Late time behaviour of the Alcubierrie warp drive metric

The Alcubeierrie Warp Drive metric looks like $$ds^2 = -dt^2+(dx-Xdt)^2+dy^2+dz^2$$ where $X = v_s(t)f(r_s)$ and $r_s = [(x-x_s(t))^2+y^2+z^2]^{1/2}$. Now, $f(r_s) approx 1 quad 0 < r < R$ and
$f(r_s) approx 0 quad r_s > R$. Obviously, this metric is time dependent. However, I would like to study the late time behaviour of this metric i.e. $tto infty$. What is the best way to take this limit?