As usual, stands for the covariant derivative associated with the four-dimensional spacetime metric . The density current is given by , representing the fluid 4-velocity and the rest-mass density in a locally inertial reference frame.
The stress-energy tensor for a non-perfect fluid is defined as
where is the specific energy density of the fluid in its rest frame, p is the pressure, and is the spatial projection tensor . In addition, and are the shear and bulk viscosities. The expansion , describing the divergence or convergence of the fluid world lines, is defined as . The symmetric, trace-free, spatial shear tensor is defined by
and, finally, is the energy flux vector.
In the following we will neglect non-adiabatic effects, such as viscosity or heat transfer, assuming the stress-energy tensor to be that of a perfect fluid
where we have introduced the relativistic specific enthalpy h defined by
Introducing an explicit coordinate chart , the previous conservation equations read
where the scalar represents a foliation of the spacetime with hypersurfaces (coordinatized by ). Additionally, is the volume element associated with the 4-metric, with , and the are the 4-dimensional Christoffel symbols.
In order to close the system, the equations of motion (1) and the continuity equation (2) must be supplemented with an equation of state (EOS) relating some fundamental thermodynamical quantities. In general, the EOS takes the form
Due to their simplicity, the most widely employed EOSs in numerical simulations are the ideal fluid EOS, , where is the adiabatic index, and the polytropic EOS (e.g., to build equilibrium stellar models), , K being the polytropic constant and N the polytropic index.
In the ``test fluid'' approximation, where the fluid self-gravity is neglected, the dynamics of the system are completely governed by Equations (1) and (2), together with the EOS (9). In those situations where such an approximation does not hold, the previous equations must be solved in conjunction with the Einstein gravitational field equations,
which describe the evolution of the geometry in a dynamical spacetime. A detailed description of the various numerical approaches to solve the Einstein equations is beyond the scope of the present article (see, e.g., Lehner  for a recent review). We briefly mention that the Einstein equations, in the presence of matter fields, can be formulated as an initial value (Cauchy) problem, using the so-called 3+1 decomposition of spacetime . More details can be found in, e.g., . Given a choice of gauge, the Einstein equations in the 3+1 formalism  split into evolution equations for the 3-metric and the extrinsic curvature (the second fundamental form), and constraint equations (the Hamiltonian and momentum constraints), which must be satisfied on every time slice. Long-term stable evolutions of the Einstein equations have recently been accomplished using various reformulations of the original 3+1 system (see, e.g., [25, 258, 4, 89] for simulations involving matter sources, and  and references therein for vacuum black-hole evolutions). Alternatively, a characteristic initial value problem formulation of the Einstein equations was developed in the 1960s by Bondi, van der Burg, and Metzner , and Sachs . This approach has gradually advanced to a state where long-term stable evolutions of caustic-free spacetimes in multi-dimensions are possible, even including matter fields (see  and references therein). A recent review of the characteristic formulation is presented in a Living Reviews article by Winicour . Examples of this formulation in general relativistic hydrodynamics are discussed in various sections of the present article.
Traditionally, most of the approaches for numerical integrations of the general relativistic hydrodynamic equations have adopted spacelike foliations of the spacetime, within the 3+1 formulation. Recently, however, covariant forms of these equations, well suited for advanced numerical methods, have also been developed. This is reviewed next in a chronological way.
|Numerical Hydrodynamics in General Relativity
José A. Font
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