Clete said:
If this is so then you should be able to demonstrate the flaw in our (Bob and I's) logic.
We both watch the sunset together in spite of the fact that one of our clocks insists that its 4 hours before the Sun should even rise.
Here's the flaw: Just because you're watching the same sunset doesn't mean the same amount of time has passed for both of you. The sun is not in either of your inertial frames, so you will agree on it's position. However, you will not agree on how long it takes to cross the sky each day.
Assume you're both keeping track of time a) by a wristwatch and b) by oil dripping from the sack, and c) by your beard length. Assume that all of these are perfectly synced when you're together (yes, your beard grows at exactly the same rate as your partners). One of you moves to the summit, one of you stays at the base.
You both watch the sun rise and set. You call your partner at the peak right after the sun has set and tell him that between sunrise and sun set there were 12 hours, 500 oil drops, and your beard grew 1 mm. However, your partner disagrees. He measured 12.1 hours, 550 oil drops, and his beard grew 1.1 mm.
So what can we learn from this? Well, first, they don't disagree on the sun setting. If they were on the phone together they would agree. However, they do disagree on when the sun set. One will say "welp, it's 6:01 and the sun just set" and the other will say "no, it's only 6:00". What does this mean? It means that time is entirely relative. One person may experience three seconds, but within that three second period (as he measures), another inertial frame may count 5 seconds. In other words, if the man at the base camp peered up at the man at the peak through a telescope and looked at his watch, it would be ticking fast. The man would also appear to be moving slightly faster than normal. However, if the man at the peak peered down through a telescope at the man at the base, he would notice that the base watch was ticking slow. The man at the base would also be moving slower than usual. It would take him longer to read a page of a book than normal, and he would have slower reflexes. Even the sun's movement across the sky is slower.
So who's watch, oil sack, and beard length is the "real" one? They both are.
You may say, "well why don't they use the sun as a clock so they both agree on how long each day is: from sunrise to sunrise". They could very well, but this will not stop one from experiencing time differently. Assuming they recalibrate everything according to the sun. They define one sunsecond as the time it takes the sun to move half a degree in the sky. So each one builds a clock that keeps track of time by the sun's motion. Their sunclocks will always read the same time, because they both see the sun in the same position.
However, they will still not agree on how long things take. For example, assume they're both watching particles decay. One will say "it took two sunseconds for my sample to decay", but the other who had an identical sample will say "no, it took 1.5 sunseconds for my sample to decay". So you see, the example Bob gave is actually very good at illustrating that relativity is much much deeper than a trick of clocks.
If you use the sun as your clock, you won't agree on how long things take. If you use wrist watches as your clock, you will agree on how long things take, but you won't agree on what time it is.
Doesn't being in a time frame with little or no gravity result in eternity and therefore pushed to the nth degree infinity? Silly question, is it not?
Good question, it's almost the opposite. If spacetime could be infinitely curved then time would also be infinitely dilated (but it can't). For example, if someone was in a place that spacetime was grossly curved (such as a black hole), they would see time severely dilated.