Appearance of Warp Bubble Internal Volume to Distant Observer

[Nugatory]

Measure with clocks how long it takes to get from one point to another.

That requires some additional assumptions that you may not be aware of. We need some rule for relating times on the two clocks (“when the in-bubble clock reads X the distant clock reads Y”) before they can be compared - otherwise we just have two unrelated intervals from two unrelated clocks. This rule is called a “simultaneity convention” and it is pretty much an arbitrary assumption; depending on what we assume we can get pretty much any answer to your question we please.

There are similar difficulties with defining distance: the distance between two points in space (remember, a point in space is a line in spacetime) depends on where they are at the same time.

Because even a person far away at rest with respect to their frame is not at rest with respect to the bubble person's frame?

There is no meaningful way of comparing the velocities of two objects far enough apart that curvature effects matter. I can be at rest in some local frame, and you can be at rest in some other local frame, but unless the two overlap (in which case they’re really just one frame) our relative velocity is undefined so we cannot be said to be or not be at rest relative to one another.

I have already suggested that you stop tossing the word “volume” around until you can precisely define it in the context of this problem. You may want to be similarly careful with “velocity” and “distance” - the ordinary meaning of these words depends on hidden assumptions about flat spacetime.

 

[Onyx]

How will you measure the speed of whatever is crossing the bubble in order to convert time taken into distance travelled? Will you have to correct different amounts depending on where the bubble is? And will distant observers in different states of motion and/or position agree on any of it?

No, because the spacetime isn't stationary so there isn't an invariant notion of what you mean by "space". All of the questions in this thread stem from this fact. You need to define "space" before you can start to answer the question, and you have considerable latitude over how to do that - probably enough that the answer to your question is either yes or no depending on how you choose your definition.

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

 

[Onyx]

We are two dozen messages into this thread, and we still don't have a definition for what you are asking.

It's clear that I don't understand what I've been asking as well as I thought.

 

[George Jones]

Summary: Questions about how a warp bubble's internal volume would appear to a distant observer at a single moment of their time.

At a single moment of coordinate time $t$, would a distant observer perceive a warp bubble's interior volume as blown up, or would it seem compressed? Looking in the catalogue of spacetimes at the static local tetrad of the Alcubierre metric, the $e^x_{(x)}$ leads me to think that a static observer would see the flat interior differently from the exterior.

It's clear that I don't understand what I've been asking as well as I thought.

Answers to your questions might be given in (the figures of) "Detailed study of null and time-like geodesics in the Alcubierre Warp spacetime" by Thomas Muller and Daniel Weiskopf,

https://arxiv.org/abs/1107.5650

 

[PeterDonis]

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

I would suggest that you stop thinking in terms of coordinates and start thinking in terms of actual physical measurements. Which means actual physical measurements that you can describe in detail.

 

[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.