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12 The Newtonian Limit and the Euler Equations

One reason relativistic fluids are needed is that they can be used to model neutron stars. However, even though neutron stars are clearly general relativistic objects, one often starts with good old Newtonian physics when considering new applications. The main reason is that effects (such as acoustic modes of oscillation) which are primarily due to the fluids that make up the star can often be understood qualitatively from Newtonian calculations. There are also certain regimes in a neutron star where the Newtonian limit is sufficient quantitatively (such as the outer layers).

There has been much progress recently in the analysis of Newtonian multiple fluid systems. Prix [91Jump To The Next Citation Point] has developed an action-based formalism, analogous to what has been used here. Carter and Chamel [22Jump To The Next Citation Point23Jump To The Next Citation Point24Jump To The Next Citation Point] have done the same except that they use a fully spacetime covariant formalism. We will be somewhat less ambitious (for example, as in [5]) by extracting the Newtonian equations as the non-relativistic limit of the fully relativistic equations. Given the results of Prix as well as of Carter and Chamel we can think of this exercise as a consistency check of our equations.

The Newtonian limit consists of writing the general relativistic field equations to order c0 where c is the speed of light. The Newtonian equations are obtained, strictly speaking, in the limit where c → ∞. To order c0 the metric becomes

2 2( 2Φ) 2 i j ( 0) ds = − c 1 + c2- dt + gij dx dx + 𝒪 c , (203 )
where the i x are Cartesian coordinates and gij is the metric of Equation (22View Equation). The gravitational potential, denoted Φ, is assumed to be small in the sense that
Φ − 1 ≪ -2 ≤ 0. (204 ) c
We will be somewhat lax by simply writing the general relativistic equations to 𝒪 (c0), and thereby avoiding some of the additional complexities that occur because the spacetime metric clearly becomes singular as c → ∞. For the reader interested in a rigorous construction of the Newtonian equations, we recommend the series of papers by Carter and Chamel [222324].

If τ is the proper time as measured along a fluid element worldline, then the curve it traces out can be written

μ i x (τ) = {t(τ),x (τ)}. (205 )
Recall that its four-velocity is given by (cf. Equation (6View Equation))
μ uμ = dx--, (206 ) dτ
but because the speed of light has been restored, uμu μ = − c2. The proper time is now given by
( 2Φ g vivj) − ds2 = c2dτ2 = c2 1 + -2-− -ij2--- dt2, (207 ) c c
where vi = dxi∕dt is the Newtonian three-velocity of the fluid. It is considered to be small in the sense that
|vi| ----≪ 1. (208 ) c
Hence, to the correct order the four-velocity components are
t Φ- v2- i i u = 1 − c2 + 2c2, u = v, (209 )
where 2 i j v = gijv v. Note that the particle number current becomes
nμ = n (uμ∕c) . (210 )
The factor of c appears because we define μ 2 n μn = − n, regardless of whether c = 1 or the meter/second value.

In order to write the single fluid Euler equations, keeping terms to the required order, it is necessary to explicitly break up the master function into its mass, kinetic, and “potential” energy ℰ parts, i.e. to write Λ as

( ) Λ = − mnc2 − 1-mg ninj∕n − ℰ (n) + 𝒪 c0 , (211 ) 2 ij
where m is the particle mass. The internal energy is small compared to the mass, meaning
--ℰ--- 0 ≤ mnc2 ≪ 1. (212 )
With this choice, a variation of Λ that leaves the metric fixed yields
( 2 ) dΛ = − mc + μ dn, (213 )
where
∂ℰ μ = --. (214 ) ∂n
The n ℬ coefficient reduces to
μ ℬn = mc2 + --. (215 ) n
In terms of these variables, the pressure p is seen to be
p = − ℰ + μn. (216 )
The Newtonian limit of the general relativistic fluid equations, in a general coordinate basis, is then
∂n ( ) 0 = ---+ ∇i nvi (217 ) ∂t
and
( ) ∂--i j i ij μ- 0 = ∂tv + v ∇jv + g ∇j Φ + m , (218 )
where ∇i is the Euclidean space covariant derivative. Of course, the gravitational potential Φ is determined by a Poisson equation with mn as source.

A Newtonian two-fluid system can be obtained in a similar fashion. As discussed in Section 10, the main difference is that we need two sets of worldlines, describable, say, by curves x μ(τx) x where τx is the proper time along a constituent’s worldline. Of course, entrainment also comes into play. Its presence implies that the relative flow of the fluids is required to specify the local thermodynamic state of the system, and that the momentum of a given fluid is not simply proportional to that fluid’s flux. This is the situation for superfluid He4 [94Jump To The Next Citation Point110Jump To The Next Citation Point], where the entropy can flow independently of the superfluid Helium atoms. Superfluid He3 can also be included in the mixture, in which case there will be a relative flow of the 3 He isotope with respect to 4 He, and relative flows of each with respect to the entropy [113].

Let us consider a two-fluid model like a mixture of He4 and He3, or neutrons and protons in a neutron star. We will denote the two fluids as fluids a and b. The magnitude squared of the difference of three-velocities

wyxi = vyi − vxi (219 )
is denoted 2 w so that the equation of state takes the form 2 ℰ = ℰ (na,nb,w ). Hence,
dℰ = μadna + μb dnb + αdw2, (220 )
where
∂ℰ || ∂ ℰ || ∂ℰ || μa = ---|| , μb = ----|| , α = ---2|| . (221 ) ∂na|nb,w2 ∂nb |na,w2 ∂w |na,nb
The α coefficient reflects the effect of entrainment on the equation of state. Similarly, entrainment causes the fluid momenta to be modified to
pxi- i α-- i mx = vx + 2ρ w yx. (222 ) x

The number density of each fluid obeys a continuity equation:

∂nx-+ ∇ (n vj) = 0. (223 ) ∂t j x x
Each fluid is also seen to satisfy an Euler-type equation, which ensures the conservation of total momentum. This equation can be written
( ) -∂-+ vjx∇j [vxi + ɛxwyix] + ∇i (Φ + ˜μx ) + ɛxwyxj ∇ivjx = 0, (224 ) ∂t
where
μx ˜μx = mx-, (225 )
i.e. ˜μx is the relevant chemical potential per unit mass, and the entrainment is included via the coefficients
ɛx = 2ρxα. (226 )
For a detailed discussion of these equations, see [916Jump To The Next Citation Point].


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