On a viable first-order formulation of relativistic viscous fluids and its applications to cosmology

Abstract

We consider a first order formulation of relativistic fluids with bulk viscosity based on a stress-energy tensor introduced by Lichnerowicz. Choosing a barotropic equation of state, we show that this theory satisfies basic physical requirements and, under the further assumption of vanishing vorticity, that the equations of motion are causal, both in the case of a fixed background and when the equations are coupled to Einstein's equations. Furthermore, Lichnerowicz's proposal does not fit into the general framework of first order theories studied by Hiscock and Lindblom, and hence their instability results do not apply. These conclusions apply to the full-fledged non-linear theory, without any equilibrium or near equilibrium assumptions. Similarities and differences between the approach explored here and other theories of relativistic viscosity, including the Mueller-Israel-Stewart formulation, are addressed. Cosmological models based on the Lichnerowicz stress-energy tensor are studied. As the topic of (relativistic) viscous fluids is also of interest outside the general relativity and cosmology communities, such as, for instance, in applications involving heavy-ion collisions, we make our presentation is largely self-contained.

Dynamical analysis of a first order theory of bulk viscosity

“…The study of the stability of such critical points allowed us to assess the past and future behaviour of the system, characterized unambiguously by the deceleration and effective equation of state parameters. The results obtained here are in accord with those presented in [27], for instance regarding the attractor behaviour of the phantom solution for α < −1/2 (critical point P 5 ), but as well highlight additional features: de Sitter-like future attractors exist for different ranges of the parameter α in the non-positive curvature case (points P 0 , P 2 and P 4 ); for α > 0, a stiff matter-dominated solution is found as a past attractor for k ≤ 0 (point P 3 ) and as both a past and a future attractor for k > 0 (points Q ± 1 ); we notice further that in general the evolution preserves the sign of the entropy production, so that positive entropy-producing initial conditions cannot evolve into negative entropy-producing states.…”

Bianchi I cosmology in the presence of a causally regularized viscous fluid

“…We can think of F as a contribution which eliminates the non-causality features of Eckart's formulation by regularizing the velocity (F will be therefore referred below as the regulator of the theory). The cosmological consequences of the Lichnerowicz description in isotropic cosmologies have been examined in [18]. In [17,22], the index of the fluid is parametrized as…”

“…The main advantages of the Lichnerowicz theory have been pointed out in [18]. A key-point consists in the fact that expression (5) reduces to the traditional description provided by Eckart upon setting F = 1.…”

“…Such a revised formulation is based on the introduction of the so-called index of the fluid, de facto a regulator scaling the four-velocity field, so defining a dynamical velocity of the fluid. This approach has been tested on some real systems, receiving interesting confirmation to its viability [17,18]. We shall stress that, being derived via a phenomenological approach, the Lichnerowicz energy-momentum tensor must be completed by the specification of an ansatz linking the index of the fluid to the thermodynamical variables of the system, so closing the dynamical problem.…”

Quaternion Algebra on 4D Superfluid Quantum Space-Time: Gravitomagnetism

“…More detail consideration of the random generalized Brownian motion has been performed by Rapoport in[75,76] 8. In a general case we should define the viscous stress tensor[24,84,85,93]. At the transition to the nonviscous superfluid we come to the noise tensor[75] fluctuating about zero.…”