# How Would an Alcubierre Drive “ride” a Wave of Spacetime?

The colloquial explanation is that the spacetime in front of a ship contracts and the spacetime behind expands. I see how one could think that this would bring you closer, but I don't see that it actually does.

Let's look at the front of the bubble. So spacetime "contracts". If you look at a differential volume being run over, I would think it contracts to a minimum, but then must expand again to pass under the ship. If the bubble is moving, but the ship inside is not accelerating (let's assume it's also motionless in its space), then spacetime must be "piling up" in a (compressed) region in front, because you can't throw away the extra spacetime.

To paraphrase A Wrinkle in Time's analogy: think of some fabric and a bug that wants to get from where it is to somewhere else. You can fold the fabric and jump across the gap, but what I'm thinking of the Alcubierre drive doing is "scrunching up" the fabric: the bug still needs to walk over the same length of material, it's just that it looks like less distance.

Is all the above bogus? I don't really know what I'm doing with the physics here, as I have more of a mathematical background. I'm not seeing a way around this without introducing some discontinuity in space, which (I don't think) was the point. I also read somewhere that the expansion/contraction analogy could be misleading; is this why?

You are thinking of spacetime as some form of elastic fluid, and with this perspective it makes sense that if you squeeze spacetime ahead of the ship it must flow round the sides and expand again behind the ship. However this is a misleading model. Spacetime can be compressed and stretched by arbitrary amounts. For example in the FLRW metric that describes our universe spacetime (or just space if you choose comoving coordinates) is expanding continuously but this doesn't mean it's crumpling up or compressing at the edge of the universe. The drive moves because spacetime just keeps compressing in front of it and expanding behind it with no limit. When you turn the drive off the spacetime doesn't rebound, it stays where it is so the spaceship stays in whatever location it has reached.

My favourite way of writing the Alcubierre metric is from Harold White's paper Warp Field Mechanics 101:

$$ds^2 = -dt^2 + (dx - v_s(t)f(r_s)dt)^2 + dy^2 + dz^2$$

where $v_s(t)$ is effectively the velocity and $f$ is just a shaping function to restrict the region where the compression/expansion occurs. If you compare this to the flat space metric:

$$ds^2 = -dt^2 + dx^2 + dy^2 + dz^2$$

It should be obvious why the drive is moving in the $x$ direction with speed $v_s$.

• "The drive moves because spacetime just keeps compressing in front of it and expanding behind it with no limit. When you turn the drive off the spacetime doesn't rebound, it stays where it is so the spaceship stays in whatever location it has reached." Yes, but then what? The spaceship needs to move outside of that region to do anything useful, moving through compressed (front) sheared (side) or stretched (back) space. My question is, doesn't this take just as long? See my third paragraph in the OP. – imallett Feb 20 '14 at 16:51