How is the highlighted portion of your statement possible? Isn't the position in the sky at any one moment determined by how long it takes to cross the sky?
Because where you have counted 2 hours I have counted 1. Thus, we will agree on the position, but you'll say it took 2 hours to get there, I'll say it took 1. In other words, I am experiencing time twice as fast as you. If I say to you "call me in an hour", my phone will ring in 30 minutes and you'll say "ok an hour has past and the sun is X". I'll say, "Sure the sun is X but it's really been only 30 minutes".
I think where you're confused is that you're not understanding what it means for me to be experiencing time faster. You're assuming that we will agree on the velocity with which the sun is moving across the sky. We won't. We don't agree on how fast the earth is rotating, because everytime you have experienced 2 seconds, I have experienced 1. Thus, I will feel the Earth rotating twice as fast as you. This means the sun will appear to be travelling twice as fast.
Even if the Sun started at the same point, if it is going across the same sky at a different rate for one than it is the other, the reported position of the Sun should immediately become out of sync just as the clocks do. And guess what, it doesn't! And that's just the exact point of the whole opening post.
That's where you and the opening post are wrong. This is something I know you're still not completely understanding. It only becomes out of sync if you assume 2 hours for you is 2 hours for me. We experience time differently. Here, I'll put it into math form for you. Assume we are measuring the apparent velocity of the sun (ignoring that it's the earth that is spinning).
v = d/t (v=velocity, d=distance, t=time)
d = vt
We know that we will always agree on distance the sun has travelled, because the sun is in the same position in the sky for us. However, I'm experiencing time twice as fast as you (according to every clock). Assume we decide to call each other when the sun reaches it's midpoint.
For Clete:
distance = 30m, time = 5 hours thus velocity = 6 meters/hour
For Johnny:
distance = 30m, time = 2.5 hours thus velocity = 12 meters/hour
So you see, we both saw the sun travelling at different speeds because we experienced time differently, nonetheless we still agree that the sun travelled 30m. You will call me and say "it's been five hours and the sun has gone 30m". I'll respond "no, it's only been 2.5 hours, but the sun has gone 30m". This is what it means to say that time is relative. That's a terrifically crude example but it illustrates that
the only way you can say things will fall out of sync is if you assume time is absolute. If you fix that time variable then sure, distance will change and we won't agree on the position of the sun . I understand that this is an extremely primitive and rough way to underline what I'm trying to convey, so all the physicists out there realize that I understand there is probably a better way, but I've got a short amount of time and I need to keep it simple for most readers.
Relativity does not have an answer for this question. That's what's got all you guys running in circles trying to find something wrong with Bob's hypothetical. But there isn't anything wrong with it, it's perfect. The only conclusion possible is that gravity does not affect time itself but only clocks.
There is no question involved, only your misunderstanding. Bob's example is an erronius example. It's a misunderstanding, that's all it is. But it's understandable from a novice. That's why initial responses were less pointed.
You keep repeating that "gravity doesn't affect time itself but only clocks" and then when I asked you to define time and how it's different from the measurement of time you worked yourself into a corner in no time. Here, just in case you forgot:
Clete: "Time "itself" does not exist. What we call time is simply duration and sequence. Clocks measure both. Something I've said about a thousand times."
Johnny: "If you have defined time as duration and sequence, and then you tell me that clocks measure duration and sequence, and clocks show a the effects of relativity, then doesn't relativity effect duration and sequence, which is what we have defined as time? So relativity affects time"
But if I have a clock that's telling me that the sunset that I am witnessing shouldn't be happing for another 15 hours, I'm throwing that clock in the trash.
Your clock only predicts the sun's setting accurately because of the extremely gravitational field the clock is in.
Now you're starting to get this thing! It would seem for the purposes of our experiment that such a clock would have turned out to be far more accurate. More accurate to the tune of a full 24 hour day in the case of Bob's opening post.
Why would it be more accurate? We'd have no way to compare times. If I wrote you cooking directions that said cook for 15 sunminutes and you did so your meal would be burnt. This is because 15 sunminutes is actually a shorter duration for me than it is for you.
The movements within our solar system give us a more correct understanding of the absolute nature of time than do the ticks of atomic clocks."
You're really not understanding here. If we used the sun as a clock in the scenario we're describing we'd be worse off and have no sense of time. We'd agree on how long has passed only because we've arbitrarily said that the sun's passage across the sky was 24 sunhours. If we both had radioactive material that decayed over a set interval and we timed it, we'd come up with different times. You'd say 15 sunminutes I'd say 30. Using a clock outside both observer's inertial frames would be disasterous and would render all communication about time meaningless because we don't experience it the same way.
What is the difference between time and the measurement of time? If you've defined time as the interval between two events, and the interval changes, how can you say that the time between two events has not changed?
Bottom line is this: the information is out there. Buy a relativity for dummies or something. That's what I had to do for quantum mechanics. But you're just embarassing yourself now.
One last question for you. If I accelerate a particle with a known decay time to near the speed of light, and it takes longer than usual to decay, how can you say that time has not dilated for that particle?
Example: My particle decays in .10 seconds at rest with me. I accelerate it to .999c and it takes 1.0 seconds to decay (by my measurement). What just happened?