The line element is written as
where is the 3-metric induced on each spacelike slice.
For a spherically symmetric spacetime the line element can be written as
m being a radial (Lagrangian) coordinate, indicating the total rest-mass enclosed inside the circumference .
The co-moving character of the coordinates leads, for a perfect fluid, to a stress-energy tensor of the form
In these coordinates the local conservation equation for the baryonic mass, Eq. (2), can be easily integrated:
The gravitational field equations, Eq. (10), and the equations of motion, Eq. (1), reduce to the following quasi-linear system of partial differential equations (see also ):
with the definitions
represents the total mass interior to radius m at time t . The final system is closed with an EOS of the form (9).
Codes based on the original formulation of May and White and on later versions (e.g. ) have been used in many non-linear simulations of supernova and neutron star collapse (see, e.g., [141, 214] and references therein), as well as in perturbative computations of spherically symmetric gravitational collapse employing the linearized Einstein theory [191, 193, 192]. In Section 4.1.1 some of these simulations are reviewed in detail. The Lagrangian character of May and White's code, together with other theoretical considerations concerning the particular coordinate gauge, has prevented its extension to multidimensional calculations. However, for one-dimensional problems, the Lagrangian approach adopted by May and White has considerable advantages with respect to an Eulerian approach with spatially fixed coordinates, mainly the lack of numerical diffusion.
the equations of motion in Wilson's formulation [226, 227] are:
with the ``transport velocity'' given by . Notice that the momentum density equation, Eq. (26), is only solved for the three spatial components, , and is obtained through the 4-velocity normalization condition .
A direct inspection of the system shows that the equations are written as a coupled set of advection equations. In doing so, the terms containing derivatives (in space or time) of the pressure are treated as source terms. This approach, hence, sidesteps an important guideline for the formulation of non-linear hyperbolic systems of equations, namely the preservation of their conservation form . This is a necessary feature to guarantee correct evolution in regions of sharp entropy generation (i.e. shocks). As a consequence, some amount of numerical dissipation must be used to stabilize the solution across discontinuities. The first attempt to solve the equations of general relativistic hydrodynamics in the original Wilson scheme  used a combination of finite difference upwind techniques with artificial viscosity terms. Such terms extended the classic treatment of shocks introduced by von Neumann and Richtmyer  into the relativistic regime (see Section 3.1.1).
Wilson's formulation has been widely used in hydrodynamical codes developed by a variety of research groups. Many different astrophysical scenarios were first investigated with these codes, including axisymmetric stellar core-collapse [149, 148, 152, 18, 211, 177, 65], accretion onto compact objects [94, 175], numerical cosmology [47, 48, 12] and, more recently, the coalescence and merger of neutron star binaries [230, 231]. This formalism has also been extensively employed, in the special relativistic limit, in numerical studies of heavy-ion collisions [229, 135]. We note that in these investigations, the original formulation of the hydrodynamic equations was slightly modified by re-defining the dynamical variables, Eq. (24), with the addition of a multiplicative factor and the introduction of the Lorentz factor, (the ``relativistic gamma''):
As mentioned before, the description of the evolution of self-gravitating matter fields in general relativity requires a joint integration of the hydrodynamic equations and the gravitational field equations (the Einstein equations). Using Wilson's formulation for the fluid dynamics, this was first considered in , building on a vacuum numerical relativity code specifically developed to investigate the head-on collision of two black holes . The resulting code was axially symmetric and aimed to integrate the coupled set of equations in the context of stellar core collapse .
More recently, Wilson's formulation has also been applied to the numerical study of the coalescence of binary neutron stars in general relativity [230, 231] (see Section 4.3). An approximation scheme for the gravitational field has been adopted in these studies, by imposing the simplifying condition that the three-geometry (the three metric) is conformally flat. The line element then reads
The curvature of the three metric is then described by a position dependent conformal factor times a flat-space Kronecker delta. Therefore, in this approximation scheme all radiation degrees of freedom are thrown away, and the field equations reduce to a set of five Poisson-like elliptic equations in flat spacetime for the lapse, the shift vector and the conformal factor. While in spherical symmetry this approximation is identical to Einstein's theory, in more general situations it has the same accuracy as the first post-Newtonian approximation .
Wilson's formulation showed some limitations in handling situations involving ultrarelativistic flows, as first pointed out by Centrella and Wilson . Norman and Winkler  performed a comprehensive numerical study of such formulation by means of special relativistic hydrodynamical simulations. Fig. 1 reproduces a plot from  in which the relative error of the density compression ratio in the relativistic shock reflection problem - the heating of a cold gas which impacts at relativistic speeds with a solid wall and bounces back - is displayed as a function of the Lorentz factor W of the incoming gas. The source of the data is . This figure shows that for Lorentz factors of about 2 (), the threshold of the ultrarelativistic limit, the relative errors are between 5% and 7% (depending on the adiabatic exponent of the gas), showing a linear growth with W .
Norman and Winkler  concluded that those large errors were mainly due to the way in which the artificial viscosity terms are included in the numerical scheme in Wilson's formulation. These terms, called collectively Q (see Section 3.1.1), are only added to the pressure terms in some cases, namely at the pressure gradient in the source of the momentum equation and at the divergence of the velocity in the source of the energy equation. However,  proposed to add the Q terms globally in order to consider the artificial viscosity as a real viscosity. Hence, the hydrodynamic equations should be rewritten for a modified stress-energy tensor of the following form:
In this way, in flat spacetime, the momentum equation takes the form:
In Wilson's formulation Q is omitted in the two terms containing the quantity . In general Q is a non-linear function of the velocity and, hence, the quantity in the momentum density of Eq. (31) is a highly non-linear function of the velocity and its derivatives. This fact, together with the explicit presence of the Lorentz factor in the convective terms of the hydrodynamic equations, as well as the pressure in the specific enthalpy, make the relativistic equations much more coupled than their Newtonian counterparts. As a result Norman and Winkler proposed the use of implicit schemes to describe more accurately such coupling. Their code, which incorporates an adaptive grid, reproduces very accurate results even for ultrarelativistic flows with Lorentz factors of about 10 in one-dimensional flat spacetime simulations.
A numerical scheme written in conservation form automatically guarantees the correct Rankine-Hugoniot (jump) conditions across discontinuities (the shock-capturing property). Writing the relativistic hydrodynamic equations as a system of conservation laws, identifying the suitable vector of unknowns and building up an approximate Riemann solver permitted the extension of state-of-the-art high-resolution shock-capturing (HRSC in the following) schemes from classical fluid dynamics into the realm of relativity .
Theoretical advances on the mathematical character of the relativistic hydrodynamic equations were achieved studying the special relativistic limit. In Minkowski spacetime, the hyperbolic character of relativistic (magneto-) hydrodynamics was exhaustively studied by Anile and collaborators (see  and references therein) by applying Friedrichs' definition of hyperbolicity  to a quasi-linear form of the system of hydrodynamic equations,
where are the Jacobian matrices of the system and are a suitable set of primitive variables (see below). System (32) will be hyperbolic in the time-direction defined by the vector field with , if the following two conditions hold: (i) and (ii) for any such that , , the eigenvalue problem has only real eigenvalues , and a complete set of right-eigenvectors . Besides verifying the hyperbolic character of the relativistic hydrodynamic equations, Anile and collaborators  obtained the explicit expressions for the eigenvalues and eigenvectors in the local rest frame, characterized by . In Font et al.  those calculations were extended to an arbitrary reference frame in which the motion of the fluid was described by the 4-velocity .
The extension to general relativity of the approach, followed in  for special relativity, was accomplished in . We will refer to the formulation of the general relativistic hydrodynamic equations presented in  as the Valencia formulation . The choice of evolved variables (conserved quantities) in this formulation differs slightly from Wilson's formulation. It comprises the rest-mass density (D), the momentum density in the j -direction (), and the total energy density (E), measured by a family of observers which are the natural extension (for a generic spacetime) of the Eulerian observers in classical fluid dynamics. Interested readers are addressed to  for their definition and geometrical foundations.
In terms of the primitive variables , the conserved quantities are written as:
where the contravariant components of the three-velocity are defined as
and W is the relativistic Lorentz factor with .
With this choice of variables the equations can be written in conservation form. Strict conservation is only possible in flat spacetime. For curved spacetimes there exist source terms, coming from the spacetime geometry, which do not contain derivatives of stress-energy tensor components. More precisely, the first-order flux-conservative hyperbolic system, well suited for numerical applications, reads:
with satisfying with . The state vector is given by
with . The vector of fluxes is
and the corresponding sources are
The local characteristic structure of the previous system of equations was presented in . The eigenvalues (characteristic speeds) of the corresponding Jacobian matrices are all real (but not distinct, one showing a threefold degeneracy) and a complete set of right-eigenvectors exists. System (37) satisfies, hence, the definition of hyperbolicity. As discussed in Section 3.1.2 below, the knowledge of the spectral information is essential in order to construct HRSC schemes based on Riemann solvers. This information can be found in  (see also ).
The range of applications considered so far in general relativity employing this formulation is still small and mostly devoted to the study of accretion flows onto black holes (see Section 4.2.2 below). In the special relativistic limit this formulation is being successfully applied to model the evolution of (ultra-) relativistic extragalactic jets (see, e.g., [129, 9]). The first numerical studies in general relativity were performed, in one spatial dimension, in , using a slightly different form of the equations. Preliminary investigations of gravitational stellar collapse were attempted by coupling the Valencia formulation to a hyperbolic formulation of the Einstein equations developed by . Some discussion of these results can be found in [123, 33]. More recently, successful evolutions of fully dynamical spacetimes in the context of adiabatic spherically symmetric stellar core-collapse have been achieved [101, 184]. We will come back to these issues in Section 4.1.1 below.
Recently, a three-dimensional, Eulerian, general relativistic hydrodynamical code, evolving the coupled system of the Einstein and hydrodynamic equations, has been developed . The formulation of the hydrodynamic equations follows the Valencia approach. The code is constructed for a completely general spacetime metric based on a Cartesian coordinate system, with arbitrarily specifiable lapse and shift conditions. In  the spectral decomposition (eigenvalues and right-eigenvectors) of the general relativistic hydrodynamic equations, valid for general spatial metrics, was derived, correcting earlier results of  for non-diagonal metrics. A complete set of left-eigenvectors has been recently presented by Ibáñez et al. . This information is summarized in Section 5.2 .
The formulation of the coupled set of equations and the numerical code reported in  were used for the construction of the milestone code ``GR3D'' for the NASA Neutron Star Grand Challenge project. For a description of the project see the website of the Washington University Gravity Group . A public domain version of the code has recently been released to the community at the same website, the source and documentation of this code can be downloaded at .
|Numerical Hydrodynamics in General Relativity
José A. Font
© Max-Planck-Gesellschaft. ISSN 1433-8351
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