The geometry
In Alcubierre’s metric, written in units of the speed of light c,
ds² = −c²dt² + (dx − vs f(rs) dt)² + dy² + dz²
The function f is 1 inside the bubble and 0 far away, with a thin wall in between. vs is the speed of the bubble and rs the distance from its centre. Inside the bubble, space is flat; the ship is at rest relative to its local space and its clock runs at the same rate as clocks far away.
Contraction ahead, expansion behind
The popular picture says space shrinks in front of the ship and grows behind it. That is what the volume expansion of Alcubierre’s geometry looks like. Natário showed in 2002 that this is not essential: a warp bubble can also slide through space without changing any volume. What matters is the “shift”, the way the bubble’s space is carried along relative to space outside.
The price
As seen by observers at rest outside, the energy density in the bubble wall is negative everywhere. This is the reason the Alcubierre drive is considered unphysical in its original form, and the starting point for all later work on reducing or avoiding negative energy.