I've been trying to learn about the speed of light and time dilation, but I'm at an impasse.
The presented facts say that if I travel around the solar system at 50% the speed of light and then come back to earth I will have experienced less local passage of time than them. I will effectively have traveled to their future. I've also read that gravity causes time dilation too.
I understand that space-time is a sort of unified thing and that affecting space affects time. It makes sense then that gravitational forces bending space will also bend time. But velocity? I can't wrap my head around it and I can't find a good explanation for it.
People cite orbiting craft and planes as proof of time dilation, since their clocks will go slower than those in the surface (and vice-versa). This certainly explains gravitational time-dilation, but not necessarily velocity. Can't the change in local time passage be caused solely by the gravitational bending of space-time?
Trying to find an answer, I came to a very recurring and frustrating example in texts that seek to explain time dilation. A man on a moving train throws a ball forward. Since he's moving with the train and the train is his point of reference, the ball to him only moves at the speed he threw it. But to a woman on the station the ball is moving at the speed of the train plus the speed it was thrown with. To some authors, this seems to open the mind to time-dilation understanding. To me it only explains the relative nature of motion. It says nothing of time.
Another example I've found: if person A speeds away from person B very quickly, A's clock will seem to advance slower from B's point of view. How is this time dilation, though? The difference can be explained by the longer time it takes light to get to B, can't it?
I realize I can't be right against the fine physicists out there, so I was hoping someone here could enlighten me. Where does the notion that velocity causes time-dilation come from?