Appearance of Warp Bubble Internal Volume to Distant Observer

[Ibix]

Specifically, would this involve redefining the spatial coordinates to something else?

Picking a coordinate system may define a notion of space, but it may not. Or it may define a notion of space in one way in some places and in a different way in others, or only define space in some regions (Schwarzschild coordinates famously have several of these issues). Or it may define a notion of space that doesn't have the properties you are expecting. Coordinates are usually chosen for mathematical convenience, so don't necessarily reflect physical concepts in the way you expect.

I have only skimmed the paper @George Jones posted, but I would recommend its approach to you. It studies geodesics of the spacetime, which let's you see how real objects behave in the warp bubble rather than trying to study the mathematical abstraction of a coordinate system.

 

[George Jones]

It studies geodesics of the spacetime, which let's you see how real objects behave in the warp bubble rather than trying to study the mathematical abstraction of a coordinate system.

Yes. The visual images of what observers (inside and outside the bubble) see are obtained using optical ray-tracing (null geodesics).

 

[Onyx]

Thank you for the suggestion, I'll take a look. Just a few more things, though: the most straightforward coordinate change I can think of is $dt=d\tau-\frac{vf}{1-v^2f^2}dx$, which takes away the off-diagonal and adds a term in front of $dx$. If by volume I just mean det$(\gamma)$$dxdydz$, then the function looks similar to $f$ plus a constant. However, I'm not sure whose time $\tau$ represents in this case, and if it applies everywhere. On the other hand, someone else told me that the trace of the extrinsic curvature tensor is a coordinate-invariant description of the curvature, so maybe taking the integral of that would give some clues? In any case, I'll check out the article.