Dropout said:
In this experiment with 2 colony ships and a planet, I purposefully started and stopped the experiment with ShipA and ShipB in equal frames of reference so nobody could say, "Simultaneouty is why this or that happens."
I think you're confused here--just because everyone agrees that they share the same reference frame, that doesn't mean you can't analyze things from the perspective of a reference frame where both are moving at the same nonzero velocity. And if their ages are the same in their own rest frame, their ages will be different in a reference frame where they are moving.
Dropout said:
It doesn't matter when, how fast, or how slow Planet C receives a radio transmission from ShipA or ShipB. A message is a message nomatter how slow it plays back, when you hear it, or how weak the signal is. If the radio message says at 0.000000000000000000000000000000001 normal speed that, "THE COLOR OF THE PEN IS BLUE!" Then the color of the pen is blue nomatter if the signal came out of a black hole, from a different galaxy, or was played backwards.
How slow it plays back is indeed irrelevant, but when you hear it is not. Suppose in 2005 you look through your telescope and see an explosion 100 light years away--would you say that in your frame, the date the explosion happened was 2005? Of course not, you'd take into account the finite speed of light and retroactively assign it a date of 1905. Then if you saw another explosion in 2025 from a distance of 120 light years away, you'd say this explosion happened in 1905, and thus that the two explosions happened simultaneously in your frame. But if an observer in a different frame also assumes that light travels at the same speed in all directions in
his frame, then he will assign different dates to these explosions, and say that they did
not happen simultaneously.
It would also be possible to assign dates to events using only local measurements made right next to the event. Suppose I am sitting on a giant ruler which is at rest relative to me, and mounted along the ruler is a series of clocks, which are all "synchronized" in my frame (more on what this means in a second). Then if an explosion happens right next to the ruler, I can just look at what marking on the ruler this explosion happened next to, and what the reading on the clock at that marking was at the moment it happened. Another observer may also be riding on a ruler that's at rest relative to
him, and which is moving parallel to my ruler alongside of it, so he can assign coordinates to the event using the same procedure. But the key here is that according to Einstein, each observer should "synchronize" the clocks along their own ruler using the
assumption that light travels at the same speed in all directions in their own frame--but if each observer uses this assumption, than each observer will see the other observer's clocks as being out-of-sync. To see this, suppose I set off a flash at the exact midpoint of two clocks. If I assume light travels at the same speed in all directions in my frame, then I should define the clocks to be "synchronized" if each one reads the same time at the moment the light from the flash hits it. But if another observer who sees the two clocks moving also assumes light moves at the same speed in all directions in
his frame, then from his point of view the light must hit the two clocks at different times, since one clock is moving
towards the point where the flash was set off and the other is moving
away from that point. So, if each clock reads the same time when the light hits it, this other observer will say the two clocks are out-of-sync (I drew some diagrams of two rulers moving alongside each other in
this thread, illustrating how each one thinks the other one's clocks are running slower and are out-of-sync, yet they are consistent in their predictions about physical events in the same local neighborhood). The end result is that each observer will get the same result for the coordinates of different events if he relies on local measurements on a system of synchronized clocks that he would if he relied on the idea I outlined in the previous paragraph, where you look at the time you received light from an event and then subtract the time the light took to get to you. Either way, if two events (such as A and B celebrating their 35th birthday) happen "at the same time" in one frame, that means that in another frame the two events happened at "different times".