Kinematic selection of the viscous stress in relativistic dissipative hydrodynamics

TL;DR

This paper derives the structure of viscous stress in relativistic hydrodynamics from a Lagrangian perspective, clarifying dependencies on kinematic quantities and analyzing frame-indifference and non-relativistic limits.

Contribution

It introduces a kinematic construction for viscous stress dependence, extending previous geometric results, and clarifies the role of acceleration and vorticity in relativistic dissipative hydrodynamics.

Findings

Viscous stress depends only on shear tensor and expansion scalar, not vorticity or acceleration.

Material frame-indifference fails for generic Killing perturbations but holds for flow-preserving isometries.

Non-relativistic limit of BDNK equations yields the deformation Laplacian universally in the viscous sector.

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.