Time travel is obviously possible. Everyone does it all the time. Just sitting around doing nothing we travel through time at a rate of 1 second per 1000 milliseconds. What’s more interesting is how we can travel through time at a different rate and how this prohibits any time paradoxes from occurring such traveling into the future to meet an older version of yourself.
As I have posted before we are always traveling at a constant speed. This is the speed of light and we cannot change that. What we can change is the direction in which we travel. Nearly all of our velocity is on the time axis but as soon as we move in the conventional sense our velocity in time slows down such that the magnitude of our velocity vector remains constant.
What this means is that every time you travel away from someone and come back you are out of synch with them in regards to time. In nearly all cases the difference is in femtoseconds and is totally irrelevant. What is be more interesting and is conceptually possible is to spend a year (in your perception) and travel two (or ten) years in time as perceived by the people you left behind.
If such a machine were built it would more properly be called a “time vehicle” than a “time machine”. To travel any interesting distance in time would require enormous amounts of energy and you cannot retrace your path or go backward in time.
Using such a machine it would be impossible to meet yourself in the future. Use of the vehicle would be exactly analogous to taking a shortcut to some geographical location your friends were traveling to. You will have traveled a shorter distance than your friends but arrive at the same location. It is not possible for you to travel both paths simultaneously.
I woke up at about 4:00 AM this morning and it took an hour or so of thinking about this before I could go back to sleep.