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I was having some issues with the definition of time/clocks in some extreme situation though - i.e. in regions of space with extreme curvature. The Caesium atom cannot be understood as a pointlike object in general, so what happens when the geometry becomes non-trivial over the area the atom covers? At first, even if we were to assume it affects its spectrum, it will affect the frequency standard of clocks based on it in just the same way (by def) such that the defining hyperfine frequency won't numerically change - because anything measured in units of itself must be constant, obviously.

But on the other hand, if the geometry becomes non-trivial, the orientation of the Caesium may become relevant and it's not unthinkable that the defining transition might split in a Stark/Zeeman-like effect but for gravity. That would render the SI definition not well defined. But if the concept of a clock becomes unclear, then so does the concept of proper time defined by it, which is tied to the temporal part of the metric tensor. This in turn means the geometry becomes blurred, yet it is fundamental to even establish what theoretically happens to Caesium in such circumstance. A recursive problem.

In physics, time is defined by its measurement: time is what a clock reads. But this needs a clarification of what even counts as a clock. Given how a clock uniquely implies a part of the metric tensor and in return the temporal part of the tensor uniquely implies what that clock measures in all circumstance, these concepts could be seen as equivalent. But there is no one clock. Nature did not choose the Caesium clock standard - we did - and i realized that there are different choices of reference oscillators usable as a standard but they exhibit different behavior. The same problem persists in geometry: given a metrizable topological manifold, there is no unique choice of a metric (meaning geometry) either but instead there are many topologically equivalent options (that aren't isometric).

Naively my first idea was to use the time of a remote clock in a friendly region: i.e. utilizing theoretical radio clocks adjusted for signal travel time. I found the Geocentric Coordinate Time (TCG; French) somewhat aligns with that idea - except I was thinking of local (platonic) devices that are able to measure the value of TCG anywhere. Taking measurements of such a TCG clock device at face value, time passes at a different rate then for Caesium clocks at various locations, since TCG simulates a Caesium clock shielded from any local gravity influence (conveniently resolving all issues that come with it). But if i were to apply the concept of proper time to these clocks, i end up with something else, yielding a different metric tensor and consequently measuring the same reality in a different geometry.

Every event has a well defined TCG time though, hence it as a valid measure of time. Besides, the definition of second specified that Caesium must be shielded from influence like electric fields, yet must not be shielded from gravity - that seems like an practical yet arbitrary choice. Or Actually it has to be "unperturbed", so technically, if extreme gravity becomes a problem, maybe even TCG is then more in line with that spec? Lacking a theoretical specification what a clock is, what is preventing us to use TCG device as an alternative clock standard? As in not just a coordinate, but a real time unit alternative to the second?

More generally, how are we supposed determine what the "right" rate is at which time passes at a location? Given two clocks, how are we supposed to decide which gives the "right" time?

Alternatively i figured that Einsteins postulates may implicitly define a time standard/clock for which they always hold - there will be only one proper time and length (i.e. metric tensor) such that Maxwell always keeps its well known form. i would think the theoretical clock this implies is always well defined even if Caesium might not be able to follow it in all circumstance. Now the issue with a pure theoretical clock is that a theory based on it cannot be compared with experimental data using another clock. one would first have to calculate the local SI Caesium transition frequency within that theory to understand how the two clocks are related and derive a transformation along each worldline. But again, that's a transformation of physics between two geometries.

How does modern physics approach this issue?