The Curl of the Vorticity, Not the Vorticity, Controls Hawking–Ellis Type IV in Warp Spacetimes

Abstract

Recent work identifies the vorticity of the ADM shift as the geometric quantity controlling theHawking-Ellis algebraic type of warp-drive stress-energy, with the Type-IV imaginary eigenvalueobeying f = kappa*omega in a controlled limit, while reporting that kappa depends on wall geometry(Le, arXiv:2602.18023). This paper shows the controlling quantity is the curl of the vorticity, andthat the reported geometry dependence of kappa is what that misidentification looks like.Under unit lapse and flat spatial slicing the ADM momentum constraint gives j = -(curl omega)/8pi.Since the mixed stress-energy tensor is block diagonal whenever j = 0, and real symmetric blocks carryreal spectra, momentum density is the only channel through which a Type-IV pair can open; theirrotational case follows in one line, recovering Santiago-Schuster-Visser and Rodal. Because the curlof the vorticity is a second-derivative quantity while the shift's first-derivative data admits onlysix O(3) invariants, no invariant of the vorticity and the shear can determine the algebraic type. Aperturbation vanishing to first order at a point holds all six fixed to machine precision whilechanging the Type-IV imaginary part by up to 395 percent and flipping the type outright in 12 of 12near-boundary cases.The two candidate laws are then tested in warpax, the open-source reference implementationaccompanying the cited work, on that work's own metrics. The published controlled family sweepsrotation amplitude, in which both quantities are linear, so it cannot discriminate; sweeping envelopewidth at fixed amplitude does. Over that sweep Im/omega varies by a factor of 58 (R^2 = 0.36) whileIm/|curl omega| is constant to four significant figures (R^2 = 0.99999998). Across a 252-evaluationscan over bubble velocity, wall thickness and off-axis position, the correlation of Im with|curl omega| is +0.998 against +0.815 for omega; for the Natario drive the correlation with omega isnegative in 8 of 12 parameter cells. At on-axis wall points symmetry forces omega = 0 exactly in threecanonical metrics that are nonetheless Type IV, which f = kappa*omega cannot accommodate. Rodal'sirrotational drive gives zero Type-IV points across all 84 probes.An analytic counterexample is given that requires no computation: for rigid rotation the vorticityis nonzero everywhere while its curl vanishes identically, so f = kappa*omega with nonzero kappapredicts Type IV at every point of a rigidly rotating spacetime, whereas the curl law predictsType I. The cited work records the same physical conclusion.Limits of validity are reported: the coefficient is a leading-order constant rather than a universalone, degrading in the thin-wall high-speed corner, and the identity excludes conformally flat metricssuch as Van den Broeck and shell constructions with radial lapse profiles.Version 2 also corrects an error in version 1, which stated the energy density was fully expressiblein the shear-vorticity invariant set; it also requires the expansion, so the complete first-derivativeset is six invariants rather than five.