Momentum Density is the Sole Channel for Hawking–Ellis Type IV in Warp Spacetimes, and It Is the Curl of the Vorticity

Recent work has identified nonzero shift vorticity as the geometric obstruction driving warp-drivebubble walls to Hawking-Ellis Type IV, while noting that the associated coupling coefficient dependson wall geometry and that no general converse was established (Le, arXiv:2602.18023). This papersupplies the mechanism, and in doing so corrects the controlling variable.Under unit lapse and flat spatial slicing the ADM momentum constraint reduces to j = -(curl omega)/8pi,so the momentum density is the curl of the vorticity. Because the mixed stress-energy tensor is blockdiagonal whenever j = 0, and real symmetric blocks cannot carry complex eigenvalues, momentum densityis the only channel through which a Type-IV eigenvalue pair can open. This yields an immediate proof ofthe known sufficient direction, that an irrotational shift gives Type I matter (Santiago, Schuster andVisser; realised explicitly by Rodal, Gen Relativ Gravit 58, 1, 2026), and shows that the operativequantity is the curl of the vorticity rather than the vorticity itself, which accounts for thegeometry-dependent coupling reported previously.The distinction has consequences. The shear and vorticity of the shift admit exactly five independentinvariants under O(3), verified complete at 300 of 300 random configurations; the curl of the vorticityis not among them and cannot be. This is exhibited constructively: a perturbation vanishing to firstorder at a point leaves all five invariants unchanged to machine precision while altering the Type-IVimaginary part by up to 395 percent, and near the boundary flips the algebraic type outright in 12 of12 cases. An orthogonal irreducible decomposition places 93.7 to 99.9 percent of the sensitivity in asingle three-dimensional piece, identified to 2e-16 as the curl of the vorticity. For isotropic spatialstress the Type-IV criterion is explicit: |j| > (rho + p)/2.Scope is stated precisely: the identity holds for unit-lapse, spatially flat metrics (Alcubierre,Natario, Lentz, Rodal) and not for conformally flat constructions such as Van Den Broeck or for shellmetrics with radial lapse profiles.