Right, so I think the answer to the question "Where in the world...", succinctly, is: "On the geoid", correct?

I don't believe GPS ticks at exactly one second at every location on the Geoid. It might be close, but since the definition is related to hydrostatic equilibrium of a rotating fluid under the influence of lumpy local gravity I'm not convinced that it happens to also be the perfect iso-tick surface at the same time. Can you cite an authoritative source for this? For example be it reference ellipsoid or geoid, I don't think the gravitation potential $\phi$ and rotational speed over either surface combine to give the same relativistic correction everywhere

@uhoh In the rotating frame, points with the same effective gravitational potential are level with each other and also have the same redshift. If this was not true, you could make a perpetual motion machine by sending energy from one point to another with photons and back mechanically.

I don't know what "effective gravitational potential" means precisely in this context (again no linked sources), but I do know that the centrifugal force used to calculate the effects of Earth's rotation on the geoid might not dilate time the same way it moves water around. While time and water both "flow", they're not the same thing.

@uhoh If two places have the same effective gravitational potential, it takes no work to move mass from one to the other. They are at the same level.

@JohnDoty OK I will call that "effective" potential a "pseudopotential" because it takes into account centripedal forces, not just gravity. Until you show otherwise, I will continue to be pretty sure that that "potential" is not the one we should use to calculate time dilation.

@uhoh it's fair to want a reference for this. The wikipedia page for TAI says that TAI (atomic time) measures proper time on the geoid: en.wikipedia.org/wiki/International_Atomic_Time. I really do think "The Geoid" is the answer to your question, but I'm not expert enough in general relativity to prove it. Wikipedia cites Guinot, B. (1986). "Is the International Atomic Time TAI a coordinate time or a proper time?". Celestial Mechanics. 38 (2): 155–161. You can check that out but it looks non-trivial.

A warning.. I went down this rabbit hole trying to understand "exactly" how do we keep time: astronomy.stackexchange.com/questions/53193/…. In my experience, when GR starts to get involved it gets really complicated.

@uhoh the geoid really is the thing you want (or at least the long-term average of the thing you want, smoothing out tidal variations). Remember the relativity part of relativity. If I put you inside a closed box, then a conventional experiment (like Cavendish's) has to give the same answer for "am I on the geoid?" as running your atomic clock for a month inside the box and seeing how much time it gained or lost.

@hobbs do you have a reference that would clarify (or just say more words about) why the geoid is the right thing? Or maybe you can give an answer? I think uhoh is prepared to accept that the geoid is the answer, but just needs more convincing.

@hobbs that doesn't work if the box is fluid and rotating; there's nothing "inertial" about standing on the rotating Earth. All these answers (so far) are shoot-from-the-hip best-easy-guess approximations.

@uhoh I think this question has shifted to being a question about time dilation on the geoid. John Doty has claimed that the geoid (the surface made by water) is ALSO a redshift isosurface. I would say this is also hinted at by the crude references so far. You seem to doubt this but without great reason. Rather, it sounds like you need to learn more about redshift isosurfaces and the geoid. I recommend you ask a question on physics stack exchange to this effect. "Is the the surface where a rotating body of water collects under the effects of gravity also a gravitational redshift isosurface?

@Jagerber48 My question has not changed. There have been several attempts to suggest a first-order approximation, and some attempts to assert that the approximations are exact. When that happens, I offer instructive counterarguments to those attempts. But my question itself has not wavered. I have a hunch within a few days or a week we'll get a good answer supported by either the math itself, or links to authoritative sources showing how it's calculated.

@uhoh For a partial explanation in terms of geometry, see Section 2.8 , "The Centrifuge and the Photon" of Misner, Thorne and Wheeler's massive "Gravitation". Also consider the Equivalence Principle: the effects of gravity and acceleration on an experiment are the same. Thus, a pair of clocks comparing redshift should constitute an instrument that should agree with a "spirit level".

I was interested enough to ask it myself: physics.stackexchange.com/questions/838388/…. John Doty's recent comment almost certainly contains the answer but I unfortunately don't have time right now to read those sources.

@uhoh Then, there's the thought experiment. If two places are at the same level, in principle it takes no work to move mass from one place to the other. Imagine shooting a photon between the two places, putting it in a box, and carrying it back to where it started. If the photon emerges from the box with a different energy that it started with, you've violated energy conservation.

@uhoh See physics.stackexchange.com/questions/126919/… where the consensus is that the geoid IS indeed a redshift isosurface. If you accept that then you should accept that "the geoid" is the answer to the question asked here. if you don't accept then, again, I suggest you make comments on that question, or make a new question on physics stack exchange where you can discuss that specific physics in detail. That will be more effective than trying to hash it out in comments here.

@Jagerber48 thanks for the link, I'll take a look If nobody derives with math nor cites an authoritative source here, I'll work on that myself, and if/when I succeed I'll post an answer here myself. It's a good question post and it deserves a good answer post. And in Stack Exchange it's the question author who decides what to accept.

@Jagerber48 I've just posted an answer. Like they say, "If you want something done right..."