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