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 ${\sigma }_{\alpha \beta }$ and the expansion scalar $\theta$ but not on the vorticity ${\omega }_{\alpha \beta }$ or the acceleration $a_{\alpha }$. 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 $h_{\alpha \beta }=g_{\alpha \beta }+u_{\alpha }{u}_{\beta }$. 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 $\delta {\mathfrak{h}}_{\alpha \beta }=+\epsilon ({\xi }_{\alpha }{a}_{\beta }+{\xi }_{\beta }{a}_{\alpha })$ 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.

6 pages