Thanks everybody for the great responses on the subject here. There's always a very high quality of discussion in this forum. (I'd have responded sooner but for a minor family medical situation)
Originally posted by Christopher:
Going by an Alcubierre warp model, it would make no difference at all. In theoretical physics, warp isn't actually motion in the conventional sense at all -- rather, you're occupying a bubble of spacetime whose topological relationship to the rest of spacetime is being altered by collapsing the space in front of it and creating new space behind it. It was actually summed up pretty well in Futurama's "Clone of My Own": "The ship doesn't move at all! It stays where it is, and moves the rest of the universe around it!"
So the effective "velocity" of the warp bubble is totally unrelated to the motion of the ship that's inside that bubble. In fact, it probably would be a very bad idea to have a lot of forward momentum when you created the bubble, because you'd just end up flying into the front of the bubble and being crushed by the extreme gravitational stresses of the space warp. Well, maybe that wouldn't happen if the warp-generating machinery was onboard your ship, because the warpfield would presumably maintain a steady distance from the generator. But your ship's kinetic energy would still have no significant effect on the performance of the drive itself, because it would only matter in relation to the "pocket universe" inside the warp bubble, not in relation to the greater universe beyond.
The only advantage of having high forward velocity when you enter warp would be that you'd still have that momentum when you left warp, due to conservation of energy. But that could be a disadvantage, because velocity is a vector quantity, with direction being part of it as well as speed. You might come out of warp travelling in the wrong direction relative to where you want to go, and thus have to waste energy braking and accelerating in the right direction.
I'll admit to a little ignorance as to the nuts and bolts of the Alcubierre warp model, but it's probably the best hard sci run at the concept I've heard of. My thinking was simply that warp drive appears to have a temporal component as described on screen, the first reference ever being captain Pike's order to go to "time warp factor 7" in "The Cage". The fact that warping down gravity wells (TOS: Tomorrow is Yesterday, ST:IV) seems to be a precursor to an actual temporal drive (the Borg may have used close earth orbit in ST:FC for the same purpose, and let's not forget what happened to the Enterprise in low orbit
around Psi 2000 in TOS: Naked Time ) suggests the warping of both space and time. If relativistic velocities in "normal" space have an effect on the rate at which time passes, I just can't help but see a possible connection.
The fact that sub-light velocity is a vector quantity can't be ignored of course. I figured you'd need to establish the absolute velocity vector, including the motion of the galaxy, star system, etc. to ensure you didn't go too far off course in exploiting a "velocity dilation effect", although this would probably be minimal over early, short interstellar hops (though if the velocity dilation effect only manifests along the forward axis of the warp drive, then the absolute velocity vector wouldn't cause any navigational deviation). Incidentally, I assume the warp propulsive effect is always directed along the longitudinal axis of any arrangement of warp coils, and would follow any change in nacelle orientation, with the sub c velocity dilation component dropping off as the warp drive axis diverges further from the sub c velocity vector.
I won't deny the many disadvantages of what I've already described as near suicidal relativistic speeds, but when the overriding concern of early interstellar exploration would probably be making transit times more manageble, it might seem worth it to those pioneering souls. In later centuries, I've assumed sub c velocities much above orbital approach speeds would be rare once warp drive tech matured, what with all that violent tactical manouvering and head on encounters with full stops in deep space.
Another incidental aspect of this whole scenario is a pretty stonking impulse drive, which is where I admit to being in the "impulse is a field drive" camp, requiring the warp mains to generate a helpful subspace field to let the impulse drive punch above its weight. The ambassador class NX-10521 prototype is the first to have a dedicated driver coil incorporated into the impulse drive for this job (TNG tech manual). You can't help noticing the significantly increased diameter of the warp nacelles on the amby - perhaps due to the now "warp optimised only" coils? Yet another assumption here would be the inability of (at least early) drives to generate concurrent impulse and warp fields, otherwise you could cram all that kinetic energy into your ship once you were "safely" in subspace.