Ok, so I'm watching myself watching myself leave. And so what? My "past selves" are just reflections of myself, I am not there anymore, it's just light that looks like me. The light is not going to change anything about the past, it's going to follow me to my next location until I stop and let it catch up.
Maybe I'm completely misunderstanding you or the problem - feel free to correct me, please. I'd really like to understand how causality is broken by FTL but could never grok it.
Yeah, that explanation was dead wrong. Here is the real one.
The phrase "at the same time" means different things in different inertial reference frames. So what is FTL to one observer, is traveling backwards in time to another. If you can warp from any reference frame, you can therefore travel backwards in time.
As for why "at the same time" changes meanings, let's adapt Einstein's example.
I'm in the middle of a moving train. You're on the train station. As we pass, lightning strikes right between us. Will the light from the lightning arrive at both ends of the train at the same time?
According to me, obviously. In my reference frame the train is not moving. Light has a constant speed. It will reach both ends at the same time.
According to you, obviously not. Light travels at constant speed, but the train is moving. Therefore the back of the train is moving towards the light, and the front is moving away. So the light gets to the back of the train before the front.
So when the person on the train says that an event in front and behind happened at the same time, the person not on the train says that the event behind happened before the event in front.
Now consider this. I'm on a spaceship with a warp drive. This is a great warp drive, travels instantly in whatever reference frame I'm in.. I warp forward a long distance - like a light year. I accelerate forward. I warp backwards and arrive back a bit behind where I started at. I decelerate, and I'm in my own past!
You left out the "accelerate forward" bit. That changes your reference frame. Without changing your reference frame, you don't get FTL to turn into time travel.
If you have FTL but on a time scale determined by one reference frame, then as I pointed out in https://news.ycombinator.com/item?id=37012594, no causality issues exist. You need FTL in 2 reference frames to create a closed time-like loop.
When you bring in gravity, you bring in general relativity. And now the phrase "at the same time as" becomes an entirely arbitrary choice of coordinate systems.
So it becomes complicated. But as long as FTL goes forward in time by some unique global definition of time, there is no causality problem.
Now do it twice.