Time on the moon

So, here's a thought:

What's the time, right now, on the moon?

This isn't a trivial question. With increased lunar exploration and potentially settlement, we're going to need GPS satellites for the moon at some point. GPS requires really good clocks.

Which means...

...everyone needs to agree what the time is on the moon.

The Apollo missions, of course, ran on  Houston time. But if we have a NASA mission, a SpaceX mission, a Chinese mission, a Japanese mission...all talking to each other? It would be temporal chaos.

So, we need to agree on a time framework for the moon. Complicating the fact is that time does not flow at quite the same speed on the moon, because gravity is lower. This isn't anything you would notice, but clocks on the moon run about 56 microseconds faster than those on Earth. Oh, and clocks on the moon don't sync with clocks in orbit either. All of this has to be taken into account.

The lunar day is 29.5 days long, so probably won't be used as a basis for the lunar clock. This results in the complication of having two days on the moon: The natural lunar day, and the Earth day. Science fiction authors never think of that.

What will we call each day?

All of these are questions we need to answer. Will the moon have its own singular timezone, which makes sense given the astronauts' day won't be the lunar day?

Time on the moon. It's not that easy...