If humans colonize Mars how will the 2 societies (Mars inhabitants and Earth inhabitants) be able to synchronize and cooperate regarding time? Since:
Martian days and years are different than the Earth's
Time passes faster on Mars since its mass doesn't curve spacetime as much as Earth's mass.
Is there a way that the Martian and Earth societies would be able to have a common unit of time?
First off, the issue with relativity is solved. No kidding! Barycentric time (TCB) is a time scale that an atomic clock would perceive were it co-moving with the sun. This, of course, is a technical phrasing which means it's as though the clock were fixed with respect to the movement of the sun, but outside of its gravity well. It ticks about 490ms/year faster than TAI, which is the atomic standard we build UTC from. Leave it to the metrologists to need a coordinated time system like TCB!
Any synchronization would naturally be done in a measure like UTC or TCB. Like o.m. I believe that, in the early days, everything would be in UTC. However, leap seconds are a pain, so I would expect something like TCB wins out in the long run.
Most mission planning can be done in seconds minutes and hours, rather than talking about days or weeks or years. The former are arbitrary time constants, while the latter are built around the revolution of the Earth. It will be easy to talk about rendezvousing in 78 hours.
As for human-to-human, I think that will happen less often than you think. People tend to form groups close to them. The Martian colonists are quickly going to develop their own culture. A key point will be when they cease to think of themselves as a colony, similar to when the US stopped being a colony and started being a nation. At that point, I'd expect them to reject UTC, define their own martian day and year, and rely on TCB for coordination from then on out.
At some point, the Martian colonists might lobby to adapt the Martian year and day, but that would be a political statement of independence, and quite impractical.
Historically, we humans have already had to solve this problem on a smaller scale: train schedules. Trains could move fast enough for long enough periods of time that sunrise and sunset was different between the beginning and end of the journey. Complicated schedules were created that specified when the train would arrive/depart a specific location in "local time."
In the case of a larger macro system, the only real question is can we create two references that are independent of or minimally dependent on relativistic properties? I'm not a nuclear physicist, but we probably already have.
The SI unit of "one second" is defined as...
Under the International System of Units (via the International Committee for Weights and Measures, or CIPM), since 1967 the second has been defined as the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom. In 1997 CIPM added that the periods would be defined for a caesium atom at rest, and approaching the theoretical temperature of absolute zero (0 K), and in 1999, it included corrections from ambient radiation. Absolute zero implies no movement, and therefore zero external radiation effects (i.e., zero local electric and magnetic fields). The second thus defined is consistent with the ephemeris second, which was based on astronomical measurements. (See History below.) The realization of the standard second is described briefly in a special publication from the National Institute of Standards and Technology, and in detail by the National Research Council of Canada.
So, frankly, all you need to do is use this (or its future equivalent) and identify the synchronizing "epoch" (not disimilar to the Unix Epoch) after which the two measurements are perfectly aligned and can be used to convert from local time to "solar universal time" (which, obviously, will be Earth time, 'cause the motherland is where it's at....)
An alternative is to use a definition like Ephemeris Time, which was created literally to solve this problem.
The time scale represented by Teph has been characterized as a relativistic coordinate time that differs from Terrestrial Time only by small periodic terms with an amplitude not exceeding 2 milliseconds of time: it is linearly related to, but distinct (by an offset and constant rate which is of the order of 0.5 s/a) from the TCB time scale adopted in 1991 as a standard by the IAU. Thus for clocks on or near the geoid, Teph (within 2 milliseconds), but not so closely TCB, can be used as approximations to Terrestrial Time, and via the standard ephemerides Teph is in widespread use.
Honestly, all you need is one whomping predictable reference point that is syncronized at the motherland's central time coordinating agency before shipping it off to any planet in the galaxy, and so long as you didn't pass too close to any black holes along the way, you're synchonized.
Note that I very much doubt any non-Earth planet would happily try to stay on Earth time anymore than any timezone on Earth has decided to stick with Greenwich Main time. It simply doesn't work. It's easier to delay transmission of your favorite movie to synch up with your local time.
The answer to "how advanced society measures X" is invariably the same: by convention.
Time, like most dimensions, is ad hoc. Its current definition is entirely based on how we measure it: the transitions between hyperfine whatchamathingies of cesium measured by cesium atomic clock. Time zero is Greenwich because the Brits used to own the world pretty much.
The logic behind it isn't geography. France went through a few time zones and it hasn't moved physically. The logic behind it is men. Honestly it doesn't matter which standard you choose, it only matters that it is universal. The story of time keeping in France, if you want to look it up, exemplifies just how made up the time on the clock is.
If you don't use UTC or similar and instead make your own standard, remember a few things.
Firstly, your readers are Earthlings, and presumably so are you. That means there is an understanding of what a minute or a day is that you can't completely overwrite. Don't mess with people's frame of reference too much if you don't want to confuse them. That also applies in-universe, people don't like drastic change happening all at once.
Secondly, that standards are built for trade, and imported through either trade or force. Paris adopted Greenwich time out of convenience, and it was told to adopt Berlin time out of lack of choice. The dominant power of the period writes the rules, and those rules can persevere well after these powers are gone, or sometimes standards die with said power. Consider the history of your world when coming up with your standard clock. Early Martians would likely follow Earth time , then develop their own rhythm of life over time. The more contact with Earth, the more prevalent Earth time will remain.
Thirdly, although standards are important, they don't have to rule one's life. Some people's day consist mostly of night despite what the clock or Sun says, aka night shifts. Even if there is a common measurement of time, your story could only require local time from the main character's perspective.
This would be quite important when calculating overtime pay for terraformers. Perhaps they'd just add 2.7%, the difference in the days.
Wikipedia Mars Time entry says that people working on Mars missions adjusted to Mars time (24 hour system with seconds 2.7% longer) because rovers require sunlight for power to perform tests and move around. Colonies would probably have similar needs. No need for years to be counted, just the days (they cal sols ). If we did that ourselves we'd be on day 736566 rather than December 3rd, 2017. It's shorter but the seasonal information is often important to us.
From there it's like inches-centimeters where we multiply by 2.54; Martian sols to Earth days add 2.7%.
I think the solution that Andy Weir chooses for Artemis is likely to be the short-term solution.
In that novel, set on the Moon, they use Nairobi time, because the original colonists were from Kenya. They ignore the oddities involved in the lunar cycle, and simply define everything based on the time in Nairobi.
On Mars, this may well be the solution early on as well. For the duration of time until the Martian society has its own character, during which presumably Mars is regularly interacting with Earth, it makes sense to be on a schedule and use the time associated with the closest Earth associates. If just one country colonizes Mars, then that is simple; if more than one do, then it may require some compromise (perhaps UTC, perhaps something else).
Once Mars develops sufficiently to be largely self reliant, where the local time is really all that matters, it will likely develop its own time zones, and simply expect people between Mars and Earth to translate, just like we do now between various countries.
The differences may not be as much as you think.
First, the gravitational time dilation can be discarded as irrelevant except perhaps for things like scientific experiments requiring extreme precision (and GPS satellites would have to be programmed for what planet they're for). You're talking a difference of something like minuscule fractions of a second per year, if that. The odd leap-second would officially handle that.
For longer periods like years, it depends if seasonal effects are an issue. Obviously, if you're near the poles, the growth and retreat of the Mars polar caps is important, but elsewhere on the planet, I'm not so sure it's that big of a deal. The main issue will be height of the sun in the sky and day length, which solar power generation will have to take into account, but that's minimized if most settlements are closer to the Martian equator. Assuming you haven't terraformed the planet, crops will be grown indoors so conditions outside have only minor significance. All this means is that the Martian year and season aren't nearly as important, so you can operate use the Earth year as a base for dating.
Also to bear in mind that because of the axial tilt and orbital eccentricity, Mars's seasons aren't even the same length between the two hemispheres on the same planet, so it makes more sense to just ignore them in terms of official time-keeping.
It's the issue of individual days where most of the problems are going to come up, and that's primarily because the Martian sol is almost, but not quite, the same as the Earth day. From a practical standpoint, the easiest method is to use what the Earth-based ground crews on Martian rovers do: extend the "official" Mars second by 2.7%, so a Martian 24 hour days is identical to a Martian day/night cycle. Maybe colonists call it as "marsec" or something.
If you do that, but keep the Earth year as a basis for longer timekeeping, then the official Martian year, instead of being 668 sols (ignoring fractions, which leap-years would take into account), would be 356 sols (equal to 365 days). This, if you wanted Martian months, conveniently divides into 12 months consisting of four quarters each with a 30 sol month, a 30 sol month, and a 29 sol month.
On the other hand, whatever system is chosen, the whole thing becomes trivial with computerized time-keeping. It's simple to program an app that converts between Earth date and time (taking into account time zones) and Martian date and time (again with time zones). All that needs to be done is for everyone to agree when the zero time for the Martian clock/calendar is.
I recommend reading Paul Krugman's quasi-famous paper, The Theory of Interstellar Trade.
It may not directly answer your question (as it focuses on the interstellar instead of intrastellar) but it is a fascinating look at how trade (and hints at contracts and financial products necessary for capital intensive commerce) would operate across unprecedented distances.
For once-off events, synchronizing between a timezone on Mars and one on Earth is not that much different from synchronizing between different timezones on earth. So I doubt that would motivate the development of a vastly different approach. Instead I'd expect to see what we see on earth: reference to some arbitrary standard like GMT, or to the timezone used by the majority of people involved, or the location of some project's original headquarters, or whatever. All of these might be employed concurrently depending on the people involved.
Things become more interesting if you have repeated events. Assume you want to have a weekly project meeting, preferably during office hours at both ends, but at least during daytime hours. On Earth this is simple: with the exception of DST transitions twice a year, relative offsets between time zones remain fixed, so a time slot that works in one week should work the week after just as well.
Not so between Earth and Mars. As the Martian solar day is about 40 minutes longer, you have a shift between the two to account for. Some weeks you'll have nice overlap, some other weeks you'll need one or even both parties to participate outside office hours. I guess most likely you'd have some software do the planning for you. But the software might come to rely on well-established patterns, and these patterns by themselves may get names. So you might decide to meet regularly “on the β3-pattern starting at 2312-03-18 13:00 GMT” or some such.
Personally I doubt this would become a common method of referring to dates, though, more like an implementation detail of the calendar software you use. Particularly as things like day of the week adjustments (to avoid extended runs of weekend meetings) would make things even more complicated, and getting inter-planet and intra-planet repeated schedules aligned would require a lot of individual adjustments anyway.
Rather than use years, you could measure time in synods. This would be an especially useful unit of time because it actually means something to the the colonists.
No colonist would care what one "martian year" it is. It doesn't matter if it's winter or summer there, because people can just assume the outdoors are going to be equally inhospitable year round.
Compare this to a synod. One synod is the time it takes to transport goods to and from Earth using a Hohmann transfer. If you're a martian and you really need a new part from Earth, you have to wait until the next synod. You could also easily keep a history of colonial growth this way:
Synod 0: first colonists arrive
Synod 1: solar panels arrive
Synod 2: more supplies, new landing pads to accomodate traffic
