4.3 Hydrodynamical evolution of neutron 4 Hydrodynamical Simulations in Relativistic 4.1 Gravitational collapse

4.2 Accretion onto black holes 

The study of relativistic accretion and black hole astrophysics is a very active field of research, both theoretically and observationally (see, e.g., [32] and references therein). On the one hand, advances in satellite instrumentation, such as the Rossi X-Ray Timing Explorer (RXTE) and the Advanced Satellite for Cosmology and Astrophysics (ASCA), are greatly stimulating and guiding theoretical research on accretion physics. The discovery of kHz quasi-periodic oscillations in low-mass X-ray binaries extends the frequency range over which these oscillations occur into timescales associated with the innermost regions of the accretion process (for a review, see [288]). Moreover, in extragalactic sources, spectroscopic evidence (broad iron emission lines) increasingly points to (rotating) black holes being the accreting central objects [282, 144, 47]. Thick accretion discs (or tori) are probably orbiting the central black holes of many astrophysical objects such as quasars and other active galactic nuclei (AGNs), some X-ray binaries, and microquasars. In addition, they are believed to exist at the central engine of cosmic GRBs.

Disk accretion theory is primarily based on the study of (viscous) stationary flows and their stability properties through linearized perturbations thereof. A well-known example is the solution consisting of isentropic constant angular momentum tori, unstable to a variety of non-axisymmetric global modes, discovered by Papaloizou and Pringle [223] (see [18] for a review of instabilities in astrophysical accretion disks). Since the pioneering work by Shakura and Sunyaev [253Jump To The Next Citation Point In The Article], thin disk models, parametrized by the so-called tex2html_wrap_inline4738 -viscosity, in which the gas rotates with Keplerian angular momentum which is transported radially by viscous stress, have been applied successfully to many astronomical objects. The thin disk model, however, is not valid for high luminosity systems, as it is unstable at high mass accretion rates. In this regime, Abramowicz et al. [3] found the so-called slim disk solution, which is stable against viscous and thermal instabilities. More recently, much theoretical work has been devoted to the problem of slow accretion, motivated by the discovery that many galactic nuclei are under-luminous (e.g., NGC 4258). Proposed accretion models involve the existence of advection-dominated accretion flows (ADAF solution; see, e.g., [202, 200]) and advection-dominated inflow-outflow solutions (ADIOS solution [33]). The importance of convection for low values of the viscosity parameter tex2html_wrap_inline4738 is currently being discussed in the so-called convection-dominated accretion flow (CDAF solution; see [130] and references therein). The importance of magnetic fields and their consequences for the stability properties of this solution are critically discussed in [17].

For a wide range of accretion problems, the Newtonian theory of gravity is adequate for the description of the background gravitational forces (see, e.g., [98]). Extensive experience with Newtonian astrophysics has shown that explorations of the relativistic regime benefit from the use of model potentials. In particular, the Paczynski-Wiita pseudo-Newtonian potential for a Schwarzschild black hole [217], gives approximations of general relativistic effects with an accuracy of tex2html_wrap_inline5340 outside the marginally stable radius (at r = 6 M, or three times the Schwarzschild radius). Nevertheless, for comprehensive numerical work a three-dimensional formalism is required, able to cover also the maximally rotating hole. In rotating spacetimes the gravitational forces cannot be captured fully with scalar potential formalisms. Additionally, geometric regions such as the ergosphere would be very hard to model without a metric description. Whereas the bulk of emission occurs in regions with almost Newtonian fields, only the observable features attributed to the inner region may crucially depend on the nature of the spacetime.

In the following, we present a summary of representative time-dependent accretion simulations in relativistic hydrodynamics. We concentrate on multi-dimensional simulations. In the limit of spherical accretion, exact stationary solutions exist for both Newtonian gravity [43] and relativistic gravity [179]. Such solutions are commonly used to calibrate time-dependent hydrodynamical codes, by analyzing whether stationarity is maintained during a numerical evolution [123Jump To The Next Citation Point In The Article, 163, 78, 244Jump To The Next Citation Point In The Article, 21].

4.2.1 Disk accretion 

Pioneering numerical efforts in the study of black hole accretion [300Jump To The Next Citation Point In The Article, 123Jump To The Next Citation Point In The Article, 120Jump To The Next Citation Point In The Article, 121Jump To The Next Citation Point In The Article] made use of the so-called frozen star paradigm of a black hole. In this framework, the time ``slicing'' of the spacetime is synchronized with that of asymptotic observers far from the hole. Within this approach, Wilson [300] first investigated numerically the time-dependent accretion of inviscid matter onto a rotating black hole. This was the first problem to which his formulation of the hydrodynamic equations, as presented in Section  2.1.2, was applied. Wilson used an axisymmetric hydrodynamical code in cylindrical coordinates to study the formation of shock waves and the X-ray emission in the strong-field regions close to the black hole horizon, and was able to follow the formation of thick accretion disks during the simulations.

Wilson's formulation has been extensively used in time-dependent numerical simulations of thick disk accretion. In a system formed by a black hole surrounded by a thick disk, the gas flows in an effective (gravitational plus centrifugal) potential, whose structure is comparable to that of a close binary. The Roche torus surrounding the black hole has a cusp-like inner edge located at the Lagrange point tex2html_wrap_inline5344, where mass transfer driven by the radial pressure gradient is possible [1]. In [123Jump To The Next Citation Point In The Article] (see also [120]), Hawley and collaborators studied, in the test fluid approximation and axisymmetry, the evolution and development of nonlinear instabilities in pressure-supported accretion disks formed as a consequence of the spiraling infall of fluid with some amount of angular momentum. The code used explicit second-order finite difference schemes with a variety of choices to integrate the transport terms of the equations (i.e., those involving changes in the state of the fluid at a specific volume in space). The code also used a staggered grid (with scalars located at the cell centers and vectors at the cell boundaries) for its suitability to finite difference the transport equations in a stable numerical way. Discontinuous solutions were stabilized with artificial viscosity terms.

With a three-dimensional extension of the axisymmetric code of Hawley et al. [122, 123Jump To The Next Citation Point In The Article], Hawley [121] studied the global hydrodynamic non-axisymmetric instabilities in thick, constant angular momentum accretion gas tori orbiting around a Schwarzschild black hole. Such simulations showed that the Papaloizu-Pringle instability saturates in a strong spiral pressure wave, not in turbulence. In addition, the simulations confirmed that accretion flows through the torus could reduce and even halt the growth of the global instability. Extensions to Kerr spacetimes have been recently reported in [62].

Yokosawa [313, 314], also using Wilson's formulation, studied the structure and dynamics of relativistic accretion disks and the transport of energy and angular momentum in magneto-hydrodynamical accretion onto a rotating black hole. In his code, the hydrodynamic equations are solved using the Flux-Corrected-Transport (FCT) scheme [46] (a second-order flux-limiter method that avoids oscillations near discontinuities by reducing the magnitude of the numerical flux), and the magnetic induction equation is solved using the constrained transport method [80]. The code contains a parametrized viscosity based on the tex2html_wrap_inline4738 -model [253]. The simulations revealed different flow patterns inside the marginally stable orbit, depending on the thickness h of the accretion disk. For thick disks with tex2html_wrap_inline5350, tex2html_wrap_inline5352 being the radius of the event horizon, the flow pattern becomes turbulent.

Igumenshchev and Beloborodov [131] have performed two-dimensional relativistic hydrodynamical simulations of inviscid transonic disk accretion onto a Kerr black hole. The hydrodynamical equations follow Wilson's formulation, but the code avoids the use of artificial viscosity. The advection terms are evaluated with an upwind algorithm that incorporates the PPM scheme [58] to compute the fluxes. Their numerical work confirms analytic expectations: (i) The structure of the innermost disk region strongly depends on the black hole spin, and (ii) the mass accretion rate goes as tex2html_wrap_inline5354, tex2html_wrap_inline5356 being the potential barrier between the inner edge of the disk and the cusp, and tex2html_wrap_inline4726 the adiabatic index.


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Figure 7: Runaway instability of an unstable thick disk: contour levels of the rest-mass density tex2html_wrap_inline4640 plotted at irregular times from t=0 to tex2html_wrap_inline5364, once the disk has almost been entirely destroyed. See [87Jump To The Next Citation Point In The Article] for details.

Furthermore, thick accretion disks orbiting black holes may be subjected to the so-called runaway instability, as first proposed by Abramowicz et al. [2]. Starting with an initial disk filling its Roche lobe, so that mass transfer is possible through the cusp located at the tex2html_wrap_inline5344 Lagrange point, two evolutions are feasible when the mass of the black hole increases: (i) Either the cusp moves inwards towards the black hole, which slows down the mass transfer, resulting in a stable situation, or (ii) the cusp moves deeper inside the disk material. In the latter case, the mass transfer speeds up, leading to the runaway instability. This instability, whose existence is still a matter of debate (see, e.g., [87Jump To The Next Citation Point In The Article] and references therein), is an important issue for most models of cosmic GRBs, where the central engine responsible for the initial energy release is such a system consisting of a thick disk surrounding a black hole.

In [87Jump To The Next Citation Point In The Article], Font and Daigne presented time-dependent simulations of the runaway instability of constant angular momentum thick disks around black holes. The study was performed using a fully relativistic hydrodynamics code based on HRSC schemes and the conservative formulation discussed in Section  2.1.3 . The self-gravity of the disk was neglected and the evolution of the central black hole was assumed to be that of a sequence of Schwarzschild black holes of varying mass. The black hole mass increase is determined by the mass accretion rate across the event horizon. In agreement with previous studies based on stationary models, [87Jump To The Next Citation Point In The Article] found that by allowing the mass of the black hole to grow the disk becomes unstable. For all disk-to-hole mass ratios considered (between 1 and 0.05), the runaway instability appears very fast on a dynamical timescale of a few orbital periods (tex2html_wrap_inline5368), typically a few tex2html_wrap_inline5370 and never exceeding tex2html_wrap_inline5372, for a tex2html_wrap_inline5374 black hole and a large range of mass fluxes (tex2html_wrap_inline5376).

An example of the simulations performed by [87Jump To The Next Citation Point In The Article] appears in Figure  7 . This figure shows eight snapshots of the time-evolution of the rest-mass density, from t =0 to tex2html_wrap_inline5380 . The contour levels are linearly spaced with tex2html_wrap_inline5382, where tex2html_wrap_inline5384 is the maximum value of the density at the center of the initial disk. In Figure  7 one can clearly follow the transition from a quasi-stationary accretion regime (panels (1) to (5)) to the rapid development of the runaway instability in about two orbital periods (panels (6) to (8)). At tex2html_wrap_inline5364, the disk has almost entirely disappeared inside the black hole whose size has, in turn, noticeably grown.

Extensions of this work to marginally stable (or even stable) constant angular momentum disks are reported in Zanotti et al. [239] (animations can be visualized at the website listed in [236Jump To The Next Citation Point In The Article]). Furthermore, recent simulations with non-constant angular momentum disks and rotating black holes [86], show that the instability is strongly suppressed when the specific angular momentum of the disk increases with the radial distance as a power law, tex2html_wrap_inline5388 . Values of tex2html_wrap_inline4738 much smaller than the one corresponding to the Keplerian angular momentum distribution suffice to supress the instability.

4.2.2 Jet formation 

Numerical simulations of relativistic jets propagating through progenitor stellar models of collapsars have been presented in [9Jump To The Next Citation Point In The Article]. The collapsar scenario, proposed by [307], is currently the most favoured model for explaining long duration GRBs. The simulations performed by [9Jump To The Next Citation Point In The Article] employ the three-dimensional code GENESIS [8] with a 2D spherical grid and equatorial plane symmetry. The gravitational field of the black hole is described by the Schwarzschild metric, and the relativistic hydrodynamic equations are solved in the test fluid approximation using a Godunov-type scheme. Aloy et al. [9] showed that the jet, initially formed by an ad hoc energy deposition of a few tex2html_wrap_inline5392 within a tex2html_wrap_inline5394 cone around the rotation axis, reaches the surface of the collapsar progenitor intact, with a maximum Lorentz factor of tex2html_wrap_inline5396 .

The most promising processes for producing relativistic jets like those observed in AGNs, microquasars, and GRBs involve the hydromagnetic centrifugal acceleration of material from the accretion disk [34], or the extraction of rotational energy from the ergosphere of a Kerr black hole [225, 36]. Koide and coworkers have performed the first MHD simulations of jet formation in general relativity [141, 140, 142Jump To The Next Citation Point In The Article]. Their code uses the 3+1 formalism of general relativistic conservation laws of particle number, momentum, and energy, and Maxwell equations with infinite electric conductivity. The MHD equations are numerically solved in the test fluid approximation (in the background geometry of Kerr spacetime) using a finite difference symmetric scheme [61]. The Kerr metric is described in Boyer-Lindquist coordinates, with a radial tortoise coordinate to enhance the resolution near the horizon.


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Figure 8: Jet formation: the twisting of magnetic field lines around a Kerr black hole (black sphere). The yellow surface is the ergosphere. The red tubes show the magnetic field lines that cross into the ergosphere. Figure taken from [142Jump To The Next Citation Point In The Article] (used with permission).

In [142Jump To The Next Citation Point In The Article], the general relativistic magneto-hydrodynamic behaviour of a plasma flowing into a rapidly rotating black hole (a =0.99995) in a large-scale magnetic field is investigated numerically. The initial magnetic field is uniform and strong, tex2html_wrap_inline5400, tex2html_wrap_inline5402 being the mass density. The initial Alfvén speed is tex2html_wrap_inline5404 . The simulation shows how the rotating black hole drags the inertial frames around in the ergosphere. The azimuthal component of the magnetic field increases because of the azimuthal twisting of the magnetic field lines, as is depicted in Figure  8 . This frame-dragging dynamo amplifies the magnetic field to a value which, by the end of the simulation, is three times larger than the initial one. The twist of the magnetic field lines propagates outwards as a torsional Alfvén wave. The magnetic tension torques the plasma inside the ergosphere in a direction opposite to that of the black hole rotation. Therefore, the angular momentum of the plasma outside receives a net increase. Even though the plasma falls into the black hole, electromagnetic energy is ejected along the magnetic field lines from the ergosphere, due to the propagation of the Alfvén wave. By total energy conservation arguments, Koide et al. [142] conclude that the ultimate result of the generation of an outward Alfvén wave is the magnetic extraction of rotational energy from the Kerr black hole, a process the authors call the MHD Penrose process. Koide and coworkers argue that such a process can be applicable to jet formation, both in AGNs and GRBs.

We note that, recently, van Putten and Levinson [292] have considered, theoretically, the evolution of an accretion torus in suspended accretion, in connection with GRBs. These authors claim that the formation of baryon-poor outflows may be associated with a dissipative gap in a differentially rotating magnetic flux tube supported by an equilibrium magnetic moment of the black hole. Numerical simulations of non-ideal MHD, incorporating radiative processes, are necessary to validate this picture.

4.2.3 Wind accretion 

The term ``wind'' or hydrodynamic accretion refers to the capture of matter by a moving object under the effects of the underlying gravitational field. The canonical astrophysical scenario in which matter is accreted in such a non-spherical way was suggested originally by Bondi, Hoyle, and Lyttleton [125, 44], who studied, using Newtonian gravity, the accretion onto a gravitating point mass moving with constant velocity through a non-relativistic gas of uniform density. The matter flow inside the accretion radius, after being decelerated by a conical shock, is ultimately captured by the central object. Such a process applies to describe mass transfer and accretion in compact X-ray binaries, in particular in the case in which the donor (giant) star lies inside its Roche lobe and loses mass via a stellar wind. This wind impacts on the orbiting compact star forming a bow-shaped shock front around it. This process is also believed to occur during the common envelope phase in the evolution of a binary system.

Numerical simulations have extended the simplified analytic models and have helped to develop a thorough understanding of the hydrodynamic accretion scenario, in its fully three-dimensional character (see, e.g., [245Jump To The Next Citation Point In The Article, 27Jump To The Next Citation Point In The Article] and references therein). The numerical investigations have revealed the formation of accretion disks and the appearance of non-trivial phenomena such as shock waves and tangential (flip-flop) instabilities.

Most of the existing numerical work has used Newtonian hydrodynamics to study the accretion onto non-relativistic stars [245]. For compact accretors such as neutron stars or black holes, the correct numerical modeling requires a general relativistic hydrodynamical description. Within the relativistic, frozen star framework, wind accretion onto ``moving'' black holes was first studied in axisymmetry by Petrich et al. [226Jump To The Next Citation Point In The Article]. In this work, Wilson's formulation of the hydrodynamic equations was adopted. The integration algorithm was borrowed from [277] with the transport terms finite-differenced following the prescription given in [123]. An artificial viscosity term of the form tex2html_wrap_inline5406, with a a being constant, was added to the pressure terms of the equations in order to stabilize the numerical scheme in regions of sharp pressure gradients.

An extensive survey of the morphology and dynamics of relativistic wind accretion past a Schwarzschild black hole was later performed by [91Jump To The Next Citation Point In The Article, 90Jump To The Next Citation Point In The Article]. This investigation differs from [226Jump To The Next Citation Point In The Article] in both the use of a conservative formulation for the hydrodynamic equations (see Section  2.1.3) and the use of advanced HRSC schemes. Axisymmetric computations were compared to [226], finding major differences in the shock location, opening angle, and accretion rates of mass and momentum. The reasons for the discrepancies are related to the use of different formulations, numerical schemes, and grid resolution, much higher in [91, 90Jump To The Next Citation Point In The Article]. Non-axisymmetric two-dimensional studies, restricted to the equatorial plane of the black hole, were discussed in [90], motivated by the non-stationary patterns found in Newtonian simulations (see, e.g., [27]). The relativistic computations revealed that initially asymptotic uniform flows always accrete onto the hole in a stationary way that closely resembles the previous axisymmetric patterns.


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Figure 9: Relativistic wind accretion onto a rapidly rotating Kerr black hole ( a=0.999 M, the hole spin is counter-clock wise) in Kerr-Schild coordinates (left panel). Isocontours of the logarithm of the density are plotted at the final stationary time t =500 M . Brighter colors (yellow-white) indicate high density regions while darker colors (blue) correspond to low density zones. The right panel shows how the flow solution looks when transformed to Boyer-Lindquist coordinates. The shock appears here totally wrapped around the horizon of the black hole. The box is 12 M units long. The simulation employed a tex2html_wrap_inline4652 -grid of tex2html_wrap_inline4654 zones. Further details are given in [94Jump To The Next Citation Point In The Article].

In [219], Papadopoulos and Font presented a procedure that simplifies the numerical integration of the general relativistic hydrodynamic equations near black holes. This procedure is based on identifying classes of coordinate systems in which the black hole metric is free of coordinate singularities at the horizon (unlike the commonly adopted Boyer-Lindquist coordinates), independent of time, and admits a spacelike decomposition. With those coordinates the innermost radial boundary can be placed inside the horizon, allowing for an unambiguous treatment of the entire (exterior) physical domain. In [94, 95] this approach was applied to the study of relativistic wind accretion onto rapidly rotating black holes. The effects of the black hole spin on the flow morphology were found to be confined to the inner regions of the black hole potential well. Within this region, the black hole angular momentum drags the flow, wrapping the shock structure around. An illustrative example is depicted in Figure  9 . The left panel of this figure corresponds to a simulation employing the Kerr-Schild form of the Kerr metric, regular at the horizon. The right panel shows how the accretion pattern would look if it were the computation performed using the more common Boyer-Lindquist coordinates. The transformation induces a noticeable wrapping of the shock around the central hole. The shock would wrap infinitely many times before reaching the horizon. As a result, the computation in these coordinates would be much more challenging than in Kerr-Schild coordinates.

4.2.4 Gravitational radiation 

Semi-analytical studies of finite-sized collections of dust, shaped in the form of stars or shells and falling isotropically onto a black hole are available in the literature [198, 118, 256, 213, 227]. These investigations approximate gravitational collapse by a dust shell of mass m falling into a Schwarzschild black hole of mass tex2html_wrap_inline5422 . These studies have shown that for a fixed amount of infalling mass, the gravitational radiation efficiency is reduced compared to the point particle limit (tex2html_wrap_inline5424), due to cancellations of the emission from distinct parts of the extended object.


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Figure 10: Movie showing the time evolution of the accretion/collapse of a quadrupolar shell onto a Schwarzshild black hole. The left panel shows isodensity contours and the right panel the associated gravitational waveform. The shell, initially centered at tex2html_wrap_inline4656, is gradually accreted by the black hole, a process that perturbs the black hole and triggers the emission of gravitational radiation. After the burst, the remaining evolution shows the decay of the black hole quasi-normal mode ringing. By the end of the simulation a spherical accretion solution is reached. Further details are given in [220Jump To The Next Citation Point In The Article]. (Editorial note: This movie has been reduced in size substantially compared to the original version provided by the author. To download the author's original version please go to this article's download page.)

In [220Jump To The Next Citation Point In The Article], such conclusions were corroborated with numerical simulations of the gravitational radiation emitted during the accretion process of an extended object onto a black hole. The first-order deviations from the exact black hole geometry were approximated using curvature perturbations induced by matter sources whose nonlinear evolution was integrated using a (nonlinear) hydrodynamics code (adopting the conservative formulation of Section  2.1.3 and HRSC schemes). All possible types of curvature perturbations are captured in the real and imaginary parts of the Weyl tensor scalar (see, e.g., [55]). In the framework of the Newman-Penrose formalism, the equations for the perturbed Weyl tensor components decouple, and when written in the frequency domain, they even separate [285]. Papadopoulos and Font [220Jump To The Next Citation Point In The Article] used the limiting case for Schwarzschild black holes, i.e., the inhomogeneous Bardeen-Press equation [23]. The simulations showed the gradual excitation of the black hole quasi-normal mode frequency by sufficiently compact shells.

An example of these simulations appears in the movie of Figure  10 . This movie shows the time evolution of the shell density (left panel) and the associated gravitational waveform during a complete accretion/collapse event. The (quadrupolar) shell is parametrized according to tex2html_wrap_inline5428 with tex2html_wrap_inline5430 and tex2html_wrap_inline5432 . Additionally, tex2html_wrap_inline5434 denotes a logarithmic radial (Schwarzschild) coordinate. The animation shows the gradual collapse of the shell onto the black hole. This process triggers the emission of gravitational radiation. In the movie, one can clearly see how the burst of the emission coincides with the most dynamical accretion phase, when the shell crosses the peak of the potential and is subsequently captured by the hole. The gravitational wave signal coincides with the quasinormal ringing frequency of the Schwarzschild black hole, 17 M . The existence of an initial burst, separated in time from the physical burst, is also noticeable in the movie. It just reflects the gravitational radiation content of the initial data (see [220] for a detailed explanation).

One-dimensional numerical simulations of a self-gravitating perfect fluid accreting onto a black hole were presented in [222Jump To The Next Citation Point In The Article], where the effects of mass accretion during the gravitational wave emission from a black hole of growing mass were explored. Using the conservative formulation outlined in Section  2.2.2 and HRSC schemes, Papadopoulos and Font [222] performed the simulations adopting an ingoing null foliation of a spherically symmetric black hole spacetime [221]. Such a foliation penetrates the black hole horizon, allowing for an unambiguous numerical treatment of the inner boundary. The essence of non-spherical gravitational perturbations was captured by adding the evolution equation for a minimally coupled massless scalar field to the (characteristic) Einstein-perfect fluid system. The simulations showed the familiar damped-oscillatory radiative decay, with both decay rate and frequencies being modulated by the mass accretion rate. Any appreciable increase in the horizon mass during the emission reflects on the instantaneous signal frequency f, which shows a prominent negative branch in the tex2html_wrap_inline5440 evolution diagram. The features of the frequency evolution pattern reveal key properties of the accretion event, such as the total accreted mass and the accretion rate.

Recently, Zanotti et al. [317Jump To The Next Citation Point In The Article] have performed hydrodynamical simulations of constant angular momentum thick disks (of typical neutron star densities) orbiting a Schwarzschild black hole. Upon the introduction of perturbations, these systems either become unstable to the runaway instability [87] or exhibit a regular oscillatory behaviour resulting in a quasi-periodic variation of the accretion rate as well as of the mass quadrupole (animations can be visualized at the website listed in [236]). Zanotti et al. [317] have found that the latter is responsible for the emission of intense gravitational radiation whose amplitude is comparable or larger than the one expected in stellar core collapse. The strength of the gravitational waves emitted and their periodicity are such that signal-to-noise ratios tex2html_wrap_inline5442 can be reached for sources at tex2html_wrap_inline5444 or tex2html_wrap_inline5446, respectively, making these new sources of gravitational waves potentially detectable.

4.3 Hydrodynamical evolution of neutron 4 Hydrodynamical Simulations in Relativistic 4.1 Gravitational collapse

image Numerical Hydrodynamics in General Relativity
José A. Font
© Max-Planck-Gesellschaft. ISSN 1433-8351
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