## PHY 442: Relativistic Hydrodynamics

3 credits | Prerequisites: PHY 223

### Course rationale

Fluid Mechanics is an effective description of the matter which flows. The scope of fluid mechanics is diverse ranging from Quark-Gluon Plasma to Astrophysics. For these applications, ideas from relativity are inseparable. Therefore, the formulae of non-relativistic fluid mechanics needed to be appropriately adjusted. There are two major paradigms, 1: Israel-Stewart formalism, 2: Carter-Lichnerowicz approach. This course will focus on the basics and concepts of these formalisms

### Course content

Essential Differential Geometry: manifold, vectors, dual vectors, tensors, covariant differentiation, geodesic, Lie differentiation, Killing vector, curvature, geodesic deviation, Fermi normal coordinates, differential forms, tetrad formalism, Cartan’s structure equation, Einstein equations; Geometry of Hypersurfaces: defining equations, normal vector, induced metric, integration on hypersurface, Gauss-Stokes theorem, tangent tensor fields, intrinsic covariant derivatives, extrinsic curvature, Gauss-Codazzi equations, initial value problem, junction conditions for thin shells; Essential Non-relativistic Fluid Mechanics: continuity equation, Euler’s equation, hydrostatistics, Bernoulli’s equation, energy-momentum flux, conservation of circulation, potential flow, incompressible fluids, drag force, waves, viscous fluids, Navier-Stokes equation, energy dissipation, waves, vortex motion; Relativistic Perfect Fluid: kinematic properties of fluids, kinematic shear, expansion and vorticity, evolution laws, mass current, energy-momentum, perfect fluid and symmetries, the Newtonian limit of the hydrodynamic equation, relativistic Bernoulli theorem, relativistic vorticity, relativistic irrotational flows, relativistic Kelvin-Helmholtz theorem, isoentropic flows, velocity-potential approach, variational principle, perfect multifluids; Relativistic Non-perfect Fluids: four-velocity of a non-perfect fluid, energy-momentum tensor of non-perfect fluid, the general form of the energy and momentum equation, the equilibrium state, classical irreversible thermodynamics, constitutive equation, the Newtonian limit: Navier-Stokes and heat conduction, causality, parabolic and hyperbolic equations, non-causality of classical irreversible thermodynamics, IsraelStewart formalism, characterstic speed of the Israel-Stewart formulation, divergences; Carter-Lichnerowicz Formalism: differential form, exterior derivative, thermodynamics, projection, Carter-Lichnerowicz equation, the canonical form for a simple fluid, Newtonian limit-Crocco equation, conservation theorem, irrotational flow; Relativistic Hydrodynamics for Spherical Stars: relativistic equation of state, two important space-time: Schwarzschild and Kerr metric, Tolman-Oppenheimer-Volkov equation, the collapse of stars, particle motion, stability, Buchdahl’s theorem, uniformly rotating stars.

### Course objectives

1. Understand the concepts of relativity and fluid mechanics.
2. Familiarize the reconciliation of general relativity and fluid dynamics.
3. Understand different formulations of relativistic fluid mechanics.
4. Use the tools of differential form to express different quantities.
5. Appy to relativistic hydrodynamics to Astrophysics and Quark-Gluon Plasma

### References

1. Relativistic Hydrodynamics by Luciano Rezzola, and Olindo Zanotti
2. Relativistic Fluids and Magneto-fluids: With Applications in Astrophysics and Plasma Physics by A. M. Anile
3. Relativistic Fluid Dynamics In and Out of Equilibrium: And Applications to Relativistic Nuclear Collisions by Paul Romatschke, and Ulrike Romatschke
4. Fundamentals of Astrophysical Fluid Dynamics: Hydrodynamics, Magnetohydrodynamics, and Radiation Hydrodynamics by Shoji Kato, and Jun Fukue
5. An introduction to relativistic hydrodynamics by Eric Gourgoulhon.