Nonrelativistic limit of first-order relativistic viscous fluids

Out-of-equilibrium effects may play an important role in the dynamics of neutron star mergers and in heavy-ion collisions. Bemfica, Disconzi, Noronha and Kovtun (BDNK) recently derived a causal, locally well posed, and modally stable relativistic fluid model that incorporates the effects of viscosity and heat diffusion. We study the nonrelativistic limit of this fluid model and show that causality for relativistic motion restricts the transport coefficients in the nonrelativistic limit. This restriction provides an upper bound on the ratio of the shear viscosity to the entropy density for fluids that can be described as relativistic within the BDNK model and can be exactly modeled using the Navier-Stokes equation in the nonrelativistic limit. Furthermore, we show that the Fourier law of heat conduction must be modified by higher gradient corrections for such fluids. We also show that the nonrelativistic limit of the BDNK equations of motion are never hyperbolic, in contrast to the nonrelativistic limit of extended variable models, whose nonrelativistic equations of motion can be hyperbolic or not depending on the scaling of the transport coefficients present in the auxiliary equations for viscous degrees of freedom.