Space colonization is going to be a pain. We've just managed to get sensible datetime APIs with instants and zoned timestamps, but they only work when time is the same everywhere. As soon as relativity comes into play we'll have to come up with a different approach.
Nothing in the solar system has a relative speed different enough to earth to make common timestamps different - the difference is well below leap second impacts.
As for simultaneity we already have to deal with that at the millisecond level. An event logged at 16:00:01.000 as far as LA is concerned maybe logged at 16:00:01.080 in London, or maybe at 16:00:00.920
I mean the clocks on different planets do tick at slightly rates. It's somewhere on the order of milliseconds per year, except for really large planets, but it does add up
Space travel will be relatively easy since requires computers. Everyone will keep track of their local clock in seconds. Then they will calculate clock for places they care about, eg Earth-time, since those are pretty predictable. Relativistic corrections will be pretty minor.
My guess is that planets will keep local day-year. Mars is close enough to Earth that throwing in extra 37 min might make sense. OTOH, most planets and moon might use Earth calendar because local day cycle is too far off human internal clocks.
They will calculate time for other places, but will be less important than time zones since every communication will have light speed lag. No video calls, but lots of email.
All planets (in the same plane of relativity) agree on the length of a second. The length of a day varies - both on earth and on mars, and is meaningless for people in orbit, but a second is a second.
Right, the passage of time is affected by how deep in a gravity well you are (i.e. how much acceleration you are experiencing). Time passes faster for satellites than for ground stations.
