A Proper Volume on Constant Hypersurface in Alcubierre Metric

[Onyx]

TL;DR Summary

Finding the proper volume in the Alcubierre metric for a constant $t$ hypersurface.

I'm wondering if there is a way to find the proper volume of the warped region of the Alcubierre spacetime for a constant $t$ hypersurface. I can do a coordinate transformation $t=τ+G(x)$, where $G(x)=\int \frac{-vf}{1-v^2f^2}dx$. This eliminates the diagonal and makes it so that the determinant of the spatial metric is $\frac{1}{1-v^2f^2}$. But this doesn't seem right for finding the volume because it is an even function over $x$ and $-x$, while I would have expected the proper volume to the rear of the bubble to be bigger than in the front. Is there some other transformation I would need to make?

 

[PeterDonis]

I would have expected the proper volume to the rear of the bubble to be bigger than in the front.

Why would you expect this?

 

[Onyx]

Because the volume elements behind the bubble are expanding while in front they are contracting. Given a bubble of $R=4$ and $\sigma=1$, I figured that integrating the spatial volume on the $t=0$ hypersurface would produce a larger volume from $x=-3$ to $x=-5$ than from $x=3$ to $x=5$, since the elements are in the process of expanding/contracting.

 

[PAllen]

What do you mean by proper volume? The volume in GR is strongly dependent on foliation. Unless there is some geometrically significant foliation (e.g. hypersurface orthogonal to a timelike KVF, or identified by homogeneity - as in cosmology, etc.) I don't understand what can by meant by "proper volume".

 

[PeterDonis]

Because the volume elements behind the bubble are expanding while in front they are contracting.

Why do you think that? (Hint: can you point at something in the actual math that says that?)

 

[Onyx]

What do you mean by proper volume? The volume in GR is strongly dependent on foliation. Unless there is some geometrically significant foliation (e.g. hypersurface orthogonal to a timelike KVF, or identified by homogeneity - as in cosmology, etc.) I don't understand what can by meant by "proper volume".

Well I guess I mean how the space around the bubble is perceived by a distant observer at constant $t$.

 

[Onyx]

Why do you think that? (Hint: can you point at something in the actual math that says that?)

I guess the only reason would be the graph of the trace of the extrinsic curvature tensor, which shows the rate of expansion in front of and behind the bubble, with the rate being negative in the front. Of course, this is different from what I've been talking about, so not the best reason.

 

[paige turner]

proper volume: how the space around the bubble is perceived by a distant observer

would it perhaps be the volume of the bubble as measured by someone at rest inside it?