The NYTimes has an article on a proposal for establishing a time standard for the moon. With the number of different nations doing activity there (South Korea?), presumably applying their local time to their work there seems to be a need.
It's further complicated because, as the article notes in passing, clocks run faster on the moon!
I had to ask Chatgpt for an explantion, which is:
"Clocks do not actually run faster on the Moon. However, they do appear to run faster when measured from Earth due to the effects of general relativity.
According to general relativity, time appears to move slower in the presence of a strong gravitational field. The strength of Earth's gravitational field is greater than that of the Moon's, so time moves more slowly on Earth than on the Moon.
This means that if you were to compare the ticking of two identical clocks, one on Earth and one on the Moon, the clock on the Moon would actually tick slightly slower than the clock on Earth. However, because the Moon is much less massive than Earth, the difference in the strength of the gravitational field between the two is relatively small.
The net effect is that when we measure the ticking of a clock on the Moon from Earth, we observe that it ticks slightly faster than an identical clock on Earth due to the time dilation effects of general relativity. This effect is small, however, and only becomes noticeable with extremely precise measurements."
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