Supposedly, if humans were to establish a colony on one of the moons of a planet (such as gas giants in our own solar system), how could we possibly design date and time system? On Earth, we define date based on rotation about Sun and time on Earth's revolution. But, with a moon of another planet, this will add another factor to date/time system. So, how can we cope with this?
PS: I am not sure about what tags this question should carry. Feel free to correct them.
I really love this question. Were we to establish a colony on our moon, we would probably want a calendar/time system which actually corresponds to physical patterns and cycles on the moon, but also something which is easy to track and keep in sync with Earth, since the two civilizations would certainly be in close contact.
In a way, we already do something this here on Earth with time zones, though time zones are just offsets within the same second/minute/hour/day system, so people throughout the world are in different times, but they stay in sync. But beyond that, our entire calendar system (we currently use the Gregorian Calendar), is mostly based on our own planet's positions relative to the sun. It just wouldn't make sense on the moon. Earth days on the moon would just be arbitrary 24-hour blocks of time; they wouldn't correlate at all to whether it's light or dark outside.
So, if we really wanted to keep some logical synchronization between earth and moon calendars, we would probably want to adjust here on earth to a lunar-based calendar. Most older civilizations did this, using a full lunar cycle to indicate a "month." But here's the (cool) thing: the moon is a little lopsided and way gravity works, the exact same side is always facing us. We never see the back of the moon. So, it's rotating at exactly the same speed as it's revolving around us, meaning a "day" on the moon lasts one full lunar cycle: just over 29 earth days. So even the idea of a day, how long a day is, and what you could accomplish in a day is so far from Earth's that it's hard to even compare the two.
Maybe one idea is to have weeks or months here on Earth which would line up with days on the moon. Earth would still have its 24-hour days and the Moon would have its 709-hour days, but at least we could predict Moon time using our own calendars. One proposal is the The Hermetic Lunar Week Calendar which continually adjusts the length of a week to keep the months in sync with the moon, so that a full month is always one lunar cycle (or lunar day). It would be weird but in a future where humans are legitimately split between Earth and Moon, it would probably make sense for both to keep a calendar like that, which takes into account the length of a day on both bodies.
Since human beings appear to have a biological sleep cycle close to an Earth day, I would strongly suspect that extraterrestrial human colonies would keep this rhythm. People living close to the poles of the Earth, where it is light all summer and dark all winter, don't stay awake for half a year and then sleep the other half. Similarly, it doesn't make sense to have a 29 Earth-day long Moon-day.
I take it you're looking for something absolute to measure time with, such as revolutions around a planet or the planet around a star? If so, you could use a "year" to be the time it takes the planet to orbit the star, or the moon to orbit the planet. Those could also be used as separate, unrelated measures of time. A "day" could be construed as the time it takes for the moon to revolve, or the time it takes for the planet to revolve. It would be interesting if the planet and moon were tidally locked, because that would mean that only one side of the moon would ever face the planet. This is also a social issue, which would have to be decided on by the occupants of such a moon.
Smaller measures of time, such as the minute and hour could still be used, because they are defined by the second, which in turn is defined by the behavior (I don't know the specifics) of a cesium atom. So the second, minute, and hour could be used anywhere - assuming you have an accurate clock or a cesium atom.
Also, I think you misunderstood the tag "space-time". Here, it refers to the combined 4-dimensional (in our universe) entity consisting of space and time, as used in special and general relativity.
