Time travel as a sudden jump seems one of the least plausible implementations, since we have no idea how to do such jumps even in just space or forward in time; and allowing for it would break a lot of physics.

More plausible alternatives include a space-time bridge, meaning both sides can follow Earth’s reference frame; or the Primer-type where one can reverse time in an isolated box in a way where you can only travel backwards along the Box’ trajectory and you have to wait inside that box for some time while you move backwards in time along that trajectory.