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Relativistic Fluid Dynamics version August 30, 2006 Relativistic fluid dynamics J.W. van Holten NIKHEF Amsterdam NL 1 Ideal relativistic fluids 1.1 Energy-momentum and fluid current An introduction to relativistic hydrodynamics can be found in ref. [1]; a recent review of perfect fluids and their generalizations is [2]. We summarize the results here following the convention that the units of space and time are chosen such that c = 1. The covariant energy-momentum tensor of an isotropic fluid at rest is ε 0 T (0) = , (1) 0 p δij where ε is the proper energy density, and p the hydrostatic pressure. Let Λ(v) represent a Lorentz transformation to a system with velocity v (measured in units of c) relative to the rest frame: 0 0 1 −vj Λ(v) = + γ (2) vivj vivj 0 δij − v2 −vi v2 where 1 γ = √ . (3) 1 − v2 In this frame the energy-momentum tensor takes the form −1 0 1 −vj 2 T (v) = Λ(v) T (0) Λ(˜ v) = p + γ (ε + p) . (4) 0 δij −vi vivj Introducing the velocity 4-vector µ 2 µ ν u = γ (1, v) , u = ηµνu u = −1, (5) the energy-momentum tensor in a frame w.r.t. which the fluid moves with velocity v can be written covariantly as Tµν = pηµν + (ε + p) uµuν. (6) The conservation laws of energy and momentum take the differential form µν µ ν ∂µT = ∂νp + ∂µ ((ε + p) u u ) = 0. (7) 1 The content of these equations becomes more explicit by splitting them in space and time components. The equation with ν = 0 becomes ∂p ∂ ε + p (ε + p) v − − ∇ · = 0, (8) ∂t ∂t 1 − v2 1 − v2 whilst the space components with ν = i become ∂ (ε + p) v (ε + p) vv 0 = ∇ p + i + ∇ · i i ∂t 1 − v2 1 − v2 (9) ∂p ε + p ∂v = ∇ p + v + i + v · ∇v . i i ∂t 1 − v2 ∂t i Here the second line is obtained by substitution of eq. (8) to eliminate the time derivatives of ε and p. The last equation can be rewritten as a relativistic form of the Euler equation: ∂v 1 − v2 ∂p + v · ∇v = − ∇p + v . (10) ∂t ε + p ∂t The fluid itself is described by a fluid-density current jµ; in the rest frame it takes the form jµ = (ρ, 0) , (11) with ρ the fluid density at rest. The definition can extended to a moving fluid, taking the general form µ 0 µ 2 µ ν 2 j = j , j = ρu , j = ηµνj j = −ρ . (12) Thus the density can be defined in an invariant way. The physical require- ment that the total amount of fluid is conserved, is described by the vanishing of the 4-divergence: µ ∂ ρ ρv ∂µj = 0 ⇔ √ + ∇ · √ = 0, (13) ∂t 1 − v2 1 − v2 the Bernouilli equation in relativistic form. 2 1.2 Thermodynamical considerations A complete description of the fluid requires specification of the relation be- tween ε, p and ρ. This is provided by the equation of state. It is often convenient to specify this in the form of expressions for the energy density and pressure in terms of the fluid density: ε = f(ρ), p = g(ρ). (14) As we will see shortly, in situations where the entropy per is constant, we can take the function g(ρ) to be the negative of the legendre transform of the energy density: p = g(ρ) = ρf 0(ρ) − f(ρ) ⇔ ε + p = ρf 0(ρ) (15) The Euler equation can then be written in the form ∂(γρv) 1 ∂p + ∇ · (γρvv) = − ∇p + v . (16) ∂t γf 0(ρ) ∂t If the fluid obeys thermodynamics (strictly speaking, in conditions of thermal equilibrium), we can derive a useful relation for the flow of entropy starting from the condition of energy-momentum conservation (7), rewritten as (ε + p) ∂ p + ∂ jµu = 0. (17) ν µ ρ ν After contraction with uν and using the conservation of the current and equation (5) expressing the fact that the four-velocity is a time-like unit vector, we get ε + p 0 = uµ∂ p − jµ∂ µ µ ρ (18) ε + p 1 ε = uµ ∂ p − ρ ∂ = −jµ p ∂ + ∂ . µ µ ρ µ ρ µ ρ Now the first law of thermodynamics for a system with one component states that dU = T dS − pdV + µdN. (19) 3 We define the specific energy, entropy and volume as the corresponding quan- tity per particle: U S V u = , s = , v = . (20) N N N Then the first law takes the alternative form T dS = NT ds + T sdN = dU + pdV − µdN (21) = N (du + p dv) + (u + pv − µ) dN, and therefore u + pv − T s − µ T ds = du + p dv + dN. (22) N Now for a 1-component fluid the Gibbs-Duhem relation implies for the Gibbs potential: G = U − TS + pV = µN; (23) as a result, the last term in parentheses in eq. (22) vanishes: u + pv − T s − µ = 0 ⇒ T ds = du + pdv. (24) The energy density ε and particle density ρ are related to the specific energy and volume by 1 ε = ρu, ρ = . (25) v Finally it then follows that 1 ε T ds = p d + d , (26) ρ ρ and ε + p f 0(ρ) = = u + pv = T s + µ. (27) ρ 0 Thus we infer that at T = 0 the chemical potential is µT =0 = f (ρ). Equation (18) is now seen to imply that the comoving time-derivative of the specific entropy vanishes: ∂s uµ∂ s = 0 ⇔ + v · ∇s = 0. (28) µ ∂t 4 We observe, that systems with equations of state satisfying (15) indeed have the special property that the specific entropy is constant: 1 ε −ρf 0 + f f 0 f T ds = pd + d = + − dρ = 0. (29) ρ ρ ρ2 ρ ρ2 N.B.: observe, that the specific entropy s is not the same as the entropy density σ: S σ = = ρs. (30) V Clearly, if the specific entropy is constant, then σ dσ s = = = constant. (31) ρ dρ Finally, the Gibbs-Duhem relation in the form G = U − TS + pV = µN ⇒ T dS − V dp + Ndµ = 0, (32) implies dp = σ dT + ρ dµ ⇔ s dT = v dp − dµ. (33) 1.3 Coupling to gravity and action principle In this section we will show, that the basic fluid equations (7) and (13), as well as the equation of state (15) can be derived from an action principle. Moreover, this action can be generalized with almost no effort to include cou- pling to the gravitational field in the contex of general relativity. Therefore we immediately proceed with the general relativistic treatment and define the action Z √ 1 S = d4x −g − R + L , (34) 16πG fluid where µ Lfluid = −j (∂µθ + iz∂¯ µz − iz∂µz¯) − f(ρ). (35) Here θ and (¯z, z) are real and complex scalar potentials respectively, and ρ is considered an composite expression of the metric and current as in (12): 2 µ ν ρ = −gµνj j . (36) 5 Variation of the action w.r.t. the (inverse) metric gives the Einstein equation 1 G = R − g R = −8πG T , (37) µν µν 2 µν µν with the energy momentum tensor the extension of (6): j j T = p g + (ε + p) u u = p g − (ε + p) µ ν , (38) µν µν µ ν µν ρ2 and the energy density and pressure being defined through (14) and (15). Hence the specific entropy (entropy per particle) is a constant in this model by construction. Furthermore, the Einstein equations imply the covariant conservation of the energy-momentum currents: µν DµT = 0. (39) Next, varying the action w.r.t. the current leads to an equation expressing the current in terms of the potentials: f 0 j = f 0u = ∂ θ + iz∂¯ z − iz∂ z,¯ (40) ρ µ µ µ µ µ whilst an extremum of the action w.r.t. variation of the scalar potentials requires µ µ µ Dµj = 0, j ∂µz = j ∂µz¯ = 0. (41) The first equation is the covariant form of the current conservation (13). Ob- serve, that in 4 space-time dimensions a conserved current has 3 indpendent degrees of freedom, which can be identified with the real and complex scalar potentials in expression (40). Therefore in 4-dimensional space-time eq. (40) represents the most general current one can write down. A typical equation of state is of the form ε = f(ρ) = αρ1+η, p = ρf 0 − f = η αρ1+η, (42) which gives a linear relation between ε and p, as expected on dimensional grounds: p = ηε. (43) Note, that by construction this equation of state satisfies the condition (29) 1 ε T ds = p d + d = 0. ρ ρ 6 For example, the standard equation of state of a gas of massless particles in 4-dimensional space-time is 1 p = ε, ⇔ ε = αρ4/3. (44) 3 Similarly, a gas of cold non-relativistic particles of mass m, with p ρm, can in first approximation be described by η = 0: p = 0 ⇔ ε = αρ. (45) In the next approximation, at finite temperature and pressure, we have 3 ε = f(ρ) = ρm + p. (46) 2 With the relations (15) this can be solved for f(ρ) to give: 2κ 3 p ε = f(ρ) = mρ + κρ5/3, p = ρ5/3, κ = 0 , (47) 3 2 5/3 ρ0 where p0 is the pressure at some reference density ρ0. The result for the pressure is well-known from classical thermodynamics. 1.4 Vorticity In non-relativistic hydrodynamics one defines the vorticity as the rotation of the velocity: ω = ∇ × v.
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