This article is from the Star Trek Tech FAQ, by Joshua Bell inexorabletash@hotmail.com with numerous contributions by others.
Alas, there is no canonical answer. The "Relativity and FTL" FAQ
offers a possibility, that the subspace field forces the ship to take
on the reference frame of subspace itself, which is a special
reference frame, circumventing the limits of Special Relativity.
Unfortunately, this still isn't an explanation of how it works. The
Tech Manual offers that each of the nested fields couple and decouple
from each other at velocities near (but less than) c. It could be that
the interaction of these fields, combined with the special frame
subspace provides, causes the ship as a whole to travel at FTL speeds.
If two nested fields have their outer edges "locked" into the special
frame, while the inner edges travel at near-c relative to one another,
this might cause the FTL effect, as an artifact of the special frame
trick. This has the added support of being almost exactly what the
Tech Manual describes, but it doesn't mention the special frame.
Since this makes for boring drama, it's unlikely we'll ever "really
know" how warp works in Star Trek.
 
Continue to: