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 [142
]):
with the definitions
satisfying
Additionally,
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. [221]) 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 [226] 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 [223
] 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 [227], building on a vacuum numerical relativity code specifically
developed to investigate the head-on collision of two black
holes [209]. The resulting code was axially symmetric and aimed to
integrate the coupled set of equations in the context of stellar
core collapse [67].
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 [108].
Wilson's formulation showed some limitations in handling
situations involving ultrarelativistic flows, as first pointed
out by Centrella and Wilson [48]. Norman and Winkler [159
] performed a comprehensive numerical study of such formulation
by means of special relativistic hydrodynamical simulations.
Fig.
1
reproduces a plot from [159
] 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 [48
]. 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 [159] 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, [159
] 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 [125].
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 [10] and references therein) by applying Friedrichs' definition of
hyperbolicity [78] 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 [10] obtained the explicit expressions for the eigenvalues and
eigenvectors in the local rest frame, characterized by
. In Font et al. [72
] 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 [72] for special relativity, was accomplished in [17
]. We will refer to the formulation of the general relativistic
hydrodynamic equations presented in [17
] 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 [17
] 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 [17]. 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 [17
] (see also [75
]).
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 [125], 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 [34
]. 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 [75]. 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 [75
] the spectral decomposition (eigenvalues and right-eigenvectors)
of the general relativistic hydrodynamic equations, valid for
general spatial metrics, was derived, correcting earlier results
of [17
] for non-diagonal metrics. A complete set of left-eigenvectors
has been recently presented by Ibáñez et al. [99]. This information is summarized in Section
5.2
.
The formulation of the coupled set of equations and the
numerical code reported in [75] 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 [1]. 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 [2].
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Numerical Hydrodynamics in General Relativity
José A. Font http://www.livingreviews.org/lrr-2000-2 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |