Since early investigations the numerical study of gravitational collapse has been neatly divided between
two main communities. First, the computational astrophysics community has traditionally focused on the
physics of the (type II/Ib/Ic) Supernova mechanism, i.e., the collapse of unstable stellar iron cores followed
by a bounce and explosion. Nowadays, numerical simulations are highly developed, as they include rotation
and detailed state-of-the-art microphysics (e.g., EOS and sophisticated treatments for neutrino transport),
along with the incorporation of methodology for the magnetic field. These studies are currently performed
routinely in multiple dimensions with advanced HRSC schemes. Progress in this direction has been
achieved, however, at the expense of accuracy in the treatment of the hydrodynamics and the gravitational
field, by using Newtonian physics, and only recently this situation has started to change (see below).
In this approach the emission of gravitational radiation is computed through the quadrapole
formula. For reviews of the current status in this direction see [271, 183, 422
] and references
therein.
On the other hand, the numerical relativity community has been more interested, since the beginning, in highlighting those aspects of the collapse problem having to do with relativistic gravity, in particular, the emission of gravitational radiation from nonspherical collapse (see [146] for a Living Reviews article on gravitational radiation from gravitational collapse). Much of the effort has focused on developing reliable numerical schemes to solve the gravitational field equations and much less emphasis has been given to the details of the microphysics of the collapse (yet again this situation is starting to change). In addition, this approach has mostly considered gravitational collapse leading to black-hole formation, employing relativistic hydrodynamics, MHD, and gravity. Both approaches, which had barely interacted over the years except for a handful of simulations in spherical symmetry, have begun a much closer interaction very recently. This has been mainly the result of the progress achieved in numerical relativity regarding long-term stable formulations of the Einstein equations, as we will illustrate in the following Section 5.1.1.
However, before proceeding any further, we must point out that the current version of this article does not cover, as earlier versions did, critical collapse. Critical phenomena in gravitational collapse were first discovered numerically by Choptuik in spherically-symmetric simulations of the massless Klein–Gordon field minimally coupled to gravity [78]. Since then, critical phenomena arising at the threshold of black-hole formation have been found in a variety of physical systems, including the perfect fluid model. The interested reader is directed to [161] and references therein for details.
At the end of their thermonuclear evolution, massive stars in the (Main Sequence) mass range from
to
develop a core composed of iron group nuclei, which becomes dynamically unstable
against gravitational collapse. The iron core collapses to a neutron star or a black hole releasing
gravitational binding energy of the order
, which is sufficient to
power a supernova explosion. The standard model of type II/Ib/Ic Supernovae involves the presence of a
strong shock wave formed at the edge between the homologous inner core and the outer core, which falls at
roughly free-fall speed. A schematic illustration of the dynamics of this process is depicted in
Figure 3
. The shock is produced after the bounce of the inner core when its density exceeds nuclear
density. Numerical simulations have tried to elucidate whether this shock is strong enough to
propagate outward through the star’s mantle and envelope, given certain initial conditions for the
material in the core (an issue subject to important uncertainties in the nuclear EOS), as well as
through the outer layers of the star. In the accepted scenario, which has emerged from numerical
simulations, the existence of the shock wave together with neutrino heating that re-energizes it
(in the delayed mechanism by which the stalled-prompt supernova shock is reactivated by an
increase in the post-shock pressure due to neutrino energy deposition [414, 44]), and convective
overturn, which rapidly transports energy into the Ledoux-unstable shocked region [81, 43]
(and which can lead to large-scale deviations from spherically-symmetric explosions), are all
necessary ingredients for any plausible explosion mechanism. The most recent simulations have also
highlighted the importance of a low-mode instability in the standing accretion shock for large-scale
explosion asymmetries, pulsar kicks, and for the development of neutrino-driven explosions
(see [422, 184
] for details on the degree of sophistication achieved in present-day supernova
modelling).
However, the path to reach such conclusions has mostly been delineated from numerical simulations in one spatial dimension. Fully relativistic simulations of microphysically-detailed core collapse off spherical symmetry are only beginning to become available, and they might well introduce some modifications. The broad picture described above has been demonstrated in multiple dimensions using sophisticated Newtonian models. Nevertheless, while studies based upon Newtonian physics are highly developed nowadays, state-of-the-art simulations still fail, broadly speaking, to generate successful supernova explosions under generic conditions.
There are only a few fully relativistic simulations with HRSC schemes available in the
literature [178, 339, 294], which, however, do not account for neutrino transport. On the other hand, there
also exist state-of-the-art neutrino-radiation–hydrodynamics codes in spherical symmetry, which incorporate
the relativistic effects of the gravitational field in an exact or approximate manner. The code of
the Oak Ridge-Basel group (AGILE-BOLTZTRAN) belongs to the former class and consists of a
general-relativistic time-implicit discrete-angle Boltzmann solver, coupled to a conservative, general
relativistic, time-implicit, hydrodynamics TVD solver. On the other hand, the VERTEX code
of the Garching group calculates neutrino transport by a variable Eddington-factor closure
of neutrino energy, number, and momentum equations, and the hydrodynamics step is based
on conservative high-order Riemann solvers, as those used in its precursor code PROMETHEUS.
However, the relativistic effects of the gravitational field are only incorporated in VERTEX through
an effective gravitational potential. Comparisons of these two supernova codes are reported
in [223].
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Despite the incomplete physics employed in adiabatic approximations of stellar core collapse it is still
useful to present them here in order to get acquainted with the complex hydrodynamics involved in the
problem. A comprehensive study of adiabatic, one-dimensional, general relativistic core collapse using
explicit upwind HRSC schemes was presented in [339] (see [424] for a similar computation using implicit
schemes). In this investigation the equations for the hydrodynamics and the geometry are written using
radial gauge polar slicing (Schwarzschild-type) coordinates. The collapse is modeled with an ideal gas EOS
with a nonconstant adiabatic index, which depends on the density as
, where
is the bounce density and
if
and
otherwise [406]. A set of animations of the
simulations presented in [339
] is included in the four movies in Figure 4
. They correspond to the
rather stiff Model B of [339
]:
and
. The initial
model is a white dwarf with a gravitational mass of
. The animations show the time
evolution of the radial profiles of the following fields: velocity (Movie 4
a), logarithm of the
rest-mass density (Movie 4
b), gravitational mass (Movie 4
c), and the square of the lapse function
(Movie 4
d).
The movies show that the shock is sharply resolved and free of spurious oscillations. The radius of the
inner core at the time of maximum compression, at which the infall velocity is maximum (), is
. The gravitational mass of the inner core at the time of maximum compression is
. The
minimum value that the quantity
reaches is
, which indicates the highly relativistic character of
these simulations (at the surface of a typical neutron star the value of the lapse function squared is
).
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Among the more detailed multi-dimensional nonrelativistic hydrodynamic simulations are those
performed by the MPA/Garching group (an online sample can be found at their website [243]). As
mentioned earlier, these include advanced microphysical EOS, state-of-the-art treatments of neutrino
physics, and they employ accurate HRSC integration schemes. To illustrate the degree of sophistication
achieved in Newtonian simulations, we present in the movie in Figure 5
an animation of the early evolution
of a core collapse supernova explosion up to 220 ms after core bounce and shock formation (Figure 5
depicts just one intermediate snapshot at 130 ms). The movie shows the evolution of the entropy within the
innermost 3000 km of the star.
The initial data used in these calculations is taken from the pre-collapse model of a blue
super-giant star in [423]. The computations begin 20 ms after core bounce from a one-dimensional model
of [65]. This model is obtained with the one-dimensional general-relativistic code mentioned
previously [64], which includes a detailed treatment of the neutrino physics and a realistic EOS, and which
is able to follow core collapse, bounce and the associated formation of the supernova shock. Because of
neutrino cooling and energy losses due to the dissociation of iron nuclei, the shock initially stalls inside the
iron core.
The movie shows how the stalling shock is “revived” by neutrinos streaming from the outer layers of the hot, nascent neutron star in the center. A negative entropy gradient builds up between the so called “gain-radius”, which is the position where the cooling of hot gas by neutrino emission switches into net energy gain by neutrino absorption, and the shock further out. Convective instabilities, which are characterized by large rising blobs of neutrino-heated matter and cool, narrow downflows, therefore, set in. The convective energy transport increases the efficiency of energy deposition in the post-shock region by transporting heated material near the shock and cooler matter near the gain radius where the heating is strongest. The freshly heated matter then rises again while the shock is distorted by the upward streaming bubbles. The reader is directed to [190] for a more detailed description of a more energetic initial model.
Multidimensional models, such as those developed by the MPA/Garching group or the Oak Ridge–Basel
group, have shown that such strong radial mixing, as in the previous movie, requires the development of
hydrodynamic instabilities near the core and even in an early stage of the explosion. In recent
years axisymmetric numerical models have highlighted the fact that the standing accretion
shock is generically unstable to nonradial deformations even when convection is absent or highly
damped, to the point that the term “standing accretion shock (advective/acoustic) instability”
(SASI) has been coined in the literature (see [184] and references therein). The SASI shows
a preferential growth of low (
and
) shock-deformation modes at the time of
shock revival, leading to large-amplitude bipolar oscillations, which have important implications
for large-scale explosion asymmetries, pulsar kicks, and the development of neutrino driven
explosions.
A new line of investigation led by Burrows et al. [70] has recently claimed successful supernova
explosions by resorting to an alternative mechanism based on acoustic power generated in the inner core.
The simulations have been performed with the Newtonian, two-dimensional radiation/hydrodynamic code
VULCAN/2D with multi-group, flux-limited diffusion. Such acoustic power is generated in two phases. It starts
in the first 100 ms following core bounce in the inner turbulent region and, most importantly, it becomes
more significant at later times, by the excitation and sonic damping of core
-mode oscillations. The
simulations of [70] for an
progenitor show that an
mode grows at late times to be
prominent around roughly 500 ms after bounce to drive a successful explosion. This mechanism has been
met with skepticism by [184].
In addition to understanding the explosion mechanism, one of the prime motivations for performing
multidimensional core-collapse simulations is to compute theoretical gravitational-radiation waveforms,
which may be confronted by experimental data currently being collected by a number of detectors. As early
as 1982, Müller [270] obtained the first numerical evidence of the low gravitational wave efficiency of the
core collapse scenario, with an energy smaller than
being radiated as gravitational waves
during core bounce. In the 1990s, the first Newtonian parameter studies of the axisymmetric collapse of
rotating polytropes were performed by [123, 425, 439
], which provided the first investigations of the
gravitational wave signals from core bounce and early post-bounce phase. In core collapse events,
gravitational waves are initially dominated by a burst associated with the hydrodynamic bounce. If
rotation is present, the post-bounce wave signal shows large amplitude oscillations associated with
pulsations in the collapsed core [439
, 328
] and (possibly) low
rotational dynamic
instabilities [306
, 305
] (
and
being the kinetic rotational energy and the potential
energy, respectively). On the other hand, it has recently been shown that gravitational waves
from convection have preeminence over the purely burst signal of the core bounce on longer
timescales [273]. In this case, the inherent long emission timescale can yield high energy in a continuous
signal.
It is well known that neutron stars have intense magnetic fields () or even larger at
birth (
). In extreme cases such as magnetars, the magnetic field can be so strong as
to affect the internal structure of the star [51]. The presence of such intense magnetic fields
render magneto-rotational core collapse simulations mandatory. We note that as early as 1979
magneto-rotational core collapse was already proposed by [272] as a plausible supernova explosion
mechanism.
In recent years, an increasing number of authors have performed axisymmetric magneto-rotational
core collapse simulations (within the ideal MHD limit) employing Newtonian treatments of
magneto-hydrodynamics, gravity, and microphysics (see, e.g., [296, 69
] and references therein). Relativistic
magneto-rotational core collapse simulations have only very recently become available (see below). The
weakest point of all existing magneto-rotational core collapse simulations to date is the fact that the
strength and distribution of the initial magnetic field in the core are unknown. If the magnetic field is
initially weak, there exist several mechanisms which may amplify it to values, which can have an impact on
the dynamics, among them differential rotation (
-dynamo) and the magneto-rotational instability
(MRI hereafter). The former transforms rotational energy into magnetic energy, winding up any
seed poloidal field into a toroidal field, while the latter, which is present as long as the radial
gradient of the angular velocity of the fluid is negative, leads to exponential growth of the field
strength.
Specific magneto-rotational effects on the gravitational wave signature were first studied in detail
by [209] and [426], who found differences with purely hydrodynamic models only for very strong initial
fields (). The most exhaustive parameter study of magneto-rotational core collapse to date has
been carried out very recently by [296
, 295
]. The axisymmetric simulations of [296
, 295] have employed
rotating polytropes, Newtonian (and modified Newtonian) gravity and hydrodynamics, and ad-hoc initial
poloidal magnetic field distributions (as no self-consistent solution is yet known). These authors have found
that for weak initial fields (
, which is the most relevant case, astrophysics-wise) there are no
differences in the collapse dynamics nor in the resulting gravitational wave signal, when comparing with
purely hydrodynamic simulations. However, strong initial fields (
) manage to slow down the core
efficiently (leading even to retrograde rotation in the proto-neutron star), which causes qualitatively
different dynamics and gravitational wave signals. For some models, [296] even find highly bipolar, jet-like
outflows. The creation and propagation of MHD jets is also studied in the recent multigroup,
radiation MHD simulations of supernova core collapse in the context of rapid rotation performed
by [69]. These simulations suggest that for rotating cores a supernova explosion might be followed
by a secondary, weak MHD jet explosion, which might be generic in the collapsar model of
GRBs.
More than twenty years later, and with no other simulations in between, the first comprehensive
numerical study of relativistic, rotational, supernova-core collapse in axisymmetry was performed by
Dimmelmeier et al. [93, 94, 95, 96
], who computed the gravitational radiation emitted in such events. The
Einstein equations were formulated using the conformally-flat metric approximation [418
] (CFC hereafter).
Correspondingly, the hydrodynamic equations were cast as the first-order flux-conservative hyperbolic
system described in Section 2.1.3. Details of the methodology and numerical code are given in [95
] (see also
Section 4.4). These simulations employed simplified models to account for the thermodynamics of the
process, in the form of a polytropic EOS conveniently modified to account for the stiffening of the matter
once nuclear matter density is reached [186]. The inclusion of relativistic effects results primarily
in the possibility of achieving deeper effective potentials. Moreover, higher densities than in
Newtonian models are reached during bounce, and the resulting proto-neutron star is more
compact.
The simulations show that the three different types of rotational supernova core collapse
(and their associated gravitational waveforms) identified in previous Newtonian simulations by
Zwerger and Müller [439] – regular collapse, multiple bounce collapse, and rapid collapse
(Types I, II, and III, respectively) – are also present in relativistic gravity, at least when the
initial models are simple rotating polytropes. As mentioned before, the gravitational wave signal
is characterized by a distinctive burst associated with the hydrodynamic bounce followed by
a strongly-damped ring-down phase of the newly-born proto-neutron star. While indeed the
maximum density reaches higher values in relativistic gravity than in Newtonian gravity, the
gravitational wave signals were found to be of comparable amplitudes. In fact, the simulations
performed by [96
] reveal that this is a general trend, the gravitational radiation emitted in
the relativistic models being strikingly similar to those previously obtained from Newtonian
simulations [439
]. On average, thus, the CFC simulations of [96
] show that the gravitational wave signals of
relativistic models have similar amplitude but somewhat higher frequency than their Newtonian
counterparts [439
].
Among the most interesting results of these investigations was the fact that in rotational core collapse,
the collapse type can change from multiple centrifugal bounce (Type II) to standard single-bounce collapse
(Type I). Centrifugal bounce was highly suppressed in relativistic gravity, yet it was still possible for simple
EOS (but see below for the situation regarding improved microphysics). The main conclusion of [96] is that
only the gravitational signal of a Galactic supernova (i.e., within a distance of about 10 kpc) might be
unambiguously detectable by first generation detectors. Signal recycling techniques in next
generation detectors may be needed, however, for successful detection of more distant core
collapse events (with an increased event rate), up to distances of the Virgo cluster. An online
waveform catalogue for all models analyzed by Dimmelmeier et al. [96
] is available at the MPA’s
website [243
].
The movie presented in Figure 6 shows the time evolution of a multiple bounce model (model A2B4G1
in the notation of [95]), with a type II gravitational wave signal. The left panel shows isocontours of the
logarithm of the density together with the corresponding velocity field distribution. The two panels on the
right show the time evolution of the gravitational wave signal (top panel) and of the central rest-mass
density. In the animation the “camera” follows the multiple bounces by zooming in and out. The panels on
the right show how each burst of gravitational radiation coincides with the time of bounce of the collapsing
core. The oscillations of the nascent proto-neutron star are therefore imprinted on the gravitational
waveform. Additional animations of the simulations performed by [96
] can be found at the MPA’s
website [243].
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Relativistic simulations with improved dynamics and gravitational waveforms were reported by [74],
who used an approximation for the field equations which they called CFC+. This approximation
incorporates additional degrees of freedom to CFC such that the spacetime metric is exact up to the second
post-Newtonian order. As in the CFC simulations of [96
] a simple (modified polytrope) EOS was used, and
the initial models were unmagnetized. The CFC simulations of [74
] cover the basic morphology and
dynamics of core collapse types studied by [96
], including the extreme case of a core with strong
differential rotation and torus-like structure. In either case no significant differences were found in
all models investigated by [74
], both regarding the dynamics and gravitational wave signals.
Therefore, second post-Newtonian corrections to CFC do not significantly improve the results when
simulating the dynamics of core collapse to a neutron star. (It remains to be seen whether
this conclusion changes in the strongest gravitational fields as, e.g., in the collapse to a black
hole.)
Comparisons of the CFC approach with fully general-relativistic simulations (employing the
BSSN formulation) have been reported by [365] in the context of axisymmetric core collapse
simulations. As in the case of CFC+, the differences found in both, the collapse dynamics and the
gravitational waveforms are minute, which highlights the suitability of CFC (and CFC+) for
performing accurate simulations of these scenarios without the need for solving the full system of
Einstein’s equations. It is also worth pointing out the detailed comparison of a comprehensive
set of models (some including collapse to black holes) reported recently by [92], to which the
interested reader is directed for further information. Owing to the excellent approximation offered
by CFC for studying core collapse, extensions to improve the microphysics of the numerical
modelling, through the incorporation of microphysical EOS and simplified neutrino treatments, as
well as extensions to account for three spatial dimensions, have been reported in the last few
months.
The most realistic simulations of rotational core collapse in general relativity so far (both in
axisymmetry and in three dimensions, but without incorporating magnetic fields) are those performed
by [305, 98
]. These simulations employ both the CFC and BSSN formulations of Einstein’s equations, and
have been carried out using state-of-the-art numerical codes, both in spherical polar coordinates (the
CoCoNut code of [97
]) and in Cartesian coordinates (the CACTUS/WHISKY codes; see [244
, 27
, 305
] and
references therein). As in the earlier works of [96
, 365
, 74
] the gravitational wave information from
these collapse simulations is extracted using (variations of) the Newtonian standard quadrapole
formula.
The initial models of [305, 98
] use nonrotating
pre-supernova models to which an
artificially-parameterized rotation is added. In addition, the relevant microphysics of core collapse is
accounted for by employing a finite-temperature nuclear EOS [351
] together with an approximate
(parameterized) neutrino treatment with deleptonization and neutrino pressure effects included, as recently
suggested by [221
]. In the simulations of [305
, 98
] the electron fraction profiles are obtained from
general-relativistic spherical models with Boltzmann neutrino transport, and are used as input data for the
time-dependent electron fraction prescription suggested by [221
] (with no analytic fit). The physics
included in the modelling provide a reasonably good approximation up until shortly after core
bounce. Therefore, the setup is valid for obtaining reliable gravitational wave signals from core
bounce.
The simulations reported by [305, 98
] allow for a straight comparison with models that have simple
(polytropic) EOS and the same rotation parameters as those of [96, 365, 74]. This sheds light on the
influence of rotation and of the EOS on the gravitational waveforms. This comparison shows that for
microphysical EOS the influence of rotation on the collapse dynamics and waveforms is somewhat less
pronounced than in the case of simple EOS. In particular, the most important result of these investigations
is the suppression of core collapse with multiple centrifugal bounces and its associated Type II gravitational
waveforms, as that shown in Figure 6
, which is explained as mainly due to the influence of deleptonization
on the collapse dynamics, which removes energy from the system. As a result, the improved simulations
of [98
, 305
] reveal an enhanced uniformity of the gravitational wave signal from core bounce
as, in particular, the low frequency signals are suppressed (as they were associated with the
multiple centrifugal bounce scenario). In addition, signals for slow and moderate rotation have
almost identical frequencies, that is, the signal peaked at a frequency of
670 Hz for the
models considered. These important results have implications for signal analysis and signal
inversion.
On the other hand, to further improve the realism of core collapse simulations in general relativity,
the incorporation of magnetic fields in numerical codes via solving the MHD equations is also
currently being undertaken [362, 75
, 76
]. The work of [362
], which is based on the conservative
formulation of the GRMHD equations discussed in Section 3.1.2, is focused on the collapse of
initially strongly magnetized cores (
). Although these values are probably
astrophysically not relevant, because the stellar evolution models of [172
] predict a poloidal field
strength of
in the progenitor, they permit one to resolve the scales needed for the MRI
to develop, since the MRI length scale grows with the magnetic field. We note that, in MRI
unstable regions, the instability grows exponentially for all length scales larger than a critical
length scale
, where
is the Alfvén speed and
is the angular velocity.
The fastest growing MRI mode develops at length scales near
on a typical time scale of
, where
is the distance to the rotation axis. Therefore, in
order to numerically capture the MRI, one has to resolve length scales of about the size of
, which can be prohibitively small to capture for initially small magnetic fields (say, on
the order of meters for a
progenitor), especially for three-dimensional computations,
where the affordable grid resolution is still limited. The results of [362
] show that the magnetic
field is mainly amplified by the wind-up of the magnetic field lines by differential rotation.
Consequently, the magnetic field is accumulated around the inner region of the proto-neutron
star, and MHD outflow forms along the rotation axis removing angular momentum from the
star.
A different approach is followed by [75, 76]. Their progenitors are chosen to be weakly magnetized
(
), which is in much better agreement with predictions from stellar evolution. Under these
conditions it is appropriate to use the “passive” magnetic field approximation, in which the magnetic field
entering into the energy-momentum tensor is negligible when compared with the fluid part, and the
components of the anisotropic term of
satisfy
. With such simplification,
the evolution of the magnetic field, governed by the induction equation, does not affect the
dynamics of the fluid, which is governed solely by the hydrodynamics equations. However, the
magnetic field evolution does depend on the fluid evolution, due to the presence of velocity
components in the induction equation. This approximation has interesting numerical advantages as
the seven eigenvalues of the GRMHD Riemann problem (entropy, Alfvén, and fast and slow
magnetosonic waves) reduce to three. Using this approach, the work of [76
] presents the first
relativistic simulations of magneto-rotational core collapse, which take into account the effects
of a microphysical EOS [351
] and a simplified neutrino treatment. These effects have been
incorporated into the code following the procedure employed by [305
] and [98
] in their unmagnetized
simulations. The results show that for the core collapse models with microphysics the saturation of
the magnetic field cannot be reached within dynamic time scales by winding up the poloidal
magnetic field into a toroidal one. Cerdá-Durán et al. [76
] have also estimated the effect of other
amplification mechanisms including the MRI and several types of dynamos, concluding that for
progenitors with astrophysically expected (i.e., weak) magnetic fields, the MRI is the only mechanism
that could amplify the magnetic field on dynamic time scales. In addition, all microphysical
models exhibit post-bounce convective overturn in regions surrounding the inner part of the
proto-neutron star. Since this has a potential impact on enhancing the MRI, it deserves further
investigation with more accurate neutrino treatment or alternative microphysical equations of
state.
The restriction to the passive magnetic-field approximation employed by [76] in studying
magneto-rotational core collapse of weakly magnetized progenitors can be justified if the MRI is indeed
inefficient. Otherwise, an “active” magnetic field approach becomes necessary (as in the work of [362]).
However, the use of active magnetic fields alone for core collapse simulations will probably not be sufficient
to model all the effects amplifying the magnetic field, since the numerical resolution needed to correctly
describe them (less than 10 m) is not affordable in current numerical simulations. In addition, most of
the relevant effects should be investigated in three dimensions, turning the computational work that lies
ahead into a daunting task.
Three-dimensional and fully general-relativistic simulations of core collapse have been performed
by [367] and [305]. The former study focuses on the likelihood of nonaxisymmetric dynamic
instabilities in the proto-neutron stars that form following core collapse, for a comprehensive set
of initial rotating polytropes with a hybrid (polytrope and thermal) EOS. In this work, the
early stage of the collapse is performed in axisymmetry and only after the stellar core becomes
compact enough the three-dimensional simulation is started, adding to the collapsed core a
bar-mode nonaxisymmetric density perturbation. The simulations show that the dynamic bar-mode
instability sets in if the progenitor already is rapidly rotating (
) and
has a high degree of differential rotation, which seems unlikely according to stellar evolution
models.
On the other hand, [305] have performed 3D simulations of general-relativistic core collapse within the
BSSN framework and a finite-temperature nuclear EOS [351
] and an approximate (parameterized) neutrino
treatment [221]. These simulations provide yet another confirmation of the satisfactory accuracy of CFC
for the core-collapse supernova problem. All models investigated in 3D are identical to those
evolved in axisymmetry with the CoCoNut code in [98
]. Typical simulation grids with nine
fixed–mesh-refinement levels and extending up to 3000 km in length were used. All models
were followed up to
20 ms after core bounce, which showed that they all stay essentially
axisymmetric through bounce, in good agreement with the models of [98]. One of the models,
labelled E20A in [305
], was further evolved until 70 ms after core bounce, as it showed the
largest gravitational wave amplitude of the sample and a value of the
ratio in the
newly-formed proto-neutron star high enough to eventually develop low
nonaxisymmetric
(bar-mode-like) deformations. This had already been found in a similar model using Newtonian physics
in [306].
The left panel of Figure 7 displays the time evolution of the normalized mode amplitudes of the four
lowest azimuthal density modes (
) in the equatorial plane, and shows that the one-armed
mode (red curve) becomes dominant from
ms after core bounce. Correspondingly, the right panel
of Figure 7
plots the gravitational wave strains
and
along the polar axis, where the late-time
increase in amplitude during the growth of the instability is visible. An animation of the entire evolution of
this collapse model is shown in Figure 8
, where the dynamics and growth of various modes are
manifest. The movie begins at
1 ms before core bounce and ends at 71 ms at which point the
simulation was terminated. Initially,
Cartesian-grid–induced features are prevalent in the
post-bounce dynamics of the proto-neutron star. These become gradually washed out as the star
goes nonaxisymmetrically unstable and develops
,
, and
features.
The mode structure is nonstationary and exhibits strong dynamic variation with radius and
time.
Ott et al. [305] find that the characteristic wave strain of this model peaks at a frequency of
930 Hz, which is the pattern frequency associated with its nonaxisymmetric unstable one-armed
structure. Such a value is slightly larger than the typical frequencies of the burst signals from
core collapse bounces (
300–800 Hz). If confirmed when incorporating additional physics
in the modelling, namely magnetic fields, and such dynamic, low
nonaxisymmetric
instabilities turn out to be a generic feature of the early phase of post-bounce dynamics, the
associated gravitational waves might be detectable out to much larger distances than the Milky
Way.
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Apart from a few one-dimensional simulations, most numerical studies of general-relativistic gravitational collapse leading to black-hole formation have used Wilson’s formulation of the hydrodynamics equations and finite difference schemes with artificial viscosity. The use of conservative formulations and HRSC schemes to study black-hole formation, both in two and three dimensions, began rather recently. The inclusion of magnetic fields in those simulations has barely been initiated. In the following we discuss only multidimensional work, directing the interested reader to earlier versions of this review that include basic coverage of one-dimensional simulations.
Evans [118], building on previous work by Dykema [111], also studied the axisymmetric
gravitational collapse problem for nonrotating matter configurations. His numerical scheme to
integrate the matter fields was more sophisticated than in previous approaches, as it included
monotonic upwind reconstruction procedures and flux limiters. Discontinuities appearing in the flow
solution were stabilized by adding artificial viscosity terms in the equations, following Wilson’s
approach.
Most of the axisymmetric studies discussed so far have adopted spherical polar coordinates. Numerical
instabilities are encountered at the origin () and at the polar axis (
), where some
fields diverge due to the coordinate singularities. Evans made important contributions toward
regularizing the gravitational field equations in such situations [118]. These coordinate problems have
deterred researchers from building three-dimensional codes in spherical coordinates. In this case,
Cartesian coordinates are adopted which, despite the desired property of being everywhere
regular, present the important drawback of not being adapted to the topology of astrophysical
systems. As a result, this has important consequences on the grid resolution requirements. To the
best of our knowledge the only extensions of axisymmetric 2+1 codes in spherical coordinates
to three dimensions has been accomplished by Stark [385] and by Dimmelmeier et al. [97].
In the former case, no applications in relativistic astrophysics have yet been presented. For
the latter, the CoCoNuT code is being applied to a number of problems such as core collapse
(discussed in the preceding section), black-hole formation, and neutron star instabilities (see
below).
Recently, Alcubierre et al. [6] proposed a method (cartoon) that does not suffer from stability problems
at coordinate singularities and in which, in essence, Cartesian coordinates are used even for axisymmetric
systems. Using this method, Shibata [354] investigated the effects of rotation on the criterion for prompt
adiabatic collapse of rigidly and differentially rotating (
) polytropes to a black hole. Collapse of
the initial approximate equilibrium models (computed by assuming a conformally-flat spatial
metric) was induced by pressure reduction. In [354] it was shown that the criterion for black-hole
formation depends strongly on the amount of angular momentum, but only weakly on its (initial)
distribution. Shibata also studied the effects of shock heating using a gamma-law EOS, concluding
that it is important in preventing prompt collapse to black holes in the case of large rotation
rates. Using the same numerical approach, Shibata and Shapiro [369] have also studied the
collapse of a uniformly rotating supermassive star in general relativity. The simulations show
that the star, initially rotating at the mass-shedding limit, collapses to form a supermassive
Kerr black hole with a spin parameter of
0.75. Roughly 90% of the mass of the system is
contained in the final black hole, while the remaining matter forms a disk orbiting around the black
hole.
Also using the cartoon approach, [110] performed long-term simulations of the gravitational collapse of
rapidly rotating neutron stars to black holes in axisymmetry, employing the code described in [109
] (see
Section 4.4). Together with the cartoon method the key novel ingredient added to the original code was an
algorithm to excise the black-hole singularity when the black hole first forms, which allowed one to extend
the evolutions far beyond what could be achieved without it. In agreement with the early works
of [277, 386
] it was found that unstable rotating stars collapse promptly to black holes only if they are
“sub-Kerr”, namely if
,
being the angular momentum and
the mass of the system.
This result was also independently obtained by [356] in a parameter study of rotational collapses of
supramassive neutron stars, employing the cartoon technique as well (but no black-hole excision).
Otherwise, for “supra-Kerr” models, [110
] found that the stars collapse to tori, which later
fragment. These types of systems have also been studied in three dimensions as we discuss
below.
Alternatively, axisymmetric codes employing the characteristic formulation of the Einstein
equations [419] do not share the axis instability problems of most 2+1 codes. Such axisymmetric
characteristic codes, conveniently coupled to hydrodynamics codes, are a promising way of
studying the axisymmetric collapse problem and, in particular, of extracting accurate information
regarding gravitational radiation. First steps in this direction have been taken by [378], where an
axisymmetric Einstein-perfect fluid code is presented, and in [379] where this code was applied
to study the core collapse of perturbed spherically-symmetric polytropes. This code achieves
global energy conservation for a strongly-perturbed neutron-star spacetime, for which the total
energy radiated away by gravitational waves corresponds to a significant fraction of the Bondi
mass.
Further recent improvements in the numerical modelling have accounted for physical effects, which may
alter the structure of differentially rotating stars on secular timescales, such as the viscosity and magnetic
fields. The former effect involves the solution of the relativistic Navier–Stokes equation, which has been
attempted by [110], both in axisymmetry and in full 3D. It has been found that the role of viscosity in a
hypermassive star generically leads to the formation of a compact, uniformly-rotating core, often unstable
to gravitational collapse, surrounded by a low-density disk. In some models viscous heating becomes
important enough to prevent the prompt collapse to a black hole, which is nevertheless the ultimate
outcome once the star cools.
On the other hand, the collapse of magnetized neutron stars in full general relativity has also been
numerically undertaken recently by [360, 105
]. Both works study the fate of a hypermassive neutron star as
it collapses to form a rotating black hole surrounded by a massive torus, a process which does not happen in
prompt timescales due to the transport of angular momentum via magnetic braking and the MRI. Analysis
of the lifetime and energetics of the black-hole–torus system formed in these computations suggests that the
collapse of hypermassive neutron stars is a possible candidate for the central engine of short gamma-ray
bursts.
In [27] the gravitational collapse of uniformly-rotating neutron stars to Kerr black holes was studied in
three dimensions, also using the cactus/whisky code for numerical relativity. Long-term simulations were
possible by excising the region of the computational domain that includes the curvature singularity when
the black hole forms and lies within the apparent horizon. It was found that for sufficiently high angular
velocities, the collapse can lead to the formation of a differentially-rotating unstable disk. Gravitational
waveforms from such simulations were reported in [28]. Despite good qualitative agreement being found
with waveforms obtained in previous axisymmetric simulations [386], the newly computed amplitudes are
about an order of magnitude smaller due to the more realistic rotation rates of the collapse
progenitors.
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More recently, [30] have succeeded in computing the gravitational waveforms from the gravitational
collapse of a neutron star to a rotating black hole well beyond what had so far been possible, even providing
information on the ring-down signal through which the formed black hole relaxes to its asymptotic,
stationary state. These new three-dimensional computations, contrary to most approaches, did not excise
the region inside the apparent horizon once the black hole formed in order to improve, in principle, the
numerical stability of the code by removing the curvature singularity. Instead, they were performed without
excision, which, combined with suitable singularity-avoiding gauge conditions and the use of
minute (artificial) Kreiss–Oliger type numerical dissipation in the spacetime evolution and gauge
equations, improved dramatically their long-term stability. An example of this remarkable result is
provided by Figure 9
, which shows a snapshot during the evolution of one of the collapse models
of [30
]1.
The figure displays the gravitational waveform train extracted by matching the numerical spacetime with
nonspherical perturbations of a Schwarzschild black hole described in terms of gauge-invariant odd and
even-parity metric perturbations, namely the lowest-order odd-parity multipole
. Moreover, in the
central parts of the figure the collapse of the lapse function and the formation of a rotating black hole are
visible. [30
] shows that (stellar mass) black-hole formation events would produce gravitational radiation
potentially detectable by the present generation of laser-interferometer detectors if the events took place at
Galactic distances.
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