(image by author)
Well, it would create a bubble of space (and time) that would surround the craft and push that forward...the craft within the bubble would never exceed the speed of light and according to the best theoretical knowledge of physics, there is no limit to the speed at which space can be bent--er, "warped"--so this changing of the shape of space could create an effect that would make a vessel travel as if it were going faster than the speed of light. Note that this "warp speed" acceleration would not produce actual inertia, so Jean-Luc Piccard saying "engage" would not cause the crew to splatter against the far wall without some kind of inertial compensation (unlike how I previously discussed this topic in my post on Gravity and Starships)...
The equations based on the best understanding of the nature of space are well understood and a number of them are referenced in Dr. White's paper. The energy requirements are huge, but far less than what were thought necessary when this concept was first discussed by physicist Miguel Alcubierre in 1994 (this thesis ideas are covered in this link). The energy is in fact comparable to the energy requirements for a realistic trip in normal space (see my discussion of Project Valkerie in my Nanite Space Weapons post).
One major drawback would be that this bubble of normal space would tend to pick up particles of normal matter as it travelled along. Then once you shut down the warp drive, those regular pieces of matter--most of them would be small particles--suddenly continue on moving with the inertia they had when you picked them up...and space is full of very fast moving particles. It would create in effect an enormous blast of high moving particles, most likely going every direction. Very dangerous for your ship...and whatever planet you happened to be visiting.
Though that's not necessarily so bad...if there were a way to direct these particles, the system could serve as a weapon. Drop out of warp drive and BOOM--no photon torpedos required. But if there were no way to direct them, that would mean you'd simply have to shut down the warp field on a regular basis to let them dissipate (hey, it would still be a lot faster than hitchhiking :).
So the Warp Drive I used to make fun of as silly mumbo jumbo turns out to be not so unrealistic after all...except for one minor detail...well, not so minor. It turns out to create the warp field in the first place you have to generate something called "Negative Energy Density." Which, while some physicists believe perhaps could exist, negative energy density has never really been observed to exist in a way that anyone even in the greatest leap of imagination could reliably generate...sort of a problem, right?
Anyway, this should by no means impair a science fiction writer's work. Just craft the tale with this itty bitty little problem (well, immensely huge problem really) already solved and you're good to go...
Oh, by the way, "negative energy density" is related to something you could call "antigravity," which, if it existed, would be some really weird and certainly story-worthy stuff. But I'll cover that next time...
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