In Star trek TOS, I take the "Five year mission" to mean 5 years aboard the Starship. Do they explain in any of the novels, or anywhere else for that matter, how many years elapsed on Earth during the five years the Enterprise spent exploring space?
Warp drive seems to defy the general theory of Relativity, in more than one way. Specifically, it seems that the relativistic effects that would normally cause a person to change at a different rate than the outside world seem to not affect anything. The only thing that would cause relativistic effects is the impulse drive. I found a forum on StarTrek.com that seems to indicate that the maximum speed that Impulse drives typically use is .25c. According to the relativity calculator, this would create a dialation of about 3%.
All in all, I'm guessing that a few days would have been gained due to relativistic effects, but nothing significant.
Only five years, Earth time.
In other words, due to warp speed and such, there is no issue with time dilation and the five years on the ship was the same as five years on Earth. Exactly five years passed.
Also, if time dilation is an issue, Starfleet would have to standardize everything, which would mean they'd have to use one standard time measurement and ships would have to sync to Starfleet Standard Time after dealing with any dilations. (But I have no idea if Starfleet ever sets their clocks back an hour i the spring for Starfleet Daylight Savings Time!)
Five years. What other years would a human-crewed ship be talking about?
Stardates were only used in the original series so that they didn't have to pin down a reallife year that it took place in. Stardates don't have a concept of "year", although some later interpretations used the thousands digit as "year".
EDIT Okay, thanks for the downvote. To clarify my answer:
- The Enterprise at the time was manned almost entirely by humans. Roddenberry originally intended the series to portray humanity's future, and the makeup of the Federation was left ambiguous throughout. This (in addition to us, the viewers, being from Earth) is why it only makes sense to have been Earth years, not Vulcan years or something else.
- As stated in Pearsonartphoto's answer, the Warp drive negates most relativistic effects, because they're not actually traveling at high speeds.
- Finally, it makes absolutely no sense for their mission to be a "5.34 ± .87 year mission, based on effect of relativity. Come back when you feel like it!" mission. The Enterprise would be keeping itself in sync with the standard time used by the Federation, and "5 years" would be defined in terms of that. If it means the crew only experiences 4.7 years, so be it - it's still a 5-year mission.
Most likely, the mission took five non-relativistic years. As mentioned in the other answers, travelling with Warp speed does not affect time consumption, but travelling with Impulse does.
As the dilatation effects are well known, the clocks onboard the ship are corrected on the fly according to the effect of the current speed. Thus, the onboard clocks are always in sync with the standard clock on Earth.
Nevertheless, due to the time travelled at Impulse speed, the crew is aging slower, leading to the biological age diverging from the actual age.