What is the significance of the velocity term in the Alcubierre metric?

[redstone]

Reading over Alcubierre's paper on his "warp" drive (http://arxiv.org/abs/gr-qc/0009013), the metric in equation 3 has a velocity term, v, that doesn't seem to be needed anywhere. Even in the one spot where it seems potentially valuable, equation 12, he just call it =1 and essentially ignores it. Also, it doesn't seem to have any mathematical connection to dx/dt (he just randomly says that's what it is after equation 5.

So I'm just wondering what it is I'm missing here? Why is the v term included at all? Is there some stronger need that requires it actually be equal to dx/dt? And finally, if v>0, doesn't that then destroy his equation 5 (i.e. 3-space would curve when a body has velocity)?

Any insight into that variable would be appreciated.

 

[redstone]

I guess nobody seems to know. So for posterity...
My own research into it, it appears to not be necessary for the metric itself to work, but when solving for the stress energy tensor, it looks like it makes terms, at least the T00 term, simpler, since there are d/dt terms that act on the x, giving new v's that cancels things out in a nice way. Haven't solved it without the v in there to verify, but guessing it would be more complicated.

 

[PAllen]

The purpose is simply to make the coordinate speed of bubble explicit. If you just used a generic function for beta-x, you would then have to solve for the coordinate speed getting some messy function. By building it in as specified, you get to pick the coordinate speed of the bubble.

As to your second question, equation 5 is preserved just fine. This equation, in context, simply says that if you consider a t=0 3-surface, you have a Euclidean spatial metric. This is clearly true for his equation (8). Just consider dt=0 in the metric.