Kinematic selection of the viscous stress in relativistic dissipative hydrodynamics

Abstract

All standard formulations of relativistic dissipative hydrodynamics, from Eckart through Israel-Stewart to the recent BDNK framework, assume that the viscous stress depends on the shear tensor $σ_{αβ}$ and the expansion scalar $θ$ but not on the vorticity $ω_{αβ}$ or the acceleration $a_α$. We derive this structure from a Lagrangian kinematic construction on Lorentzian spacetimes, extending a recent result on Riemannian manifolds. The spatial strain rate, constructed from the rate of change of spatial inner products of Lie-dragged connecting vectors, is the spatially projected Lie derivative of the projected metric $h_{αβ} = g_{αβ} + u_αu_β$. The acceleration terms drop out exactly under spatial projection, and the vorticity cancels by symmetry. We show that material frame-indifference fails for generic Killing perturbations by an amount $δ\mathfrak{h}_{αβ} = +ε(ξ_αa_β+ ξ_βa_α)$ proportional to the acceleration, and is restored only for flow-preserving isometries. We prove that the non-relativistic limit of the BDNK equations gives the deformation Laplacian universally in the viscous sector, with the BDNK parameter dependence identified by Hegade K R, Ripley, and Yunes arising entirely from the thermal (heat-flux) sector. As an application, we derive the Weinberg gravitational-wave damping formula directly from the kinematic strain rate in a perturbed FRW spacetime.