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What is the exact mechanism of the warp drive that propels it forward? I know that it creates negative pressure behind it and a positive pressure in front of it but how does this actually propels it forward.

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There isn't any time-evolution dynamics for warp bubbles that would single out straight-line motion. In fact, there can't be a cause-and-effect relationship between the position of the bubble at different times because it would violate light-cone causality. For the same reason, you can't create or control the bubble locally, so it isn't really a "drive". If this method of travel is possible at all (a very big if), you would have to set up the conditions for the bubble across the whole route in advance, at slower-than-light speed. The bubble would then follow the shape of the track you'd laid down, presumably, but no one has come up with a detailed proposal for how that would work.

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The movement is not coming from the pressure, indeed there are contractionless warp drives too which are still able to move.

For superluminal drives, you are forced to create the necessary energy density distribution in advance, so that at the right time the warp can be propelled along the trajectory you designed superluminally, but without requiring tachionic matter. The warp is by definition the metric generated by that energy distribution. By varying the energy profile you can in principle obtain an arbitrary acceleration profile. See for instance chapter 3.2 of this paper.

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