B GR Hydrodynamical and MHD equations

In what follows, we will adopt the stress energy tensor of an ideal relativistic fluid,

μν μ ν μν T = ρhu u + g P , (60 )
where ρ, P, and uμ are the rest mass density, pressure, and fluid 4-velocity, respectively, and
h = 1 + 𝜖 + P ∕ρ (61 )
is the relativistic specific enthalpy, with 𝜖 the specific internal energy of the fluid.

The equations of ideal GR hydrodynamics [186] may be derived from the local GR conservation laws of mass and energy-momentum:

∇ Jμ = 0, ∇ Tμν = 0, (62 ) μ μ
where ∇ μ denotes the covariant derivative with respect to the 4-metric, and Jμ = ρuμ is the mass current.

The 3-velocity i v can be calculated in the form

ui βi vi = ---+ --, (63 ) W α
where
W ≡ αu0 = (1 − vivi)−1∕2 (64 )
is the Lorentz factor. The contravariant 4-velocity is then given by:
( ) 0 W i i βi u = ---, u = W v − --- , (65 ) α α
and the covariant 4-velocity is:
u0 = W (viβi − α), ui = W vi. (66 )

To cast the equations of GR hydrodynamics as a first-order hyperbolic flux-conservative system for the conserved variables D, Si, and τ, defined in terms of the primitive variables ρ,𝜖,vi, we define

D = √ γ-ρW, (67 ) √ -- Si = γ ρhW 2vi, (68 ) τ = √ γ-(ρhW 2 − P) − D, (69 )
where γ is the determinant of γij. The evolution system then becomes
∂U ∂Fi ----+ ---i = S, (70 ) ∂t ∂x
with
U = [D, Sj, τ], i [ i i i i i] F = α[ Dv&tidle;,Sj(&tidle;v + δjP,τ &tidle;v + P) v , μν ∂g νj λ S = α 0, T ---μ-− Γμνgλj , ( ∂x ) ] μ0 ∂ lnα- μν 0 α T ∂x μ − T Γ μν . (71 )
Here, &tidle;vi = vi − βi∕α and Γ λμν are the 4-Christoffel symbols.

Magnetic fields may be included in the formalism, in the ideal MHD limit under which we assume infinite conductivity, by adding three new evolution equations and modifying those above to include magnetic stress-energy contributions of the form

( 1 ) T EμMν = b2 uμu ν + -gμν − bμbν, (72 ) 2
where the magnetic field seen by a comoving observer, bμ is defined in terms of the dual Faraday tensor ∗ νμ F by the condition
bμ = ∗ F νμu , (73 ) ν
where b2 = bμbμ represents twice the magnetic pressure. With magnetic terms included, we may rewrite the stress-energy tensor in a familiar form by introducing magnetically modified pressure and enthalpy contributions:
2 2 μν ∗ μ ν ∗ μν ∗ P-+-b-- ∗ b- T = ρh u u + P g ; h ≡ 1 + 𝜖 + ρ , P = P + 2 (74 )
and redefine the conserved momentum and energy variables i S and τ accordingly:
Si = √ γρh ∗W 2vi − αb0bi, (75 ) √ --( ∗ 2 ∗ 0 2) τ = γ ρh W − P − (αb ) − D. (76 )
Defining the (primitive) magnetic field 3-vector as
i ∗ μi ∗ 0i B = − F nμ = α F (77 )
and the conserved variable ℬi = √ γBi, which are related to the comoving magnetic field 4-vector bμ through the relations
W Bkv b0 = ------k, (78 ) α ( ) i Bi- k i βi- b = W + W (B vk) v − α , (79 ) bi = Bi-+ W (Bkvk)vi, (80 ) W 2 BiBi- i 2 b = W 2 + (B vi), (81 )
we may rewrite the conservative evolution scheme in the form
U = [D, S ,τ,ℬk ], j⌊ ⌋ D &tidle;vi, i | Sj&tidle;vi + δijP∗ − bjℬi∕W, | F = α × |⌈ τ&tidle;vi + P ∗vi − αb0Bi∕W, |⌉, (82 ) k i i k ⌊ ℬ &tidle;v − ℬ &tidle;v ⌋ 0, | μν (∂gνj λ ) | S = α × || T ( ∂xμ − Γμνgλj , ) || , (83 ) ⌈ α T μ0∂∂lnxμα− TμνΓ 0μν ,⌉ ⃗0
where the magnetic field evolution equation is just the relativistic version of the induction equation. An external mechanism to enforce the divergence-free nature of the magnetic field, ∂iℬi must also be applied.


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