How is the FTL drive supposed to work in Star Trek?
In our known universe, people traveling close to or at the speed of light would experience time roughly 1000x slower than people moving at "normal" astronomic speeds. Equivalently stated, people on a spaceship traveling around the galaxy near light-speed for one year, as referenced by a timepiece on their ship, would come back to Earth to find that 1000 years have passed.
This obviously does not happen in the Star Trek universe; traveling at Warp 9, which would be something like 729c, there is apparently no perceived time difference when returning to port. If there were, then if the NX-01 had set out for Alpha Centauri traveling Warp 1, by the time it got there the NCC-1701-Z would probably have been gone for millenia, and human life would either have evolved to pure energy or been annihilated. Even impulse engines can propel a small craft at extremely high sublight speeds, where the effects of relativity would become evident.
How does the Star Trek universe explain the lack of effects of special relativity across the spectrum of FTL transit methods? Or is that just conveniently ignored in canon?