Geometric constraints on vorticity and helicity in relativistic ideal flows

This paper investigates geometric constraints on vorticity and helicity in ideal, isentropic, compressible relativistic flows. The natural dynamical variable in this setting is the relativistic fluid momentum p→=(γw/c2)V→, whose curl Ω→t=∇→×p→ is the relativistic fluid vorticity: it encodes both the kinematic and the thermal inertia of the fluid element, satisfies a frozen-in evolution equation, and reduces smoothly to the Helmholtz vorticity ∇→×V→ in the non-relativistic limit. This object admits a further unification with the electromagnetic field into a magnetofluid tensor; the present work, however, is concerned with the purely hydrodynamic regime and the geometric content of Ω→t itself. We analyze the interplay between Ω→t, the Biot–Savart helicity, and the physical helicity, and derive three hierarchical conditions. First, if Ω→t≡0→, then the scalar density V→·(∇→×V→) vanishes everywhere; by the Frobenius theorem, this yields, on the open set {V→≠0→}, a local foliation by surfaces orthogonal to V→ (a global foliation requires additional global/topological hypotheses). Second, if the conserved relativistic physical helicity vanishes, HRPh=0, then there exists at least one point in each time slice at which V→·(∇→×V→)=0. Third, in the ultra-relativistic matter-dominated regime (kBT≫m0c2), where p→=ζγTV→, the condition Ω→t≡0→ forces vortex lines to lie on the level surfaces of γT. We emphasize that the expansion of Ω→t contains an additional term ∇→(γw/c2)×V→ with no non-relativistic counterpart; it is precisely this term that makes the present geometric phenomena genuinely relativistic. In particular, Ω→t can vanish while ∇→×V→ does not, leading to the orthogonality of V→ and ∇→×V→ (special condition I) and, in the ultra-relativistic regime, the confinement of vortex lines to level sets of γT (special condition III). These results provide analytically checkable constraints for ideal relativistic-flow models and numerical simulations.