Cause of time dilation when net gravity is zero

[stevendaryl]

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.

 

[1977ub]

I really think that it's very misleading/false to say that time dilation is more "absolute" in GR than in SR. The time dilation for clocks at different heights in a gravitational field is no more "absolute" than the time dilation for clocks at different locations within an accelerated spaceship. The latter can be computed using SR. So I don't agree at all with the claim that GR makes time dilation more real or absolute or anything than SR.

For linear non-accelerated SR, the world views of the the different observers cannot be combined into a single narrative. In the accelerating ship, observers at the front and back agree on whose clock is moving faster, just like in GR where observers at different altitudes agree which clock is faster.

 

[PeterDonis]

focusing on comparison of elapsed times for different paths is always applicable.

That's true, but such a comparison is also independent of your choice of coordinates in all the cases we've discussed in this thread. Either the two paths have a pair of events in common, or there is a coordinate-independent way of picking out which events on each path "correspond" for purposes of the comparison. It seems to me that a blanket statement that "time dilation depends on your choice of coordinates" obfuscates these coordinate-independent facts.

Part of the problem (which has been discussed here on PF before) is that the term "time dilation" is overloaded. We really need at least three separate terms: (1) one for the coordinate-dependent quantity ##d\tau / dt##, (2) one for the coordinate-independent comparison of elapsed times between "corresponding" events on different paths, and (3) one for the coordinate-independent frequency shifts in light signals between events of emission and reception. Unfortunately there are no such distinct terms in the literature.

I really do object to talk about GR somehow making time dilation into something more "objective" than SR.

Please note that such talk is not what I was arguing for. I was arguing for being clear about what quantities depend on your choice of coordinates and what quantities do not. You are perfectly correct that one can pick out coordinate-independent quantities in an accelerated rocket in flat spacetime that correspond, in a useful sense, to the "gravitational time dilation" (the second of the three things I described above) in a stationary curved spacetime. And it seems to me that a blanket statement that "time dilation depends on your choice of coordinates" obfuscates that as well.