This is the exact z=0 plane of the same Alcubierre metric as the 3D version — nothing here is a flattened approximation. The shape function contracts space ahead of the ship and expands it behind:
f(r_s) = tanh(σ(r_s+R)) − tanh(σ(r_s−R))
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2 tanh(σR)
Its field equations demand a stress-energy source with negative energy density concentrated in the thin wall, strongest perpendicular to the direction of travel (angle φ from the ship's heading, measured in this x–y slice):
T⁰⁰(r_s,φ) = − c⁴/(32πG) · v_s² sin²φ · (df/dr_s)²
M·c² ≈ ∫ |T⁰⁰| dV (integrated over the full 3D wall shell)
The heatmap traces a ring around the ship at radius ≈ R: white/orange = strong negative-energy demand (the flanks, φ=90°/270°), dim blue = negligible (dead ahead/astern, φ=0°/180°). The rotating scan line is a visual aid only — the physics is static for fixed sliders. The panel still integrates T⁰⁰ over the full 3D shell volume every time a slider moves (the same integral the 3D simulator uses) and reports the total mass-equivalent |M| = E/c², then divides it by a real astronomical mass — for an everyday bubble radius pushed even a few times past light speed this ratio typically runs into the thousands or far beyond, the actual reason the drive stays theoretical: no known matter has negative energy density, let alone this much of it.
- vs — bubble's coordinate velocity in units of c.
- R — bubble radius in metres; σ — how sharply the wall transitions from f=1 to f=0 (a sharper wall concentrates the same total curvature into less volume, raising the peak density).