- #1

Onyx

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- 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?