Can an underdensity in space act as a negative matter density powering a warp drive? Consider a homogeneous isotropic universe filled with a perfect fluid with density $\rho$ and pressure $P = \rho/w$. E.g. for $w=-1$ we get a universe equivalent to one with "vacuum energy" or with the cosmological constant.
However, consider now that it is really a perfect fluid with which we can manipulate and $w$ is more or less arbitrary. Considering General relativity, could a relative underdensity of the fluid in the shape of a ring act as exotic matter powering a warp drive of an Alcubierre type?
Note that I want to ask only about the formal solution of Einstein equations and whether a similar metric to the one proposed by Alcubierre can be achieved without negative matter density.

This question is one of the questions breaking up the discussion of this question into smaller pieces.
 A: No, there is nothing like an "underdensity of space" and there is no medium that could power the "warp drive".
The cosmological constant may perhaps be considered an example of a "perfect fluid" because $p,\rho$ fully specify its state. But they actually overdetermine it. For an environment to be called the cosmological constant, $w=p/\rho$ has to be equal to $-1$ so the value of $w$ surely cannot be "manipulated". Otherwise the stress-energy tensor isn't proportional to the metric tensor – otherwise the "medium" isn't cosmological constant.
Also, the value of $p$ and $\rho$ – and we always have $p=-\rho$, I repeat – is constant in time and space. That's why this concept introduced by Einstein was called the cosmological constant. The very word "constant" means that its values cannot be modified in any way.
There exist genuine or hypothetical materials with $-1\leq w \leq +1$, with the vicinity of $w=+1$ being controversial ("the black hole gas"). For example, the cosmic domain walls and cosmic strings would have $w=-2/3$ and $w=-1/3$, respectively. Radiation has $w=+1/3$ while the dust has $w=0$, of course. However, one can't achieve $|w|\gt 1$ which would violate the null energy condition. The speed of sound in that environment would have to be greater than the speed of light which is not allowed by relativity.
Spacetime geometries typically classified as "Alcubierre warp drive" generally violate various energy conditions. In all the explicit versions, they also violate the null energy condition which is the strongest argument implying that these solutions aren't physically allowed. There are various general ways to argue that the energy conditions have to be upheld. A violation of energy conditions generally implies that the vacuum would have to be highly unstable which it demonstrably is not; preserving the Lorentz invariance while energy conditions are violated means that causality is violated and one may modify the past which is a logical inconsistency.
