Alcubierre Warp Bubble 2D Cross-Section: Spacetime Contraction & Expansion

Warp Bubble Parameters

Coordinate velocity v_s 4.0 c

Bubble radius R 6.0

Wall thickness σ 1.0

Live Readouts

Distance travelled

0.0

Coordinate speed

4.0 c

θ front (contraction)

0.00

θ rear (expansion)

0.00

Local proper accel.

0.00 g

Shape f(0)

1.00

How it works

The Alcubierre metric describes a bubble of flat spacetime carrying a ship, surrounded by a thin wall where space itself is contracted ahead and expanded behind. This 2D view is the exact z=0 cross-section of the same 3D metric — nothing here is a flattened approximation:

ds² = -dt² + [dx - v_s f(r_s) dt]² + dy² + dz²

f(r_s) = tanh(σ(r_s+R)) − tanh(σ(r_s−R))
         ───────────────────────────────
                  2 tanh(σR)

r_s = √[(x−x_s(t))² + y²]   (z=0 slice)

f(r_s) ≈ 1 deep inside the bubble (flat, comoving interior) and f(r_s) ≈ 0 far away (space at rest). The shift vector β^x = −v_s f(r_s) drags "Eulerian" grid points along x as the wall sweeps past — that is exactly what the warped grid lines in the scene trace out.

The local volume expansion scalar in the wall is:

θ = v_s · (x−x_s)/r_s · df/dr_s

θ < 0 ahead of the ship (space contracts to meet it), θ > 0 behind (space expands to push it away) — verified numerically for the default parameters (v_s=4c, R=6, σ=1): θ_front ≈ −2.00, θ_rear ≈ +2.00. Because the ship sits in a locally flat patch of the bubble, its proper acceleration stays ≈ 0 — no g-forces — even though the bubble's coordinate speed v_s can exceed c. This requires a wall with negative energy density (violates the classical energy conditions), which is why the drive stays theoretical: no known material has that property.

v_s — bubble's coordinate velocity in units of c.
R — bubble radius; σ — how sharply the wall transitions from f=1 to f=0.
Field toggle — colours grid markers by local θ (blue = contraction, orange = expansion).
Drag the main view to pan; scroll to zoom. The lower strip plots f(r) and θ(r) live against radial distance from the ship.

Ship / bubble wall Contraction (θ<0) Expansion (θ>0)

drag to pan · scroll to zoom