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 rest frame specific internal energy density of the fluid, 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, and 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 are the 4-dimensional Christoffel symbols.
The system formed by 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
In the `test-fluid' approximation, where the fluid self-gravity is neglected, the dynamics of the system is completely governed by Eqs. (1) and (2), together with the EOS (9). In those situations where such approximation does not hold, the previous equations must be solved in conjunction with the Einstein gravitational field equations,
which describe the evolution of a dynamical spacetime. The formulation of the Einstein equations as an initial value (Cauchy) problem, in the presence of matter fields, adopting the so-called 3+1 decomposition of the spacetime  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 , and constraint equations, the Hamiltonian and momentum constraints, that must be satisfied at every time slice. Alternatively, a characteristic initial value problem formulation of the Einstein equations was developed by Bondi, van der Burg and Metzner , and Sachs . A recent review of the characteristic formulation is presented in a Living Reviews article by Winicour .
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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