Zero expansion is not free: the sharp minimum of Eulerian negative energy in Natário warp drives

Abstract

Natário's warp drive replaces the contraction and expansion of the Alcubierre bubble by a divergence-free shift. We determine exactly how much Eulerian negative energy this construction requires. For the Lobo-Visser volume integral $\mathcal{E}_{\rm LV}=-\int_{Σ_t}ρ_{\rm E}\,d^3x$ on the standard flat slice, with unit lapse, a flat interior of radius $R$, and exact matching to a uniform exterior stream of speed $v$ at $R+Δ$, we prove a lower bound for every admissible shift and find the exact minimum in closed form at every aspect ratio. As the wall thins, $\mathcal{E}_{\min}\sim v^2R^4/(4Δ^3)$ with sharp constant $1/4$, two powers of $R/Δ$ above the earlier estimate $v^2R^2/Δ$ applied to both Natário and Alcubierre bubbles. Incompressibility forces the flux displaced by the bubble to return through the wall, and the tangential shear of this return current dominates the energy. The full variational problem is equivalent to classical Stokes flow between concentric spheres. Its unique minimizer tends to a minimal-curvature cubic in the thin-wall limit. An exact flux identity and independent quadrature support the analytic result. The bounded quantity is slice and observer dependent and is distinct from the ADM mass.