harrylin said:
Evidently an elaboration is necessary: I wrote ceteris paribus and that
includes relativistically moving observers. Surely all observers agree that clocks tick slower near co-moving masses; this is the actual physics of what you will measure with real instruments.
Else you would get the kind of contradictions that the OP imagined.
I don't see how taking "all things equal" applies here, especially not to the point I'm trying to make about the requirement for a stationary observer to define a notion of simultaneity in order to be able to compare clocks.
In general, if you have two observers A and B in relative motion, they can't agree whether A's clock is faster or whether B's clock is faster. This is true even in flat space-time, and remains true in curved space-time. This is because they have different notions of simultaneity.
Graphically, I think of it rather like the attached diagram below (which is for flat space-time) where one observer uses red lines to compare the clocks, and the other the green lines. (Add: I tried to upload this, don't see it, and it won't let me re-up).
The purpose of the stationary observer is to define the "set of lines" on the space-time diagram that define simultaneity which is a pre-requisite in order to be able to compare the clock's rate.
I've also seen a LOT of confusion in the past over this issue, related to the issue of what a relativistically infalling observer to a black hole sees. If one takes the "time slows down" argument seriously, one would think that an infalling observer sees time "slow down" just like a stationary one, and that since a stationary observer sees the future of the universe play out in a flash as, so does the infalling observer.
The actual situation is that the infalling observer does NOT see the future of the universe play out in a flash. IIRC, a detailed calculation shows that for an observer free-falling from infinity, the clock at infinity appear to be slower than his own from his (infalling) viewpoint, not faster (as judged by the doppler shift of received signals that the infalling observer receives).
That's happens because the infalling observer has a different notion of how to compare clocks (i.e a different notion of simultaneity) than the stationary observer - though one can explain it in other terms. The whole "rate of time" idea doesn't really hold up reliably when you have speeding observers. In flat spacetime, or curved, it's pretty common to see the "slow/slow" case, where both observers think the other's clock is slow.