A was what Alcubierre called the lapse function which in his paper he used for it, and g was 1-f which he used f for f. In regard to the lapse function I just chose a different letter to represent it. The function I independently introduce is h, which is roughly the square of g, and in regions where g is a constant is exactly g. What Alcubierre did was write down his choice of spacetime that would yield a warp drive to calculate what matter state would correspond to yielding it. The choice of warp drive to write down is not unique. The exotic part of it that didn't correspond to the usual state of matter was the energy density term as it was negative and spread all throughout the warp. He published the energy density as observed by a Euclarian observer probably because the publisher didn't want readers to know how the ship frame value is decreased by the lapse function, which I'm sure Alcubierre knew because he included the lapse function in the paper. In my choice that exotic matter is contained "inside" the warp bubble. I believe it can be produced by the Casimir effect. The physicality of the momentum like term I just showed does not prefer a sign. As for the outer part of the hull a ceramic is the best material I know of to help shield the ship, although the warp itself tends to deflect forward incoming geodesics around it. Essentially the warp itself works like a deflector shield.