Alcubierre warp drive
1. History
2. Topology
The Alcubierre warp drive spacetime has the topology $\mathbb{R^n}$.
3. Metrics and coordinates
Alcubierre coordinates
The original coordinates of the Alcubierre metric are taken from the point of view off an observer outside the bubble.
$$ds^2 = -dt^2 + (dx - v_s(t) f(r_s)) + dy^2 + dz^2$$With some curve $x_s(t)$, which defines the following :
$$v_s(t) = \frac{dx_s(t)}{dt}$$ $$r_s(t) = [(x - x_s(t))^2 + y^2 + z^2]^{\frac 12}$$Hiscock coordinates
$$ds^2 = -A(r) (dt - \frac{v_0 (1 - f(r))}{A(r)} dr)^2 + \frac{dr^2}A(r){}$$4. Tensor quantities
5. Symmetries
6. Stress-energy tensor
7. Curves
8. Equations
9. Causal structure
The Alcubierre warp drive is globally hyperbolic and does not have any singularities.