If you could decrease the distance between here and the nearest star by, say, four light years, wouldn't it take four years for that distance to disappear? For this reason, wouldn't so called "warp drive" be impractical?
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This is a fairly complicated question to answer. The basic idea behind the answer is that, since we are using general relativity in which the metric is dynamical, there is no good definition of "it is four years away".
Saying that something is four years away means that at a time $t_0$, the shortest path to that destination intersects it at $t_1 = t_0 + 4$ years, but this is defined by some geodesic $\gamma$ such that $$\int_{t_0}^{t_1} d\tau = 4 \ \text{years} $$ where $d\tau$ is the proper time, which depends on the metric. At the time $t_0$, before the warp drive is constructed, we do not know what that metric is. It is only 4 years if we continue to use flat space in future predictions.
The question then becomes, if it is 4 years for any metric we can construct. The answer is once again complicated.
The Alcubierre metric is fairly simple for that : the bubble can be built in some arbitrary region and go at an arbitrary speed. There is no need to wait any period of time a priori. But then again, there is a trick : the Alcubierre metric requires matter that violates the dominant energy condition to work, and in particular the flux energy condition : the flux of energy already goes faster than light.
This isn't necessarily a big problem (there are plenty of other problems with it, of course). Some phenomenon do break the flux energy condition, like the Casimir effect.
To avoid this sort of issue, another type of warp drive metric was proposed, the Krasnikov tube. The Krasnikov tube does take 4 years (or longer) to build, laying matter that violates the weak energy condition but not the flux energy condition along the way. but, due to the way it is constructed, while the trip to the star takes 4 years or more, the trip back can be arbitrarily short, and can bring you arbitrarily close to your departure date.
Proving it is slightly tricky but this does not violate causality. Although this does mean that a system of two Krasnikov tubes will definately violate causality if positioned the right way.
