I've been working on a (mostly) hard sci-fi setting and I've been trying to figure out how to give humanity a standard method of time keeping to give everyone, in-universe and irl, a frame of reference for various interstellar timelines.

What I've come up with is that the base/stock design of ships, stations, and orbital rings utilize an atomic clock that has saved the oscillations recorded in a part of Earth and automatically readjusts to the dilations and gravity of a different area in order to constantly display the time back on Earth. While the clock isn't that reliable for anything shorter than an hour, it is mainly used to keep track of days, months, and years.

I know that the rule of cool always wins and what matters in the end is consistency and sound logic; but how plausible would this all be realistically? Is technology like this possible with our current understanding of physics?