It's not too bad. Relativity says that no frame of reference is special.

• On earth, a second looks like a second, but a second on the moon looks too quick.

• On the moon, the second looks like a second, but a second on earth looks too slow.

Both are actually correct. The simplest solution is to declare 1 to be the base reference. In this case, the earth second. Any lunar colonies will just have to accept that their second is slightly longer than they think it should be.

If it helps, the difference is tiny. A second is 6.5x10^-10 seconds longer. This works out to 56 microseconds per 24 hours. It won't affect much for a long time. About the only thing affected would be a lunar GPS.

It's well understood math, but it's "only" relativistic orbital mechanics.

It boils down to a pretty consistent number, but how you get there is related to the weight of the moon, how far it is from earth, and how fast it's going.
Since the moon is going different speeds at different places in it's orbit, the number actually changes slightly over the month.

They're just using the average though, since it makes life far easier. We use the average for earth too, since clocks move differently at different altitudes or distances from the equator.

So, one observer will see those oscillations happen faster than the other?

Not quite. In each observer's frame of reference, time appears to pass the same; it's only when you try to reconcile the between two objects that are not at rest with respect to each other does relativity show up.

Basically, when you bring someone back to Earth, the observers will find that their watches don't match up even though both observers experience time passing the same way as normal (because the oberserver is by definition at rest with respect to their own frame of reference).

TL; DR: Relativity is a pain in the ass and makes no sense in everyday terms.

edit: disclaimer - I am not a physicist and have not taken physics classes in a decade plus, but I do teach science at a college. I'm going mostly on half-remembered lectures and some random one-off discussions I've had with my buddy in the physics department over the past few years.

It's that relativity thing where each person will see the oscillations happening correctly, but when they look at what the other person did, the answer will seem wrong.

The difference is small enough that it really only matters if you're NASA and building moon GPS. MPS?