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Relativistic Fluid Dynamics Living Reviews in Relativity (2021) 24:3 https://doi.org/10.1007/s41114-021-00031-6 REVIEW ARTICLE Relativistic fluid dynamics: physics for many different scales Nils Andersson1 · Gregory L. Comer2 Received: 1 June 2020 / Accepted: 22 January 2021 © The Author(s) 2021 Abstract The relativistic fluid is a highly successful model used to describe the dynam- ics 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 astrophysi- cal observations—an understanding of relativistic features can lead to insight into physics on the microscopic scale. Relativistic fluids have been used to model sys- tems 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 theo- retical 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 deriva- tion 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 This article is a revised version of https://doi.org/10.12942/lrr-2007-1. Change summary: Major revision, updated and expanded. Change details: This revision represents a significant revision/update of the review. We have expanded the discussion to make it more pedagogical (introducing new sections on thermodynamics and matter equations of state as well as general variational principles), added sections to account for more physics (electromagnetism, elasticity and heat conductivity) and made contact with state of the art numerical relativity simulations and a range of issues (from cosmology to AdS/CFT and field theory models). The discussion of the relevant literature has been expanded by the addition of more than 300 new references. B Nils Andersson [email protected] Gregory L. Comer [email protected] 1 Mathematical Sciences and STAG Research Centre, University of Southampton, Southampton SO17 1BJ, UK 2 Department of Physics, Saint Louis University, St. Louis, MO 63156-0907, USA 0123456789().: V,-vol 123 3 Page 2 of 251 N. Andersson, G. L. Comer 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. Keywords Fluid dynamics · Relativistic hydrodynamics · Relativistic astrophysics · Variational methods · Field theory Contents 1 Setting the stage ........................................... 3 1.1 A brief history of fluids .................................... 5 1.2 Why are fluid models useful? ................................. 7 1.3 Notation and conventions ................................... 10 2 Thermodynamics and equations of state .............................. 11 2.1 Fundamental, or Euler, relation ................................ 11 2.2 Case study: neutron stars ................................... 13 3 Physics in a curved spacetime ................................... 19 3.1 The metric and spacetime curvature ............................. 20 3.2 Parallel transport and the covariant derivative ........................ 22 3.3 The Lie derivative and spacetime symmetries ........................ 27 3.4 Spacetime curvature ..................................... 32 3.5 The Einstein field equations ................................. 33 4 Variational analysis ......................................... 35 4.1 A simple starting point: the point particle .......................... 36 4.2 More general Lagrangians .................................. 38 4.3 Electromagnetism ....................................... 42 4.4 The Einstein field equations ................................. 44 4.5 The stress-energy tensor as obtained from the action principle ............... 46 5 Case study: single fluids ...................................... 47 5.1 General stress decomposition ................................. 47 5.2 “Off-the-shelf” analysis ................................... 49 5.3 Conservation laws ...................................... 54 5.4 A couple of steps towards relative flows ........................... 56 5.5 From microscopic models to the equation of state ...................... 58 6 Variational approach for a single-fluid system ........................... 59 6.1 The action principle ...................................... 60 6.2 Lagrangian perturbations ................................... 66 6.3 Working with the matter space ................................ 67 6.4 A step towards field theory .................................. 69 7 Newtonian limit and Lagrangian perturbations ........................... 71 7.1 The Newtonian limit ..................................... 71 7.2 Local dynamics ........................................ 74 7.3 Newtonian fluid perturbations ................................ 76 7.4 The CFS instability ...................................... 78 7.5 The relativistic problem .................................... 83 8 A step towards multi-fluids ..................................... 84 8.1 The two-constituent, single fluid ............................... 86 8.2 Speed of sound (again) .................................... 88 8.3 Multi-component cosmology ................................. 92 9 The “pull-back” formalism for two fluids ............................. 94 10 Waves in multi-fluid systems .................................... 98 10.1 Two-fluid case ........................................ 98 10.2 The two-stream instability .................................. 101 11 Numerical simulations: fluid dynamics in a live spacetime .................... 107 123 Relativistic fluid dynamics: physics for many different… Page 3 of 251 3 11.1 Spacetime foliation ...................................... 108 11.2 Perfect fluids ......................................... 111 11.3 Conservative to primitive ................................... 115 11.4 The state of the art ...................................... 116 12 Relativistic elasticity ........................................ 117 12.1 The matter space metric ................................... 118 12.2 Elastic variations ....................................... 120 12.3 Lagrangian perturbations of an unstrained medium ..................... 125 13 Superfluidity ............................................ 127 13.1 Bose–Einstein condensates .................................. 128 13.2 Helium: the original two-fluid model ............................. 132 13.3 Relativistic models ...................................... 137 13.4 Vortices and mutual friction ................................. 142 13.5 The Kalb–Ramond variation ................................. 144 13.6 String fluids .......................................... 148 13.7 Vortex dynamics ....................................... 151 14 Perspectives on electromagnetism ................................. 156 14.1 The Lorentz force ....................................... 156 14.2 Maxwell in the fluid frame .................................. 158 14.3 Variational approach for coupled charged fluids ....................... 161 14.4 The foliation equations .................................... 163 14.5 Electron dynamics and Ohm’s law .............................. 166 14.6 Tetrad formulation ...................................... 171 14.7 A brief status report of magnetic field models ........................ 178 15 The problem with heat ....................................... 179 15.1 The “standard” approach ................................... 180 15.2 Case study: neutron star cooling ............................... 182 15.3 The multi-fluid view ..................................... 184 15.4 A linear model and the second sound ............................. 189 16 Modelling dissipation ........................................ 197 16.1 Eckart versus Landau–Lifshitz ................................ 199 16.2 The Israel–Stewart approach ................................. 202 16.3 Application: heavy-ion collisions ............................... 204 16.4 The fluid-gravity correspondence ............................... 207 16.5 Completing the derivative expansion ............................. 210 16.6 Carter’s canonical framework ................................. 211 16.7 Add a bit of chemistry... ................................... 216 16.8 Towards a dissipative action principle ............................ 218 16.9 A reactive/resistive example ................................. 222 16.10Adding dissipative stresses .................................. 226 16.11A few comments ....................................... 229 17 Concluding remarks ......................................... 230 The volume tensor in n-dimensions ................................... 231 The matter space Levi-Civita symbol .................................. 232 References ................................................ 233 1 Setting the stage If one performs
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