500 years from now, if we're still alive, enough of us will be living in space colonies rather than on the Earth's surface that the Earth's rotation will not necessarily be immediately relevant to everyone's lives.
Regardless, we've had maybe half a minute of drift since the 1970's. If the drift is 1 minute in the 2000's, 1 second per year in the 2100's, 2 seconds per year in the 2200's, 3 seconds per year in the 2300's, etc., then by 2500, the cumulative drift would add up to...60 + 100 + 200 + 300 + 400 + 500 or 1560 seconds of total cumulative drift by 2600. That's less than half an hour. If we didn't bother accounting for leap seconds, UTC midnight would be only half an hour off from astronomical midnight in 600 years, which is roughly the same error inherent in most time zones.
The day would shift half an hour in the next 600 years. That’s half as much as it shifts twice a year for DST, except it would happen gradually over a period of time roughly equal to the time between the fall of the Eastern Roman Empire and today.
I was sort of nodding to the people who claim they have a 25 hour sleep cycle. Once we are in space and using artificial light, you can choose the sleep cycle that suits you. But predicting things in 500 year is just Sci Fi I guess!
Regardless, we've had maybe half a minute of drift since the 1970's. If the drift is 1 minute in the 2000's, 1 second per year in the 2100's, 2 seconds per year in the 2200's, 3 seconds per year in the 2300's, etc., then by 2500, the cumulative drift would add up to...60 + 100 + 200 + 300 + 400 + 500 or 1560 seconds of total cumulative drift by 2600. That's less than half an hour. If we didn't bother accounting for leap seconds, UTC midnight would be only half an hour off from astronomical midnight in 600 years, which is roughly the same error inherent in most time zones.