## 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

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

### References

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