This is a problem I've wondered about at times: just what standard time system is in use? It's not so much a problem aboard a ship, but when ships of different races meet, it becomes more problematic. Even more interesting is when standard time units like "hours" are brought up in first contact episodes; of course it would bog down the episodes the same way that the language and food problems would, but some kind of exchange of standards must have taken place.
It is a different matter when alien worlds come into play: each planet would have its own unique rotational period and orbital period around its sun. There is a whole problem of adjustment to different day lengths when moving from one planet to another, or from ship to alien world. Within the Federation, people on every member/colony world are all on entirely different time scales! It's jet lag squared!
We may assume that "hours," "days," etc refer to those on Earth. This would continue the proud tradition of homocentrism/terracentrism in Trek, but makes the most sense in light of who's watching it! However, it would have greatly simplified the issue if the terms "Earth hours," "Earth Days," etc were used in place of "hours," "days," etc. This solution would not only elucicate and clarify the issue, it would be so easy to implement as to incur zero notable cost!
Standard time in Trek
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Standard time in Trek
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I have noticed that - it would be hard not to - and I've always forced myself to assume that the term went through the UT to become one specifically related to the units we know. In other words, the UT wouldn't translate "1965 4-4-2 GTO" into the alien word that merely means "automobile." "Hour" would end up being translated as "the unit of time, x long in our native units, which is derived from approx. 1/24 of Earth's rotational period." If that happens, it's only natural that "hour" is the half of that conversation we hear, because the "English" side is the one we get from our TV.
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I got the high gloss luster
I'll massacre your ass as fast
as Bull offed Custer