Alcubierre metric

Example

Use units where the gravitational constant and the speed of light equal $1$. Let $x_s \colon \R \to \R$ be a smooth function, the path of the bubble centre, with velocity $v_s = x_s'$. Let $f \colon [0, \infty) \to [0, 1]$ be a smooth function with $f(s) = 1$ for small $s$ and $f(s) = 0$ for large $s$. Set $r_s = \sqrt{(x - x_s(t))^2 + y^2 + z^2}$. On $\R^4$ with coordinates $(t, x, y, z)$, the Alcubierre metric is $$g = -dt^2 + \left(dx - v_s(t) f(r_s)\, dt\right)^2 + dy^2 + dz^2.$$

Check of the main features: where $f(r_s) = 0$, far from the bubble, $g$ is the flat Minkowski metric. Where $f(r_s) = 1$, inside the bubble, $g = -dt^2 + (dx - v_s\, dt)^2 + dy^2 + dz^2$, which is flat Minkowski space in coordinates moving with velocity $v_s$. The world line $x = x_s(t)$, $y = z = 0$ of the bubble centre has tangent $\partial_t + v_s\partial_x$, and $g(\partial_t + v_s\partial_x, \partial_t + v_s\partial_x) = -1$. So it is a timelike curve parametrized by proper time, and one checks that it is a geodesic. No condition limits $v_s$, so the bubble may move faster than light relative to the distant flat region. Yet the observer inside never moves faster than light locally.

The metric is defined first, and the matter is read off from the Einstein field equations. Consider the observers with $4$-velocity $n = \partial_t + v_s f\, \partial_x$, normal to the slices $t = \text{const}$. The energy density they measure is $$T_{ab} n^a n^b = -\frac{1}{8\pi}\, \frac{v_s^2\left(y^2 + z^2\right)}{4 r_s^2} \left(f'(r_s)\right)^2 \le 0,$$ and it is strictly negative where $f' \ne 0$ off the axis. So the weak energy condition fails in the wall of the bubble.