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2 Equations of General-Relativistic Hydrodynamics

The general-relativistic hydrodynamics equations consist of the local conservation laws of the stress-energy tensor, μν T (the Bianchi identities) and of the matter current density, μ J (the continuity equation):

μν ∇ μT = 0, (1 )
μ ∇ μJ = 0. (2 )
As usual, ∇ μ stands for the covariant derivative associated with the four-dimensional spacetime metric gμν. The density current is given by μ μ J = ρu, μ u representing the fluid 4-velocity and ρ the proper rest-mass density.

The stress-energy tensor for a nonperfect (unmagnetized) fluid is defined as (see, e.g., [264])

Tμν = ρ(1 + ɛ)uμu ν + (p − ζθ)h μν − 2 ησμν + qμuν + qνuμ, (3 )
where ɛ is the specific energy density of the fluid in its rest frame, p is the pressure, and hμν is the spatial projection tensor hμν = uμu ν + gμν. In addition, η and ζ are the shear and bulk viscosities. The expansion θ, describing the divergence or convergence of the fluid world lines, is defined as θ = ∇ uμ μ. The symmetric, trace-free, spatial shear tensor σ μν is defined by
μν 1- μ αν ν αμ 1- μν σ = 2(∇ αu h + ∇ αu h ) − 3θh , (4 )
and, finally, μ q is the energy flux vector.

In the following, we will neglect nonadiabatic effects, such as viscosity and heat transfer, assuming the stress-energy tensor to be that of a perfect fluid

T μν = ρhu μuν + pgμν, (5 )
where we have introduced the relativistic specific enthalpy, h, defined by
p- h = 1 + ɛ + ρ . (6 )

Introducing an explicit coordinate chart (x0,xi), the previous conservation equations read

-∂--√−-gJ μ = 0, (7 ) ∂xμ -∂-√ --- μν √ --- ν μλ ∂xμ − gT = − − g ΓμλT , (8 )
where the scalar 0 x represents a foliation of the spacetime with hypersurfaces (coordinatized by i x). Additionally, √ --- − g is the volume element associated with the 4-metric, with g = det(gμν), and the objects Γ νμλ are the 4-dimensional Christoffel symbols.

In order to close the system, the equations of motion (1View Equation) and the continuity equation (2View Equation) must be supplemented with an equation of state (EOS) relating some fundamental thermodynamic quantities. In general, the EOS takes the form

p = p(ρ,ɛ). (9 )

The available EOSs have become sophisticated enough to take into account physical and chemical processes such as molecular interactions, quantization, relativistic effects, nuclear physics, etc. Nevertheless, due to their simplicity, the most widely employed EOSs in numerical simulations in astrophysics are the ideal fluid EOS, p = (Γ − 1)ρɛ, where Γ is the adiabatic index, and the polytropic EOS (e.g., to build equilibrium stellar models), p = K ρΓ ≡ K ρ1+1∕N, K being the polytropic constant and N the polytropic index. State-of-the-art, microphysical EOSs that describe the interior of compact stars at nuclear matter densities have also been developed. While they are being increasingly used in the numerical modelling of relativistic stars, the true EOS of neutron-star interiors remains still largely unknown, as reproducing the associated densities and temperatures is not amenable to laboratory experimentation.

In the “test-fluid” approximation, where the fluid self-gravity is neglected, the dynamics of the system is completely governed by Equations (1View Equation) and (2View Equation), together with the EOS (9View Equation). In those situations in which such an approximation does not hold, the previous equations must be solved in conjunction with the Einstein gravitational-field equations,

μν μν G = 8πT , (10 )
which describe the evolution of the geometry in a dynamic spacetime. A detailed description of the various numerical approaches to solving the Einstein equations is beyond the scope of the present article (see, e.g., [217Jump To The Next Citation Point41] for recent reviews). We briefly mention that the Einstein equations, both in vacuum as well as in the presence of matter fields, can be formulated as an initial value (Cauchy) problem, using the 3+1 decomposition of spacetime [25Jump To The Next Citation Point]. More details can be found in, e.g., [431158Jump To The Next Citation Point]. Given a choice of gauge, the Einstein equations in the 3+1 formalism split into evolution equations for the 3-metric γij and the extrinsic curvature Kij (the second fundamental form), and constraint equations (the Hamiltonian and momentum constraints), which must be satisfied on every time slice.

The Arnowitt–Deser–Misner (ADM) 3+1 metric equations have been shown over the years to be intrinsically unstable for long-term numerical simulations, especially for those dealing with black-hole spacetimes. Recently, there have been diverse attempts to reformulate those equations into forms better suited for numerical investigations (see [363Jump To The Next Citation Point39Jump To The Next Citation Point5Jump To The Next Citation Point] and references therein). Among the various approaches proposed, the Baumgarte–Shapiro–Shibata–Nakamura (BSSN) reformulation of the ADM system [363Jump To The Next Citation Point39Jump To The Next Citation Point] is very appropriate for long-term stable numerical work. BSSN makes use of a conformal decomposition of the 3-metric, −4φ &tidle;γij = e γij and the trace-free part of the extrinsic curvature, Aij = Kij − γijK ∕3, with the conformal factor φ chosen to satisfy e4φ = γ1∕3 ≡ det(γij)1∕3. In this formulation, as shown by [39Jump To The Next Citation Point], in addition to the evolution equations for the conformal three-metric &tidle;γ ij and the conformal-traceless extrinsic-curvature variables &tidle; Aij, there are evolution equations for the conformal factor φ, the trace of the extrinsic curvature K and the “conformal connection functions” &tidle;Γ i = &tidle;γjk&tidle;Γ ijk. Indeed, BSSN (or slight modifications thereof) is currently the standard 3+1 formulation for most numerical relativity groups [5158Jump To The Next Citation Point]. Long-term stable applications include strongly-gravitating systems such as neutron stars (both, isolated and in binary systems) and single and binary black holes. Such binary–black-hole evolutions, possibly the grandest challenge of numerical relativity ever, since the beginning of the field, have only been possible in the last few years (see, e.g., [323Jump To The Next Citation Point] and references therein).

Alternatively, a characteristic initial-value–problem formulation of the Einstein equations was developed in the 1960s by Bondi, van der Burg, and Metzner [59], and by Sachs [344]. This approach has gradually advanced to a state where long-term stable evolutions of caustic-free spacetimes in multiple dimensions are possible, even including matter fields (see [217] and references therein). A comprehensive review of the characteristic formulation is presented in a Living Reviews article by Winicour [419Jump To The Next Citation Point]. Examples of this formulation in general-relativistic hydrodynamics are discussed in various sections of the current article.

Traditionally, most of the approaches for numerical integrations of the general-relativistic hydrodynamics equations have adopted spacelike foliations of the spacetime, within the 3+1 formulation. More recently, however, covariant forms of these equations, well suited for advanced numerical methods, have also been developed. This is reviewed next (Section 2.1) in a chronological way.

 2.1 Spacelike (3+1) approaches
  2.1.1 1+1 Lagrangian formulation (May and White)
  2.1.2 3+1 Eulerian formulation (Wilson)
  2.1.3 3+1 conservative Eulerian formulation (Ibáñez and colleagues)
 2.2 Covariant approaches
  2.2.1 Eulderink and Mellema
  2.2.2 Papadopoulos and Font
 2.3 Going further: beyond ideal hydrodynamics

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