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Sure, we can compromise; they can have their own timezone, but it has a constant time value.
const moonTime = DateTime.Utc.MoonTimeAs in, it is perpetually 4:20 PM on the moon?
nice.
To the moon 🚀 🚀 🚀
That sounds... iffy. Thing is that UTC lags more and more behind TAI as UTC takes the earth's rotation into account, introducing leap seconds so that all the timezones don't slowly drift across the globe. Moon people care preciously little about the earth's rotation around its own axis, more relevant is its own day/night cycle which (because tidal lock) is an earth month. The system might just be stable enough so that UTC can simultaneously sync to that, you'd have to ask an astronomer.
Actually, no, forget it: The moon moves quite fast relatively to the earth's surface, more than enough for relativistic effects to apply -- they also apply to GPS satellites, stuff simply wouldn't work if those things ran on Newtonian maths. Sooner or later it's going to need adjustments due to that.
Well TAI stands for International Atomic Time and "international" generally pertains to Earth-bound locations.
Coordinated Universal Time sounds like it has a bigger inclusivity scope
Otherwise we'd have to rename TAI to "Intergalactic Atomic Time"