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PeterDonis said:These are all coordinate-independent invariant facts about the scenario, and they are the facts that people are referring to when they talk about one clock "running slower" than the other.
Why I object to this language is that it really jerks the student around. The first introduction to SR very often talks about time dilation in a way that sounds as if there is an objective criterion for saying that one clock is running slower than another. Students have trouble squaring this with the claim that all inertial observers are equivalent. So you have to explain that "time dilation" is a coordinate effect, so one clock can be running slower than another according to one coordinate system and running faster according to another.
Now, switch to GR. People seem to be saying that, unlike with SR, time dilation is objective and coordinate-independent in GR. That's really ridiculous, since GR has no more absolute notion of one clock running slower than another than SR does. Yes, in a situation of a timelike Killing Vector Field, you can use that field to compare different clocks in a way that is coordinate-independent. But that is NOT a difference between GR and SR. Flat spacetime also has a bunch of timelike Killing Vector Fields. For two clocks accelerating inside a rocket undergoing constant proper acceleration and Born rigid motion, the exact same reasoning about a coordinate-independent way of comparing their "clock rates". There's a Killing Vector field associated with Rindler motion, and blah blah blah.
So I really do object to talk about GR somehow making time dilation into something more "objective" than SR. That just doesn't seem true to me. And it also seems unnecessary and misleading.