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The models/depictions I've seen of warp bubbles show space compressed ahead of the bubble and expanded behind, so that the space inside the bubble moves with respect to the space outside. If that is so, then what is happening at the sides? It would seem that there is some sort of shear taking place between the space inside and outside. Is that correct, or is that based on a misinterpretation of the model?

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I don't really know the answer, so this is just speculation. "In front" of the bubble could span side to side, and "behind" the bubble could be the remaining 180 degrees. There is no "side" in the sense you mean, though the sides require negative energy. – markovchain Aug 6 '13 at 17:06
There aren't really sides, there are places where the spheres overlap, touch or don't meet based on their radius. – user6972 Aug 7 '13 at 7:22

From Wikipedia:

Alcubierre has shown that a ship using an Alcubierre drive travels on a free-fall geodesic even while the warp bubble is accelerating: its crew would be in free fall while accelerating without experiencing accelerational g-forces. Enormous tidal forces, however, would be present near the edges of the flat-space volume because of the large space curvature there, but suitable specification of the metric would keep them very small within the volume occupied by the ship.

See also:

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I don't see how this answers the question. – John Rennie Jan 3 '15 at 11:57
He asked what would happen along the sides of the bubble, and according to that source, tidal forces would, indeed, be present, as he assumed. – AdamHovorka Jan 3 '15 at 21:27
@AdamHovorka There would be obvious tidal forces at the front and back edges, where the space is highly warped; but Anthony is asking specifically about the "sides" of the bubble (90 degrees from the axis of travel) where the space may or may not be highly warped. The sides of the bubble might be spatially stationary, just as the surface of a spinning tire is stationary at its contact with the ground, even when the rest of it is spinning rapidly. That's the sort of behavior Anthony wants to know about. – Asher May 15 '15 at 8:20

the outside of the bubble is causally disconnected from what happens inside the bubble. So that means that the space around the bubble doesn't even know there is a bubble. Even if you had the exotic matter to do the trick, you would have to lay it out as a cylinder between home and destination before you can make the trip. So, as an interstellar propulsion, is without a doubt useless

this is why the Alcubierre drive is a no-go; compared to this problem, the lack of exotic matter (aka negative mass) is just a minor detail

If you had the exotic matter, you could probably try to do this on a lab, arranging the field beforehand. But of course, you don't have exotic matter either

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According to this page:… there is the suggestion that exotic matter is not actually required. – Anthony X Jun 19 '15 at 1:05

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