Particle creation, clock variables, and relabelling symmetry in pull-back variational fluids

Abstract

Abstract Particle-number conservation is an off-shell kinematical identity of the standard pull-back variational formulation when the matter-space three-form depends only on the comoving labels of a constituent. We revisit the relabelling argument when the matter-space density also carries particle-creation information. The conservative contribution to the Eulerian current variation then cannot be considered alone: the variation of the creation sector must be included. For the three constructions studied here---a proper-time functional, an independent local scalar clock, and interacting matter spaces---the full current variation is invariant under flow-aligned relabellings even when $\Gamma_N\equiv\nabla_a n^a\neq0$. The constructions differ instead in locality, field content, and dynamical closure. A proper-time clock introduces no additional field, but its variation contains an acceleration-dependent history term and its metric dependence produces a non-local longitudinal stress contribution. An independent scalar clock is local but requires its own dynamics or constraint; a freely varied algebraic clock forces $\Gamma_N=0$. Interacting matter spaces provide a local description of inter-species conversion through the relative dynamics of the partner fluids. On FLRW backgrounds the two single-fluid realizations can reproduce the same homogeneous creation history, but the background law does not determine its perturbative completion. The gauge-invariant mismatch $\Delta_\Gamma=\delta\Gamma-\beta\delta\theta$ controls the non-adiabatic creation pressure in the effective one-fluid description, and its value depends jointly on the clock dynamics and the constitutive completion of the creation law. Thus particle creation is compatible with pull-back relabelling symmetry, provided the additional creation-sector structure needed for closure is specified.