[unholiness]

That's right. While a day passes on earth, a day minus 58.721 μs passes on the moon (which is moving faster than the Earth).

This multiplies, so after a million days a clock on the moon will read 58 seconds behind a clock on the earth.

[pdonis]

It's the other way around: while a day passes on Earth, a day plus 58.721 microseconds passes on the Moon. The Moon clock gradually gets ahead of the Earth clock.

In an Earth-centered inertial frame, the Moon is moving faster than the Earth, but it is also at a very high altitude, and the altitude effect, which speeds up the Moon clock relative to Earth, is much larger than the speed effect, which slows it down.

(Note that the above only takes into account the effects of the Earth's gravity. The paper also takes into account the effects of the Moon's gravity, which don't change the above answer qualitatively but do add small corrections numerically, so the 58.721 microseconds per day is not the actual value the paper ends up with.)

[alganet]

Cool, thanks!

I'm assuming this affects clocks (and things) but not time, right? Time itself is no different on the moon (it's not the future there).

I know this must be true otherwise we would be surrounded by time travellers by now. So, where does this cancels out?

My intuition says that if we have two clocks, each clock with a display and a laser pointing to each other, and we put one of them on the moon and the other on earth, someone observing it from a third equidistant point would see both lasers blinking at the same rate.

If that's true (I don't know if it is) both of your answers are kinda right, aren't they? From earth, the moon clock ticks slower compared to earth clocks, therefore it lags behind. From the moon, the moon clock ticks faster compared to earth clocks, therefore it skips ahead, and vice-versa. I am not sure though, I feel like I'm missing something.

[mr_mitm]

Moves faster than than the Earth relative to what? Isn't it the gravitational time dilation that is the relevant effect here?

[alganet]

Thanks!