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I was wondering how Alcubierre derived the metric for the warp drive? Sources have said it's based on Einstein's field equations, but how did he go from this to the metric?

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Alcubierre started with the metric and used the Einstein equation to calculate what stress energy tensor was required.

The Einstein equation tells us:

$$ R_{\mu\nu} - \tfrac{1}{2}R g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu} $$

Normally we start with a known stress-energy tensor $T_{\mu\nu}$ and we're trying to solve the equation to find the metric. This is in general exceedingly hard. However if you start with a metric it's easy to calculate the Ricci tensor and scalar so the left hand side of the equation is easy to calculate, and therefore the matching stress-energy tensor is easy to calculate.

The only trouble is that doing things this way round will usually produce an unphysicial stress-energy tensor e.g. one that involves exotic matter. And indeed this is exactly what happens for the Alcubierre metric - it requires a ring of exotic matter.

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The metric for the Alcubierre warp drive was constructed by considering the properties that it should obey, and not the matter source (which is why it's fairly unphysical).

The two ingredients used in it are :

  1. A bump function, so that the warp drive is localized in a specific region (and that bump function moves, so that the inside may move along with it)
  2. A widening of the lightcone in that bump function, so that, compared to the outside, the speed of light is "larger".

Given these two characteristics, we get the properties we want for a warp bubble. It is possible to also get variants by changing them, for instance the Krasnikov tunnel does not have a travelling bump function, but still has a widening of the light cone. This is why it is "static", compared to the warp drive.

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