Why doesn't time get slower and slower in a g-field?

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g-field as in gravitational field (character limit)

As far as I know, special relativity says that observers traveling fast experience slower time than observers at rest. So if an observer were to accelerate in a space ship, his time would get slower and slower relative to ours.

But the observer in the space ship cannot tell the difference between a reference frame at rest in a gravitational field and an accelerating reference frame, and neither can we. So why isn't our time getting slower and slower, just like his? As far as I've learned, physics is the same in a reference frame that is accelerating and one at rest in a gravitational field (Einstein's elevator thought experiment).

Sorry if this is a dumb question, and thanks in advance.
 

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jfizzix
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Actually, because we're in a gravity well (i.e., the surface of Earth), time does pass by just a teensy bit more slowly for us than for objects at rest higher up (say, on top of a mountain). This is a small effect for Earth, but it's a lot more significant near the surface of a neutron star (or the event horizon of a black hole).
 
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PeterDonis
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special relativity says that observers traveling fast experience slower time than observers at rest.
No, that's not what SR says. All observers experience the same flow of time, one second per second. What can vary is how much experienced time elapses for observers following different paths through spacetime. But to know how much time has elapsed along different paths, you need to have some way of picking out "corresponding" events on the different paths. The simplest way is for the paths to meet again after separating; then the paths have a pair of events in common (where the two observers separate, and where they meet up again), and can just compare their experienced times between those two events.

if an observer were to accelerate in a space ship, his time would get slower and slower relative to ours.
And our time would get slower and slower relative to his. Time dilation in this sense is relative. And if the two of us kept moving apart, there would be no invariant way of picking out "corresponding" events on our paths through spacetime, so there would be no way of saying that either one of us was "really" aging faster or slower than the other.

the observer in the space ship cannot tell the difference between a reference frame at rest in a gravitational field and an accelerating reference frame
True.

why isn't our time getting slower and slower, just like his?
Because his isn't. See above.

What does happen, as jfizzix pointed out, is that if we have two observers, both at rest in a gravitational field, but at different altitudes, then they will age at different rates; the one at a lower altitude will age more slowly. The observers can make this comparison in an invariant way by exchanging round-trip light signals; the observer at the lower altitude will experience less elapsed time for a round trip of light between the two. This works because the observers are at rest relative to each other. (Note that the fact that they are at rest relative to each other is a key difference between this scenario and the scenario where two observers are in empty space and one accelerates away from the other.)
 

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