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, Equation (2), can be easily integrated to yield the metric potential b :
The gravitational field equations, Equation (10), and the equations of motion, Equation (1), reduce to the following quasi-linear system of partial differential equations (see also ):
with the definitions and , satisfying . Additionally,
represents the total mass interior to radius m at time t . The final system, Equations (17), is closed with an EOS of the form given by Equation (9).
Hydrodynamics codes based on the original formulation of May and White and on later versions (e.g., ) have been used in many nonlinear simulations of supernova and neutron star collapse (see, e.g., [184, 280] and references therein), as well as in perturbative computations of spherically symmetric gravitational collapse within the framework of the linearized Einstein equations [251, 252]. In Section 4.1.1 below, some of these simulations are discussed in detail. An interesting analysis of the above formulation in the context of gravitational collapse is provided by Miller and Sciama . By comparing the Newtonian and relativistic equations, these authors showed that the net acceleration of the infalling mass shells is larger in general relativity than in Newtonian gravity. The Lagrangian character of May and White's formulation, together with other theoretical considerations concerning the particular coordinate gauge, has prevented its extension to multi-dimensional 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, most notably the lack of numerical diffusion.
the equations of motion in Wilson's formulation [300, 301] are
with the ``transport velocity'' given by . We note that in the original formulation  the momentum density equation, Equation (21), 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 nonlinear hyperbolic systems of equations, namely the preservation of their conservation form . This is a necessary condition to guarantee correct evolution in regions of sharp entropy generation (i.e., shocks). Furthermore, some amount of numerical dissipation must be used to stabilize the solution across discontinuities. In this spirit, the first attempt to solve the equations of general relativistic hydrodynamics in the original Wilson's scheme  used a combination of finite difference upwind techniques with artificial viscosity terms. Such terms adapted the classic treatment of shock waves introduced by von Neumann and Richtmyer  to 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 [195, 193, 199, 22, 276, 228, 79], accretion onto compact objects [122, 226], numerical cosmology [53, 54, 12] and, more recently, the coalescence of neutron star binaries [303, 304, 169]. This formalism has also been employed, in the special relativistic limit, in numerical studies of heavy-ion collisions [302, 175]. We note that in most of these investigations, the original formulation of the hydrodynamic equations was slightly modified by re-defining the dynamical variables, Equation (19), with the addition of a multiplicative factor (the lapse function) and the introduction of the Lorentz factor, :
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, such coupled simulations were 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 been applied to the numerical study of the coalescence of binary neutron stars in general relativity [303, 304, 169] (see Section 4.3.2). These studies adopted an approximation scheme for the gravitational field, by imposing the simplifying condition that the 3-geometry (the 3-metric ) is conformally flat . The line element, Equation (11), then reads
The curvature of the 3-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 removed, while 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 approach is no longer an approximation, being identical to Einstein's theory, beyond spherical symmetry its quality degrades. In  it was shown by means of numerical simulations of extremely relativistic disks of dust that 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 assessment of such formulation by means of special relativistic hydrodynamical simulations. Figure 1 reproduces a plot from  in which the relative error of the density compression ratio in the so-called 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 (), which is 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, commonly called Q in the literature (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, Equation (21), and at the divergence of the velocity in the source of the energy equation, Equation (22). However,  proposed to add the Q terms in a relativistically consistent way, 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, for instance, the momentum equation takes the following form (in flat spacetime):
In Wilson's original formulation, Q is omitted in the two terms containing the quantity . In general, Q is a nonlinear function of the velocity and, hence, the quantity in the momentum density of Equation (26) is a highly nonlinear 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 as a way to describe more accurately such coupling. Their code, which in addition incorporates an adaptive grid, reproduces very accurate results even for ultrarelativistic flows with Lorentz factors of about 10 in one-dimensional, flat spacetime simulations.
Very recently, Anninos and Fragile  have compared state-of-the-art artificial viscosity schemes and high-order non-oscillatory central schemes (see Section 3.1.3) using Wilson's formulation for the former class of schemes and a conservative formulation (similar to the one considered in [221, 218]; Section 2.2.2) for the latter. Using a three-dimensional Cartesian code, these authors found that earlier results for artificial viscosity schemes in shock tube tests or shock reflection tests are not improved, i.e., the numerical solution becomes increasingly unstable for shock velocities greater than about . On the other hand, results for the shock reflection problem with a second-order finite difference central scheme show the suitability of such a scheme to handle ultrarelativistic flows, the underlying reason being, most likely, the use of a conservative formulation of the hydrodynamic equations rather than the particular scheme employed (see Section 3.1.3). Tests concerning spherical accretion onto a Schwarzschild black hole using both schemes yield the maximum relative errors near the event horizon, as large as % for the central scheme.
If a numerical scheme written in conservation form converges, it automatically guarantees the correct Rankine-Hugoniot (jump) conditions across discontinuities - the shock-capturing property (see, e.g., ). 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 schemes (HRSC in the following) from classical fluid dynamics into the realm of relativity .
Theoretical advances on the mathematical character of the relativistic hydrodynamic equations were first achieved studying the special relativistic limit. In Minkowski spacetime, the hyperbolic character of relativistic hydrodynamics and magneto-hydrodynamics (MHD) 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 is a suitable set of primitive (physical) variables (see below). The system (27) is 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 is described by the 4-velocity .
The approach followed in  for the equations of special relativistic hydrodynamics was extended to general relativity in . The choice of evolved variables (conserved quantities) in the 3+1 Eulerian formulation developed by Banyuls et al.  differs slightly from that of 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 directed to  for more complete definitions and geometrical foundations.
In terms of the so-called primitive variables , the conserved quantities are written as
where the contravariant components of the 3-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, arising from the spacetime geometry. However, these terms 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 as a result of the assumed directional splitting approach), and a complete set of right-eigenvectors exists. System (30) satisfies, hence, the definition of hyperbolicity. As it will become apparent 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 the above formulation of the hydrodynamic equations, Equation (30, 31, 32, 33), is still small and mostly devoted to the study of stellar core collapse and accretion flows onto black holes (see Sections 4.1.1 and 4.2 below). In the special relativistic limit this formulation is being successfully applied to simulate the evolution of (ultra-)relativistic extragalactic jets, using numerical models of increasing complexity (see, e.g., [167, 8]). The first applications 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 above formulation of the hydrodynamic equations to a hyperbolic formulation of the Einstein equations developed by . These results are discussed in [161, 38]. More recently, successful evolutions of fully dynamical spacetimes in the context of adiabatic stellar core collapse, both in spherical symmetry and in axisymmetry, have been achieved [129, 244, 67]. These investigations are considered in Section 4.1.1 below.
An ambitious three-dimensional, Eulerian code which evolves the coupled system of Einstein and hydrodynamics equations was developed by Font et al.  (see Section 3.3.2). The formulation of the hydrodynamic equations in this code follows the conservative Eulerian approach discussed in this section. 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, extending earlier results of  for non-diagonal metrics. A complete set of left-eigenvectors was presented by Ibáñez et al. . Due to the paramount importance of the characteristic structure of the equations in the design of upwind HRSC schemes based upon Riemann solvers, we summarize all necessary information in Section 5.2 of this article.
|Numerical Hydrodynamics in General Relativity
José A. Font
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