From what I understand, the time rates (I'm not speaking about absolute times) of all clocks on earth's surface are synchronized. This means that, say, a mobile phone's clock is generally not beating the mobile phone's proper time – any synchronization would be out as soon as we swing the phone around – although the difference is probably undetectable with today's technology.
I assume this synchronized time rate is the proper-time rate of some observer. My question is: which one? For instance: An observer standing still on Earth at zero sea level and zero latitude? Or an observer at zero sea level and zero latitude that's standing still with respect to the distant stars? Or an observer at infinity (the $t$ coordinate in Schwarzschild's metric)? (My question is different from this question, which is about how synchronization is achieved in practice).
There are several nice works discussing effects of time rate differences on and around the earth, for example with relation to the GPS system (on which there are many questions here). I give a list below. But in none of them I managed to find an explicit answer to my question.
Glad if you can share some references where this is discussed!
[Edit: here are some useful references from @PM2Ring's informative answer:
- Petit, Wolf: Relativistic theory for time comparisons: a review
- McCarthy, Seidelmann: Time: From Earth Rotation to Atomic Physics (also here)
- Guinot: Is the International Atomic Time TAI a coordinate time or a proper time?