Edit
The theory on time travel is incomplete. We need the unifying theory first if we are to make any assessment about General Relativity's and Special Relativity's projections about inertial frames.
Special Relativity says that the fundamental laws of physics should be the same across inertial frames. I suspect that our next great Physical theory will do to this idea what Relativity did to Newtonian physics. The answer will be, "well, most of the time".
The classic and dramatic example is if an astronaut left earth and zipped around the galaxy several times at the speed of light. When he/she returned there perception of time would be that they experienced very little of it, while the earth would seem to them to have elapsed over a great deal of time. Is this technically time travel? From a psychological point of view it is absolutely time travel, but from a physics point of view, not really. We all experience time dilation in unnoticeable amounts during our life. Someone who is at a higher altitude than you will experience time a tiny, tiny, tiny amount slower than you.
To my mind, time travel means I can go forwards and backwards in time. Time dilation, from a a psychological point of view, is about slowing one thing down relative to another. Like this:
Let's not forget, the brightest minds in the world, self admittedly, don't know that much about how the Universe actually works. There are theoretical objections to time travel.
Travelling back in time is another issue. Supposedly time reverses when moving faster than the speed of light, but we don't how to do that.
And we're doing it all the time.
Sorry, but isn't that fundamentally impossible?
https://news.ycombinator.com/item?id=4534359
https://news.ycombinator.com/item?id=6238297
but the OP is talking about traveling forward through time by going near light speed, which isn't really relevant to a warp drive.