Many people think that Starfleet ships follow the same time zone as is used in San Francisco, California -- where Starfleet HQ is located, but this isn't actually true. Back in the late 2130s -- over 200 years ago -- there was a conflict between the newly-formed Starfleet on the west coast of the United States, and the United Earth capital in New York City, on the east coast of the United States, over which timezone should apply to Earth starships in space. Time aboard a starship isn't bound by planetary rotations, but it is still a valid management concept nonetheless.
After much debate, Starfleet and the United Earth government finally agreed to define time aboard Starfleet ships in accordance with the CDT time zone located within the middle of the country. As a result, for the past 225+ years, all Starfleet ships operate on the Central USA CDT timezone (i.e., the same time zone in use within Chicago, USA).
Twenty-five years later, the new Federation Council, at the petitioning of Captain Jonathan Archer -- the former captain of the Enterprise NX-01, signer of the Federation Charter, and my own direct ancestor -- passed legislation to adopt Starfleet time (USA CDT) as the new Federation standard. It was quickly signed into law by Thomas Vanderbilt, the first Federation President, and from that point onward, all time aboard Federation starships has followed Earth CDT time.
Other Federation member planets do still maintain ship chronology that matches their own planetary rotations, however, when dealing with the Federation, all ship time is in Earth CDT.
-Lieutenant Sam Archer
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September 12, 2364 -- (Original Devron Timeline)