Relativistic fluid dynamics: physics for many different scales

Abstract

The relativistic fluid is a highly successful model used to describe the dynamics of many-particle systems moving at high velocities and/or in strong gravity. It takes as input physics from microscopic scales and yields as output predictions of bulk, macroscopic motion. By inverting the process—e.g., drawing on astrophysical observations—an understanding of relativistic features can lead to insight into physics on the microscopic scale. Relativistic fluids have been used to model systems as “small” as colliding heavy ions in laboratory experiments, and as large as the Universe itself, with “intermediate” sized objects like neutron stars being considered along the way. The purpose of this review is to discuss the mathematical and theoretical physics underpinnings of the relativistic (multi-) fluid model. We focus on the variational principle approach championed by Brandon Carter and collaborators, in which a crucial element is to distinguish the momenta that are conjugate to the particle number density currents. This approach differs from the “standard” text-book derivation of the equations of motion from the divergence of the stress-energy tensor in that one explicitly obtains the relativistic Euler equation as an “integrability” condition on the relativistic vorticity. We discuss the conservation laws and the equations of motion in detail, and provide a number of (in our opinion) interesting and relevant applications of the general theory. The formalism provides a foundation for complex models, e.g., including electromagnetism, superfluidity and elasticity—all of which are relevant for state of the art neutron-star modelling.

Rotation and deformation of strangeon stars in the Lennard-Jones model

“…But if the shear modulus is actually larger, then we need to consider the corrections due to the elasticity. For the slow rotation approximation, Carter & Quintana (1975) developed Hartle-Thorne approximation and gave the method to construct an elastic rotational solid star based on the elastic perturbation theory in general relativity (Carter 1973;Andersson & Comer 2021). To the first order of Ω, the star remains to be spherical and Quintana (1976) found that the moment of inertia is the same as that of fluid case.…”

Formulating bulk viscosity for neutron star simulations

“…Because the underlying model is required to be causal, and hence based on Cattaneo-type laws, it is natural to set the discussion within the Extended Irreversible Thermodynamics (EIT) framework [22]. We here provide a streamlined discussion, and refer to the monograph [23] and references therein (see also [38] for a recent analysis focused specifically on bulk viscosity).…”

Dynamical tides in superfluid neutron stars

“…From this result it is evident that the two fluids are coupled even in the absence of entrainment. We need to consider the fact that they live in the same gravitational potentialgravity cannot distinguish neutrons from protons/electronsand the equation of state may also lead to 'chemical coupling', arising when the chemical potential of one species depends on the presence of particles of the other kind (Mendell 1991a,b;Lee 1995;Andersson and Comer 2001;Prix and Rieutord 2004;Andersson and Comer 2021).…”

“…Adding important realism to the model, let us now account for entrainment (Prix 2004;Andersson and Comer 2021). This effect introduces an additional (non-dissipative) coupling mechanism to the two-fluid model, an essential aspect if we want to describe the dynamics of the neutron star crust, where Bragg scattering off of the nuclear lattice makes the 'free' neutrons less mobile than one might otherwise have expected (Carter, Chamel and Haensel 2005;Chamel 2005Chamel , 2006.…”