As per the 20th Amendment, the terms of US presidents begin and end at "noon of the 20th day of January", but no location is specified.
One possible interpretation is that the transition occurs per time zone, one by one, similar to us watching New Years fireworks across the world spread over a whole day. That would create a weird situation of different parts of the US being under different rule for a few hours.
Another possible interpretation is that the time zone of the White House is what counts. As (to my knowledge) it is not prescribed that the transition ceremony has to take place in DC (or that there is a big ceremony at all), it might happen that a President gets sworn in, say, in Hawai'i at 7am local time
A third possible interpretation is that the time zone of wherever the President is right now counts. But in that case, the old President might be in a different time zone and claim that their term has not ended yet when the new President is already in office in another zone (or with locations swapped, there'd be an interregnum)
All interpretations have some major or minor problems, in my opinion the second one being least problematic. But since the amendment does not specify it directly, I wonder if some kind of explicit clarification exists in any other form?