I said, I have to take off for now but I would like to return to this some other time.
It takes AMT 25 years to get 20ly closer to Earth, so he passes NMT at that time. OF course, 25 years AMT time is only 15 years Earth time, so it's Earth 2015 then? (just my guess)
@barrycarter no, he doesn't assign the Earth anything. In the AMT's frame the Earth's trajectory is a straight line that passes through the point (0, -6).
@JohnRennie Rephrasing: there exists a time when AMT passes NMT. AMT assigns this as t=0. At that exact instant, he also looks at Earth, adjusts for light travel time and says: "The Earth is at position x, and time t". What are x and t?
The time on AMT clock agrees with the time on NMT's clock when they meet, but only when they meet. However at this moment the Earth clock doesn't agree with AMT's clock but does agree with NMT's clock.
That's because the time axes of the AMT and NMT are rotated wrt each other
OK, but here's the paradox. NMT, stirred by a fraternal love for this brother, abandons his sedentary ways and accelerates to 0.8c. Now, he must see the exact same thing his brother sees, right?
The question is easily answered by drawing a spacetime diagram in the Earth's rest frame:
I've scaled the time axis so that a light ray travels at 45º. You start at $x=10$ light years so the light ray emitted from the Earth at $t=-10$ years reaches you just as you start.
The red line shows yo...
When the AMT passes the NMT what the AMT would see is the light that left the Earth at t = -10.
OK, let's go back a step. If you're in a reference frame, how do you assign times to remote objects? I thought the answer was: what you see right now plus light travel time (as you compute it). Not true?
So if we can agree on a clock synchronisation at some point (typically a meeting point) we can calculate elapsed time along the trajectory in either direction from our clock synchronisation point.
So, if I'm moving at 0.8 c wrt Earth and I'm 6 ly away, and see (with my eyes) Earth is at t=0, what time do I say those light rays left Earth to strike my eyeballs now?
Does that mean at what spacetime point in my coordinates was the light ray emitted, or what was the time shown by clocks on Earth when the light ray was emitted. The two are different.
@JohnRennie So they both see a big sign saying "t=0" from Earth. Now, we ask both: when the light rays that just hit your eyes left Earth, where was Earth, and what time was it on Earth
In the AMT's frame the light was emitted when the the Earth was at x' = -30 and t' = -30. At that time the Earth's clock read -10. The light reached the AMT when the Earth was at x' = -6 and at that moment the Earth's clock read +8 years.
The ratio 18/30 is just the time dilation factor of 0.6
What we should really do is draw the spacetime diagram in the AMT's rest frame and label up the Earth's trajectory with the proper times. However I'm not doing that now.
The NMT and Earth agree on their times, so at the moment the light ray was emitted the simultaneous point on the NMT's trajectory was (t = -10, x = 0). We need to transform that point into the AMT's frame.
I have no problem plugging and chugging numbers to get answers.... but when I started to think about what those answers actually mean in the real world... blargh.
Even after more years than I care to think about I still find myself confused about what's actually happening with SR. If you want to go forward we need to draw the spacetime diagram in the AMT's frame.
@barrycarter : I wouldn't love it I'm afraid Barry. I'm afraid to say I think you're making to much of a meal of this SR stuff. I've answered your questions, it's simpler than you think. By the way, you have to say my name three times to summon me.
@JohnDuffield I thought finding an inconsistency in the currently accepted theory of relativity would elate you. Unforunately, the evil Rennie has won this round.
I am now torn between doing actual productive work and following this black hole of relativity.
@JohnDuffield (before we get too far, we may want to "get a room"). Yes, but your interpretation of it varies from the generally accepted interpretation.
The question is easily answered by drawing a spacetime diagram in the Earth's rest frame:
I've scaled the time axis so that a light ray travels at 45º. You start at $x=10$ light years so the light ray emitted from the Earth at $t=-10$ years reaches you just as you start.
The red line shows yo...
