Travel 10x the Speed of Light: Warp Drive Explained

[PeterDonis]

Does that mean you assume future to be predefined?

GR is a deterministic theory, so whenever we are talking about a GR model, the entire spacetime is "predefined".

we have a region of spacetime between Earth and Alpha Centauri and t<0 with flat spacetime. Another region of spacetime between Earth and Alpha Centauri and 0<t<T is highly curved. That's what I mean with the change of spacetime in my postings above.

But this is not a "change of spacetime"; it's just a fact about the spacetime geometry that the curvature is not the same everywhere. And thinking of it as a "change" is leading you to make incorrect inferences; see below.

Why does the region of spacetime at Alpha Centauri change (from the view of a local observer) from flat to highly curved within t<=T<4.3 years?

Spacetime doesn't change. See above.

If what you really mean is "a region of space between Earth and Alpha Centauri changes from being flat to being highly curved", this sort of thinking works OK for nearly flat spacetimes, but it doesn't work for highly curved spacetimes like the warp drive spacetime. See below.

it can't be caused by Kirks decision to travel to Alpha Centauri because this information would take at least 4.3 years to reach Alpha Centauri.

It would if spacetime were flat everywhere, yes. But this spacetime is not flat everywhere. The region of the spacetime that is curved (because of the warp drive being on) is curved in a way that allows the causal consequences of Kirk's decision to reach Alpha Centauri when clocks on Alpha Centauri read much less than 4.3 years. There's no way (that I'm aware of) to correctly model this as "space changing with time"; that type of model is simply an approximation, that works OK in flat or nearly flat spacetimes, but doesn't work in the kind of highly curved spacetime that you get when a warp drive is present.

(The reason the "space changing with time" model is only an approximation is that it leads you to a chicken-and-egg problem. A particular event can only causally influence events in its future light cone; but which events are in its future light cone depends on the geometry of spacetime. So thinking of the geometry of spacetime itself as "propagating" at a speed that's limited by the light cones is a logical circle that can't be closed: the light cones are determined by the very thing whose "propagation" is supposed to be determined by the light cones. The only way to rigorously avoid this problem is to look at the entire 4-d spacetime geometry "all at once" as a self-contained solution to the Einstein Field Equation. But in cases where the spacetime curvature is small enough everywhere, you can think of the light cones as being "fixed" to a good approximation, and then think about how curvature propagates within those approximate light cones, and get answers that are close enough for many purposes. A warp drive spacetime is one of those where the curvature is not small enough everywhere for this to work.)

 

[DrStupid]

If what you really mean is "a region of space between Earth and Alpha Centauri changes from being flat to being highly curved", this sort of thinking works OK for nearly flat spacetimes, but it doesn't work for highly curved spacetimes like the warp drive spacetime.

What I mean is this:

Then there will be a region of spacetime to the past of the warp drive starting which is flat; this region will be bounded by a surface of simultaneity in the mutual rest frame of Earth and Alpha Centauri, which we will call the surface t=0.
[...]
But spacetime in the region between the event of the ship leaving Earth and the event of it arriving on Alpha Centauri is highly curved

For t<0 the spacetime between Earth and Alpha Centauri is flat and for 0<t<T it is highly curved. All I currently understand is that you do not like the term "change of spacetime" for this fact about the spacetime geometry but I do not want to discuss semantics. I try to understand why these regions of spacetime are different - especially if (and if yes how) this can be caused by an event at Earth and t=0.

So thinking of the geometry of spacetime itself as "propagating" at a speed that's limited by the light cones is a logical circle that can't be closed: the light cones are determined by the very thing whose "propagation" is supposed to be determined by the light cones. The only way to rigorously avoid this problem is to look at the entire 4-d spacetime geometry "all at once" as a self-contained solution to the Einstein Field Equation.

Yes, this avoids the problem but it doesn't solve it. You still need to explain how the spacetime geometry around the spaceship during the flight results from the decision of the captain. For the case that you deny his freewill just replace him by a mechanism similar to Schrödinger's cat (which is widely agreed to be really randomly).

 

[JesseM]

This is one of those scenarios that you simply can't think about as "space" changing with "time"; you have to think about it as a 4-d solution right from the start.

Are you talking specifically about a spacetime containing closed timelike curves? A spacetime with a single warp bubble doesn't contain any (I think Alcubierre showed this in his original paper, though with multiple bubbles moving in different directions there are CTCs), so in that case can't one foliate the spacetime into a series of 3D hypersurfaces, and use the ADM formalism to describe how spatial curvature of the 3D hypersurfaces changes with time?

 

[PeterDonis]

this avoids the problem but it doesn't solve it. You still need to explain how the spacetime geometry around the spaceship during the flight results from the decision of the captain

And that's what the Alcubierre solution to the Einstein Field Equation does. It shows how the causal structure of the spacetime permits the events that take place in the region of spacetime along the ship's path from Earth to Alpha Centauri to all be causally connected to the event where the captain makes his decision. Unfortunately I don't have a handy quick visualization of how this works, but that doesn't mean it's not a valid solution.

 

[PeterDonis]

Are you talking specifically about a spacetime containing closed timelike curves?

No, just a single Alcubierre drive. (I'd rather not open the additional can of worms that comes into play when you have multiple warp drives in relative motion. )

can't one foliate the spacetime into a series of 3D hypersurfaces, and use the ADM formalism to describe how spatial curvature of the 3D hypersurfaces changes with time?

Sure. But there's no guarantee that the "speed of propagation" of changes in the spatial curvature will not appear to be "faster than light" in this formulation, just as the apparent "speed of recession" of galaxies from one another in the universe can be "faster than light". In both cases, the key physical reason is that the spacetime (or at least the portion of it in question) is highly non-stationary, so there is no way to foliate it with 3D hypersurfaces that all have the same geometry, and the "speed of changes in the geometry" from one hypersurface to another can in principle be arbitrarily fast. (If they are, then the light cones will also appear to "expand" arbitrarily fast; so you can still use the light cones to verify that nothing actually moves faster than light in the sense that matters. But it all will still appear highly counterintuitive.)

 

[DrStupid]

And that's what the Alcubierre solution to the Einstein Field Equation does. It shows how the causal structure of the spacetime permits the events that take place in the region of spacetime along the ship's path from Earth to Alpha Centauri to all be causally connected to the event where the captain makes his decision.

That sounds like I need to take a closer look at the Alcubierre solution. I wasn't aware that it includes the creation of the warp bubble.