Evolutions of polytropic models of spherical neutron stars
(i.e. with EOS
,
*K*
being the polytropic constant) using relativistic hydrodynamics
can be used as test-bed computations for multidimensional codes.
One-dimensional hydrodynamical studies of relativistic stars have
been performed by [87], employing pseudo-spectral methods, and by [184] with HRSC schemes. These investigations adopted radial gauge
polar slicing coordinates in which the general relativistic
equations are expressed in a simple way which resembles Newtonian
hydrodynamics. Gourgoulhon [87] used a numerical code to detect, dynamically, the zero value of
the fundamental mode of a neutron star against radial
oscillations. Romero et al. [184] highlighted the accuracy of HRSC schemes by finding,
numerically, a change in the stability behavior of two slightly
different initial neutron star models: A model with mass
is stable and a model of
is unstable.

Three-dimensional hydrodynamical evolutions of relativistic,
self-gravitating neutron stars have been considered in [75,
202,
8,
203,
205]. In [75] a new efficient parallel general relativistic hydrodynamical
code was presented and thoroughly tested. In this code the
Valencia hydrodynamical (conservative) formulation was adopted
and a variety of state-of-the-art Riemann solvers were
implemented, including Roe's solver [183] and Marquina's flux formula [57]. The Einstein equations were formulated using two different
approaches: (i) the standard ADM formalism and (ii) a hyperbolic
formulation developed in [35]. The code was subjected to a series of convergence tests, with
(twelve) different combinations of the spacetime and
hydrodynamics finite differencing schemes, demonstrating the
consistency of the discrete equations with the differential
equations. The simulations performed in [75] include, among others, the evolution of equilibrium
configurations of compact stars (solutions to the TOV equations),
and the evolution of relativistically boosted TOV stars (*v*
=0.87
*c*) transversing diagonally across the computational domain. In the
long term plan this code is designed to simulate the inspiral
coalescence of two neutron stars in general relativity. The more
``academic'' scenario of a head-on collision has been already
analyzed in [140] and is briefly described below.

The simulations presented in [202, 203, 205, 8] are performed using a reformulation of the ADM equations, first proposed by Shibata and Nakamura [204], and recently taken up by Baumgarte and Shapiro [23]. This formulation is based upon a conformal decomposition of the three metric and extrinsic curvature. Additionally, three ``conformal connection functions'' are introduced so that the principal part of the conformal Ricci tensor is an elliptic operator acting on the components of the conformal three metric. In this way the evolution equations reduce to a coupled set of non-linear, inhomogeneous wave equations for the conformal three metric, which are coupled to evolution equations for the (gauge) connection functions. Further details can be found in [23]. This formulation has proven to be much more long-term stable (though somehow less accurate, see [8]) than standard ADM. The tests analyzed include evolutions of weak gravitational waves [23], self-gravitating matter configurations with prescribed (analytic) time evolution [22], spherical dust collapse [202], strong gravitational waves (which even collapse to black holes) [8], black holes [8], and boson stars [8]. Hydrodynamical evolutions of neutron stars are extensively considered in [202, 203, 205].

Nowadays, most of the current efforts in developing codes in
relativistic astrophysics are strongly motivated by the
simulation of the coalescence of compact binaries. These
scenarios are considerered the most promising sources of
gravitational radiation to be detected by the planned laser
interferometers going online worldwide in the next few years. The
computation of the gravitational waveform during the most
dynamical phase of the coalescence stage depends crucially on
hydrodynamical finite-size effects. This phase begins once the
stars, initially in quasi-equilibrium orbits of gradually smaller
orbital radius, due to the emission of gravitational waves, reach
the innermost stable circular orbit (see the schematic plot in
Fig.
10). From here on, the final merger of the two objects takes place
in a dynamical timescale and lasts for a few milliseconds. A
treatment of the gravitational radiation as a perturbation in the
quadrupole approximation would be valid as long as
and
simultaneously,
*M*
being the mass of the binary,
*R*
the neutron star radius and
*r*
the separation of the two stars. As the stars gradually approach
each other and merge, both inequalities are less valid and fully
relativistic calculations become necessary.

The accurate simulation of a binary neutron star coalescence is however one of the most challenging tasks in numerical relativity. These scenarios involve strong gravitational fields, matter motion with (ultra-) relativistic speeds and/or strong shock waves. The difficulties of a successful numerical integration are exacerbated by the intrinsic multidimensional character of the problem and by the inherent complexities in Einstein's theory of gravity, e.g. coordinate degrees of freedom and the possible formation of curvature singularities (e.g. collapse of matter configurations to black holes).

For these reasons it is not surprising that most of the
available simulations have been attempted in the Newtonian (and
post-Newtonian) framework. Most of these studies employ
Lagrangian particle methods such as SPH, and only a few have
considered (less viscous) high-order finite difference methods
such as PPM [186]. The interested reader is referred to the recent review by
Rasio and Shapiro [180] and an upcoming
*Living Reviews*
article by Swesty [213], for detailed descriptions of the current status of Newtonian
simulations.

Concerning relativistic simulations Wilson's hydrodynamical
formulation has been applied to the study of neutron star binary
coalescence in [230,
231] under the assumption of a conformally flat spacetime, which
leads to a considerable simplification of the gravitational field
equations. In this case they reduce to a coupled set of elliptic
(Poisson-like) equations for the lapse function, shift vector and
conformal factor. Their simulations revealed, unexpectedly, the
appearance of a ``binary-induced collapse instability'' of the
neutron stars, prior to the eventual collapse of the final merged
object, with the central density of each star increasing by an
amount proportional to 1/
*r*
. The numerical results of Wilson and collaborators have received
considerable attention in the literature, and their unexpected
outcome has been strongly criticized by many authors on
theoretical grounds (see references in [180] for an updated discussion). In particular, a radial stability
analysis carried out by [21] showed that fully relativistic, corotating binary
configurations are stable against collapse to black holes all the
way down to the innermost stable circular orbit. More recently
Flanagan [69] has pointed out the use of an incorrect form of the momentum
constraint equation in the simulations performed by Wilson and
collaborators [230,
231], which gives rise to a first post-Newtonian-order error in the
scheme, showing analytically that this error can cause the
observed increase of the central densities obtained in the
simulations. However, in revised hydrodynamical simulations
performed by Mathews and Wilson [131], which incorporate the correction identified by Flanagan [69], it was found that the compression effect is not completely
eliminated, although its magnitude significantly diminishes at a
given angular momentum. Reliable numerical simulations with the
full set of Einstein equations will ultimately clarify these
results.

Nakamura and co-workers have been pursueing a programme to simulate neutron star binary coalescence in general relativity since the late 80's (see, e.g., [150]). A three-dimensional code employing the full set of Einstein equations and self-gravitating matter fields has been developed [164]. In this code the complete set of equations, spacetime and hydrodynamics, are finite-differenced on a uniform Cartesian grid using van Leer's scheme [219] with TVD flux limiters. Shock waves are spread out using a tensor artificial viscosity algorithm. The hydrodynamic equations follow Wilson's formulation and the ADM formalism is adopted for the Einstein equations. This code has been tested by the study of the gravitational collapse of a rotating polytrope to a black hole (comparing to the axisymmetric computation of [211]) and applied to the coalescence of a binary neutron star system. Further work to achieve long term stability is currently under way [164].

The most advanced simulations of neutron star coalescence in
full general relativity are those reported recently by
Shibata [202,
205]. His code is able to simulate a coalescence event for a long
time from the innermost circular orbit up to the formation of a
final merged object (either a black hole or a neutron star). The
code shares many features with that of Oohara and Nakamura [164]: The hydrodynamic equations are formulated following Wilson's
approach and they are solved using van Leer's [219] second order finite difference scheme with artificial
viscosity. The most important difference concerns the
reformulation of the ADM Einstein equations into a conformal
traceless system, as mentioned previously. This formulation was
originally introduced by Shibata and Nakamura [204] and recently slightly modified by [23]. In [202] Shibata computed the merger of two models of corotating binary
neutron stars of
in contact and in approximate quasi-equilibrium orbits. The
central density of each star in the two models is
(mildly relativistic) and
in geometrized units. The quasi-equilibrium models are
constructed assuming a polytropic EOS with
*K*
=10. For both initial models, the neutron stars begin to merge
forming spiral arms at half the orbital period
*P*
of the quasi-equilibrium states, and by
the final object is a rapidly and differentially rotating highly
flattened neutron star. For the more relativistic model the
central density of the merged object is
, which is nearly the maximum allowed density along the sequence
of stable neutron stars of
*K*
=10 and
. Hence, a black hole could be formed directly in the merger of
initially more massive neutron stars. For the cases considered by
Shibata, the new star is strongly supported by rapid rotation (,
*J*
being the angular momentum and
the gravitational mass) and could eventually collapse to a black
hole once sufficient angular momentum has dissipated through,
e.g., neutrino emission or gravitational radiation.

Recently, Miller et al. [140] have studied the head-on collision of two neutron stars by
means of time-dependent relativistic simulations using the code
of Font et al. [75]. These simulations are aimed at investigating whether the
collapse of the final object occurs in prompt timescales (i.e. a
few milliseconds) or delayed (after neutrino cooling) timescales
(i.e. a few seconds). In [195] it was argued that in a head-on collision event sufficient
thermal pressure is generated to support the remnant in
quasi-static equilibrium against (prompt) collapse prior to slow
cooling via neutrino emission (delayed collapse). Nevertheless,
in [140] prompt collapse to a black hole was found in the head-on
collision of two
neutron stars modeled by a polytropic EOS with
and
. The stars are initially separated by a proper distance of
and are boosted towards one another at a speed of
(the Newtonian infall velocity). The simulation employed a
Cartesian grid of
points. The time evolution of this simulation can be followed in
the Quicktime movie in Fig.
11
. This animation simultaneously shows the rest-mass density and
the internal energy evolution during the on-axis collision. The
formation of the black hole in prompt timescales is demonstrated
by the sudden appearance of the apparent horizon at
(*t*
=63.194 in code units). The violet dotted circles indicate the
trapped photons. The animation also shows a moderately
relativistic shock wave (Lorentz factor of about 1.2) appearing
at
(code units; yellow-white colors) which eventually is followed
by two opposite moving shocks (along the infalling
*z*
direction) which propagate along the atmosphere surrounding the
black hole.

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 |