Perfect Fluids  

In General > s.a. gas.
* Idea: A fluid with isotropic pressure, no viscosity and (in general relativity) no heat conduction; It is described by a four-velocity ua, number density ρ, energy density μ, pressure p, specific entropy s, temperature T, relativistic enthalpy h; Of the scalar quantities, only two are independent, to define the rest use the equation of state and the first law of thermodynamics; If μ = μ(ρ, s), then

h = ∂μ / ∂ρ = (μ + p)/ρ ,      T = (∂μ / ∂s) / ρ ,       and       p = ρ hμ .

* Stress-energy tensor: Of the form

Tab = (μ + p) ua ub + p gab ,

derivable from the Lagrangian L = − μ, where the variations are subject to constraints; The field equations are stress-energy conservation, ∇a Tab = 0, together with either the continuity equation ∇a(ρ ua) = 0, or uaa s = 0 [@ Coll et al a1902, hydrodynamic approach].
* 3+1 Equations: The conservation of stress-energy, ∇a Tab = 0, gives, projected parallel and perpendicular to ua, respectively,

a(μ ua) = − pa ua       and       (μ + p) uaa vb = − qaba p

(conservation of mass-energy and momentum, respectively); The Newtonian limit of the second equation is the Euler equation.

Types, Couplings and Applications > s.a. relativistic cosmological models and matter.
* Isentropic: One with an equation of state of the form p = p(μ) only; In this case, one can introduce the elastic potential (or internal energy) ε and the density ρ, with μ = ρ (1 + ε); For example, the ideal gas.
@ Null dust: Kijowski et al PRD(90); Bičák & Kuchař PRD(97)gq [canonical general relativity].
@ Other types: Quevedo & Sussman CQG(95)gq/94 [non-isentropic, thermodynamics]; Jackiw et al JPA(04)hp [with internal symmetry and supersymmetry]; Majd & Momeni IJMPE(11)-a0903 [p = −ρ, statistical derivation of density of states]; Jacak & Steinberg PT(10)may [quarks and gluons in heavy-ion collisions]; > s.a. Chaplygin Gas; Logotropic Fluid; Phantom Field.
> In general relativity: see canonical general relativity with matter; relativistic cosmology and cosmological models, in particular FLRW spacetimes; 3-dimensional and numerical relativity; solutions, solutions with symmetries, spherically symmetric, and generation methods.

@ General: Thorne in(67); Misner in(68); in Hawking & Ellis 73, 69ff; in Misner et al 73; Smarr et al in(80); Anile 89.
@ Variational formulation: Schutz & Sorkin AP(77); Kopczyński AP(90); Dimitrov gq/99, gq/99; Elze et al JPG(99)hp; Marsden et al JGP(01) [general continuum mechanics]; Ootsuka et al CQG(16)-a1605 [relativistic].
@ Lagrangian, Hamiltonian: Bao et al CMP(85); Bombelli & Torrence CQG(90); Brown CQG(93), AP(96)gq/94; Langlois in(94); Peitz & Appl MNRAS(98)gq/97 [3+1]; Morrison RMP(98); Antoniou & Pronko TMP(04)ht/01 [and Clebsch variables]; Manoff in(01)gq/02, gq/03 [Lagrangian]; Bhat mp/04 [compressible]; Kolev DCDSA(07)-a0711 [Poisson brackets]; Roberts a0910 [Clebsch potential approach]; Faraoni PRD(09)-a0912 [inequivalence of Lagrangians with non-minimal gravity coupling]; Minazzoli & Harko PRD(12)-a1209 [barotropic fluid, Lagrangian]; Frisch & Villone EPJH(14)-a1402 [3D ideal incompressible flow, Cauchy's formulation]; Wongjun PRD(17)-a1602 [Lagrangian]; Minguzzi a1606 [action]; Datta a1807 [and analog spacetime]; Mendoza & Silva IJGMP-a2011.
@ Covariant: Bruni & Sopuerta CQG(03)gq [long-wavelength].
@ Numerical implementation: Papadopoulos & Font PRD(00)gq/99 [spherical]; Baumgarte et al AIP(99)gq [relativistic, 3D].
@ Relativistic: Schutz PRD(71) [Hamiltonian]; Calzetta CQG(98)gq/97 [fluctuating]; Walton ap/05 [symmetric hyperbolic Euler equations]; Gourgoulhon EAS(06)gq-ln; Borshch & Zhdanov Sigma(07)-a0709-proc [exact non-stationary non-homogeneous flows]; Banerjee et al EPJC(15)-a1409 [Hamiltonian, in equal-time and light-cone coordinate systems]; > s.a. Geometrization.
@ Quantization: Roberts MPAG(98)gq; Endlich et al JHEP(11)-a1011 [canonical]; Gripaios & Sutherland PRL(15)-a1406 [as a low-energy, effective field theory].
@ Related topics: Zloshchastiev APPB(99)gq/98 [models for Lorentzian geometry]; Ghosh ht/01 [as gauge theory], ht/01 [conserved quantities]; Verozub IJMPD(08)gq/07 [particles as moving along geodesics]; Mannheim et al GRG(10) [on perfect fluids as general matter models].
> Related topics: see viscosity; Weyssenhoff Fluid [particles with intrinsic spin].

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