2.1 Spacelike (3+1) approachesNumerical Hydrodynamics in General Relativity1 Introduction

2 Equations of General Relativistic Hydrodynamics 

The general relativistic hydrodynamic equations consist of the local conservation laws of the stress-energy tensor, tex2html_wrap_inline3563 (the Bianchi identities) and of the matter current density, tex2html_wrap_inline3565 (the continuity equation):



As usual tex2html_wrap_inline3567 stands for the covariant derivative associated with the four-dimensional spacetime metric tex2html_wrap_inline3569 . The density current is given by tex2html_wrap_inline3571, tex2html_wrap_inline3573 representing the fluid 4-velocity and tex2html_wrap_inline3575 the rest-mass density in a locally inertial reference frame.

The stress-energy tensor for a non-perfect fluid is defined as


where tex2html_wrap_inline3577 is the rest frame specific internal energy density of the fluid, p is the pressure and tex2html_wrap_inline3581 is the spatial projection tensor tex2html_wrap_inline3583 . In addition, tex2html_wrap_inline3585 and tex2html_wrap_inline3587 are the shear and bulk viscosities. The expansion tex2html_wrap_inline3589, describing the divergence or convergence of the fluid world lines is defined as tex2html_wrap_inline3591 . The symmetric, trace-free, and spatial shear tensor tex2html_wrap_inline3593, is defined by


and, finally, tex2html_wrap_inline3595 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 tex2html_wrap_inline3599 the previous conservation equations read



where the scalar tex2html_wrap_inline3601 represents a foliation of the spacetime with hypersurfaces (coordinatized by tex2html_wrap_inline3603). Additionally, tex2html_wrap_inline3605 is the volume element associated with the 4-metric, with tex2html_wrap_inline3607, and tex2html_wrap_inline3609 are the 4-dimensional Christoffel symbols.

The system formed by the equations of motion (1Popup Equation) and the continuity equation (2Popup Equation) 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. (1Popup Equation) and (2Popup Equation), together with the EOS (9Popup Equation). 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 [14Jump To The Next Citation Point In The Article] can be found in, e.g., [239]. Given a choice of gauge, the Einstein equations in the 3+1 formalism [14Jump To The Next Citation Point In The Article] split into evolution equations for the 3-metric tex2html_wrap_inline3611 and the extrinsic curvature tex2html_wrap_inline3613, 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 [40], and Sachs [187]. A recent review of the characteristic formulation is presented in a Living Reviews article by Winicour [232Jump To The Next Citation Point In The 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.

2.1 Spacelike (3+1) approachesNumerical Hydrodynamics in General Relativity1 Introduction

image Numerical Hydrodynamics in General Relativity
José A. Font
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