The synchronized clocks on earth's surface: at which observer's rate are they beating?

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:

]

I assume this synchronized time rate is the proper-time rate of some observer.

Yes and no. ;) UTC is derived from TAI, International Atomic Time

a high-precision atomic coordinate time standard based on the notional passage of proper time on Earth's geoid. TAI is a weighted average of the time kept by over 450 atomic clocks in over 80 national laboratories worldwide. It is a continuous scale of time, without leap seconds, and it is the principal realisation of Terrestrial Time (with a fixed offset of epoch).

TAI / Terrestrial Time is notionally the proper time of an observer at sea level, but it's not actually defined in terms of an observer at some location. Instead, it's defined as a scaled version of TCG, Geocentric Coordinate Time, which

is equivalent to the proper time experienced by a clock at rest in a coordinate frame co-moving with the center of the Earth: that is, a clock that performs exactly the same movements as the Earth but is outside the Earth's gravity well. It is therefore not influenced by the gravitational time dilation caused by the Earth.

Since TCG is defined in terms of an observer at rest relative to the Earth's center it is not affected by special relativity time dilation due to Earth's rotation. The conversion from TCB to TAI is a simple linear equation, so TAI is also oblivious to SR time dilation variations; it assumes a constant rate difference due to gravitational time dilation.

Incidentally, the Lorentz time dilation factor due to the Earth's rotation speed at the equator is $$\gamma \approx 1 + 1.2×10^{-12}$$.

Another important precision time scale is TCB, Barycentric Coordinate Time

It is equivalent to the proper time experienced by a clock at rest in a coordinate frame co-moving with the barycenter (center of mass) of the Solar System.

Closely related to TCB is TDB, Barycentric Dynamical Time, which is used by JPL for

calculating orbits and astronomical ephemerides of planets, asteroids, comets and interplanetary spacecraft in the Solar System.

High precision conversion between TAI and TDB is rather elaborate, since it takes into account the gravitational field of all the major Solar System bodies. For details, see The JPL Planetary and Lunar Ephemerides DE440 and DE441, Park et al (2021). https://doi.org/10.3847/1538-3881/abd414

If you really want to dive into the rabbit-hole of UTC and leap seconds, see A brief history of time scales, by Steve Allen of the Lick Observatory.

• Fantastic, thank you! I'll check your last reference and the ones given on the wikipedia pages. Please feel free to add any other references not on wikipedia. Dec 9, 2023 at 10:11
• PS: the usual problem with the wikipedia pages is that many statements are given without proof or reference. For instance "It is equivalent to the proper time experienced by a clock at rest in a coordinate frame co-moving with the barycenter (center of mass) of the Solar System..." :( Dec 9, 2023 at 10:12
• @pglpm There are lots of details on the IERS site, eg iers.org/IERS/EN/Science/Recommendations/resolutionB5.html However, that site is being upgraded and its internal search is currently disabled. There are other sites with details on IAU resolutions, eg iaufs.org/res.html Dec 9, 2023 at 10:35