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_inline4676 (the Bianchi identities) and of the matter current density tex2html_wrap_inline4678 (the continuity equation):



As usual, tex2html_wrap_inline4680 stands for the covariant derivative associated with the four-dimensional spacetime metric tex2html_wrap_inline4682 . The density current is given by tex2html_wrap_inline4684, tex2html_wrap_inline4686 representing the fluid 4-velocity and tex2html_wrap_inline4640 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_inline4690 is the specific energy density of the fluid in its rest frame, p is the pressure, and tex2html_wrap_inline4694 is the spatial projection tensor tex2html_wrap_inline4696 . In addition, tex2html_wrap_inline4698 and tex2html_wrap_inline4700 are the shear and bulk viscosities. The expansion tex2html_wrap_inline4702, describing the divergence or convergence of the fluid world lines, is defined as tex2html_wrap_inline4704 . The symmetric, trace-free, spatial shear tensor tex2html_wrap_inline4706 is defined by


and, finally, tex2html_wrap_inline4708 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_inline4712, the previous conservation equations read



where the scalar tex2html_wrap_inline4714 represents a foliation of the spacetime with hypersurfaces (coordinatized by tex2html_wrap_inline4716). Additionally, tex2html_wrap_inline4718 is the volume element associated with the 4-metric, with tex2html_wrap_inline4720, and the tex2html_wrap_inline4722 are the 4-dimensional Christoffel symbols.

In order to close the system, 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


Due to their simplicity, the most widely employed EOSs in numerical simulations are the ideal fluid EOS, tex2html_wrap_inline4724, where tex2html_wrap_inline4726 is the adiabatic index, and the polytropic EOS (e.g., to build equilibrium stellar models), tex2html_wrap_inline4728, 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 (1Popup Equation) and (2Popup Equation), together with the EOS (9Popup Equation). 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 [151Jump To The Next Citation Point In The Article] 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 [15Jump To The Next Citation Point In The Article]. More details can be found in, e.g., [315]. Given a choice of gauge, the Einstein equations in the 3+1 formalism [15Jump To The Next Citation Point In The Article] split into evolution equations for the 3-metric tex2html_wrap_inline4734 and the extrinsic curvature tex2html_wrap_inline4736 (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., [25Jump To The Next Citation Point In The Article, 258Jump To The Next Citation Point In The Article, 4Jump To The Next Citation Point In The Article, 89Jump To The Next Citation Point In The Article] for simulations involving matter sources, and [7Jump To The Next Citation Point In The Article] 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 [45], and Sachs [247]. 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 [151] and references therein). A recent review of the characteristic formulation is presented in a Living Reviews article by Winicour [305Jump To The Next Citation Point In The Article]. 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.

2.1 Spacelike 3+1 approachesNumerical Hydrodynamics in General Relativity1 Introduction

image Numerical Hydrodynamics in General Relativity
José A. Font
© Max-Planck-Gesellschaft. ISSN 1433-8351
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