We carry out numerical simulations of the gravitational collapse of a rotating perfect fluid with the ultrarelativistic equation of state $P=κρ$, in axisymmetry in $2+1$ spacetime dimensions with $Λ<0$. We show that for $κ\lesssim 0.42$, the critical phenomena are type I and the critical solution is stationary. The picture for $κ\gtrsim 0.43$ is more delicate: for small angular momenta, we find type II phenomena and the critical solution is quasistationary, contracting adiabatically. The spin-to-mass ratio of the critical solution increases as it contracts, and hence so does that of the black hole created at the end as we fine-tune to the black-hole threshold. Forming extremal black holes is avoided because the contraction of the critical solution smoothly ends as extremality is approached.
In this paper we present a simple but non-simplistic model of gravitational collapse with thermal emission of pre-Hawking radiation. We apply Einstein equations to a time-dependant spherically symmetric metric and an ultrarelativistic stress-energy tensor. In our model, particles either radially approach the center of the star as collapsing matter, or radially flee from it.
We carry out numerical simulations of the gravitational collapse of a perfect fluid with the ultrarelativistic equation of state $P=κρ$, in spherical symmetry in $2+1$ spacetime dimensions with $Λ<0$. At the threshold of prompt collapse, we find type II critical phenomena (apparent horizon mass and maximum curvature scale as powers of distance from the threshold) for $κ\gtrsim 0.43$, and type I critical phenomena (lifetime scales as logarithm of distance from the threshold) for $κ\lesssim 0.42$. The type I critical solution is static, while the type II critical solution is not self-similar (as…
In this paper we investigate the critical collapse of an ultrarelativistic perfect fluid with the equation of state $P=(Γ-1)ρ$ in the limit of $Γ\to 1$. We calculate the limiting continuously self similar (CSS) solution and the limiting scaling exponent by exploiting self-similarity of the solution. We also solve the complete set of equations governing the gravitational collapse numerically for $(Γ-1) = 10^{-2},...,10^{-6}$ and compare them with the CSS solutions. We also investigate the supercritical regime and discuss the hypothesis of naked singularity formation in a generic gravitational c…
This is the first of two papers examining the critical collapse of spherically symmetric perfect fluids with the equation of state P = (Gamma -1)rho. Here we present the equations of motion and describe a computer code capable of simulating the extremely relativistic flows encountered in critical solutions for Gamma <= 2. The fluid equations are solved using a high-resolution shock-capturing scheme based on a linearized Riemann solver.
We present a new numerical code designed to solve the Einstein field equations for axisymmetric spacetimes. The long term goal of this project is to construct a code that will be capable of studying many problems of interest in axisymmetry, including gravitational collapse, critical phenomena, investigations of cosmic censorship, and head-on black hole collisions. Our objective here is to detail the (2+1)+1 formalism we use to arrive at the corresponding system of equations and the numerical methods we use to solve them. We are able to obtain stable evolution, despite the singular nature of th…
Motivated by a recent paper by the Potsdam numerical relativity group, we have constructed a new numerical code for hydrodynamic simulation of axisymmetric systems in full general relativity. In this code, we solve the Einstein field equation using Cartesian coordinates with appropriate boundary conditions. On the other hand, the hydrodynamic equations are solved in cylindrical coordinates. Using this code, we perform simulations to study axisymmetric collapse of rotating stars, which thereby become black holes or new compact stars, in full general relativity. To investigate the effects of rot…
We numerically investigate the dynamics near black hole formation of solutions to the Einstein--Vlasov system in axisymmetry. Our results are obtained using a particle-in-cell and finite difference code based on the $(2+1)+1$ formulation of the Einstein field equations in axisymmetry. Solutions are launched from non-stationary initial data and exhibit type I critical behaviour. In particular, we find lifetime scaling in solutions containing black holes, and support that the critical solutions are stationary. Our results contain examples of solutions that form black holes, perform damped oscill…
We report a new critical solution found at the threshold of axisymmetric gravitational collapse of a complex scalar field with angular momentum. To carry angular momentum the scalar field cannot be axisymmetric; however, its azimuthal dependence is defined so that the resulting stress energy tensor and spacetime metric are axisymmetric. The critical solution found is non-spherical, discretely self-similar with an echoing exponent of 0.42 (+- 4%), and exhibits a scaling exponent of 0.11 (+- 10%) in near critical collapse. Our simulations suggest that the solution is universal (within the impose…
The inequality $\sqrt{J}\leq m$ is proved for vacuum, asymptotically flat, maximal and axisymmetric data close to extreme Kerr data. The physical significance of this inequality and its relation to the standard picture of the gravitational collapse are discussed.
In numerical relativity, spacetimes involving compact strongly gravitating objects are constructed as numerical solutions of Einstein's equations. Success of such a process strongly depends on the availability of appropriate coordinates, which are typically constructed dynamically. A very robust coordinate choice is a so-called moving puncture gauge, commonly used for numerical simulations of black hole spacetimes. Nevertheless it is known to fail for evolving near-critical Brill wave data. We construct a new `quasi-maximal' slicing condition and demonstrate that it exhibits better behavior fo…
Axisymmetric numerical simulations of rotating stellar core collapse to a neutron star are performed in the framework of full general relativity. The so-called Cartoon method, in which the Einstein field equations are solved in the Cartesian coordinates and the axisymmetric condition is imposed around the $y=0$ plane, is adopted. The hydrodynamic equations are solved in the cylindrical coordinates (on the $y=0$ plane in the Cartesian coordinates) using a high-resolution shock-capturing scheme with the maximum grid size $(2500,2500)$. A parametric equation of state is adopted to model collapsin…
We demonstrate that evolutions of three-dimensional, strongly non-linear gravitational waves can be followed in numerical relativity, hence allowing many interesting studies of both fundamental and observational consequences. We study the evolution of time-symmetric, axisymmetric {\it and} non-axisymmetric Brill waves, including waves so strong that they collapse to form black holes under their own self-gravity. The critical amplitude for black hole formation is determined. The gravitational waves emitted in the black hole formation process are compared to those emitted in the head-on collisio…
We report a new implementation for axisymmetric simulation in full general relativity. In this implementation, the Einstein equations are solved using the Nakamura-Shibata formulation with the so-called cartoon method to impose an axisymmetric boundary condition, and the general relativistic hydrodynamic equations are solved using a high-resolution shock-capturing scheme based on an approximate Riemann solver. As tests, we performed the following simulations: (i) long-term evolution of non-rotating and rapidly rotating neutron stars, (ii) long-term evolution of neutron stars of a high-amplitud…
We present results from axisymmetric stellar core collapse simulations in general relativity. Our hydrodynamics code has proved robust and accurate enough to allow for a detailed analysis of the global dynamics of the collapse. Contrary to traditional approaches based on the 3+1 formulation of the gravitational field equations, our framework uses a foliation based on a family of outgoing light cones, emanating from a regular center, and terminating at future null infinity. Such a coordinate system is well adapted to the study of interesting dynamical spacetimes in relativistic astrophysics suc…
We present results from a numerical study of critical gravitational collapse of axisymmetric distributions of massless scalar field energy. We find threshold behavior that can be described by the spherically symmetric critical solution with axisymmetric perturbations. However, we see indications of a growing, non-spherical mode about the spherically symmetric critical solution. The effect of this instability is that the small asymmetry present in what would otherwise be a spherically symmetric self-similar solution grows. This growth continues until a bifurcation occurs and two distinct region…
We have carried out an extensive set of two-dimensional, axisymmetric, purely-hydrodynamic calculations of rotational stellar core collapse with a realistic, finite-temperature nuclear equation of state and realistic massive star progenitor models. For each of the total number of 72 different simulations we performed, the gravitational wave signature was extracted via the quadrupole formula in the slow-motion, weak-field approximation. We investigate the consequences of variation in the initial ratio of rotational kinetic energy to gravitational potential energy and in the initial degree of di…
Gravitational waves from core collapse of rotating very-massive stars: 3D numerical relativity computation
The Pennsylvania State University（宾夕法尼亚州立大学）；Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut)（马克斯·普朗克引力物理学研究所（阿尔伯特·爱因斯坦研究所））；Kyoto University（京都大学）；Tohoku University（东北大学）
We investigate spherically-symmetric, general relativistic systems of collapsing perfect fluid distributions. We consider neutron star models that are driven to collapse by the addition of an initially "in-going" velocity profile to the nominally static star solution. The neutron star models we use are Tolman-Oppenheimer-Volkoff solutions with an initially isentropic, gamma-law equation of state. The initial values of 1) the amplitude of the velocity profile, and 2) the central density of the star, span a parameter space, and we focus only on that region that gives rise to Type II critical beh…
We present a new pseudospectral code, bamps, for numerical relativity written with the evolution of collapsing gravitational waves in mind. We employ the first order generalized harmonic gauge formulation. The relevant theory is reviewed and the numerical method is critically examined and specialized for the task at hand. In particular we investigate formulation parameters, gauge and constraint preserving boundary conditions well-suited to non-vanishing gauge source functions. Different types of axisymmetric twist-free moment of time symmetry gravitational wave initial data are discussed. A tr…
(Abridged.) The accretion-induced collapse (AIC) of a white dwarf (WD) may lead to the formation of a protoneutron star and a collapse-driven supernova explosion. This process represents a path alternative to thermonuclear disruption of accreting white dwarfs in Type Ia supernovae. Neutrino and gravitational-wave (GW) observations may provide crucial information necessary to reveal a potential AIC. Motivated by the need for systematic predictions of the GW signature of AIC, we present results from an extensive set of general-relativistic AIC simulations using a microphysical finite-temperature…
We present a new theory for the gravitational wave signatures of core-collapse supernovae. Previous studies identified axisymmetric rotating core collapse, core bounce, postbounce convection, and anisotropic neutrino emission as the primary processes and phases for the radiation of gravitational waves. Our results, which are based on axisymmetric, Newtonian radiation-hydrodynamics supernova simulations (Burrows et al. 2006), indicate that the dominant emission process of gravitational waves in core-collapse supernovae may be the oscillations of the protoneutron star core. The oscillations are …
Jin et al reported that axisymmetric simulations of NS-like objects with polytropic EOS undergo critical gravitational collapse. As the critical collapse observed via fine-tuning of the adiabatic index $Γ$, they conjecture that critical phenomena may occur in realistic astrophysical scenarios. To clarify the implications this numerical observation has on realistic astrophysical scenarios, here, we perform dynamical analysis on the structure of the critical collapse observed in the former work. We report the time scales and oscillation frequencies exhibited by the critical solution and compare …
This paper is concerned with the Einstein equations in axisymmetric vacuum spacetimes. We consider numerical evolution schemes that solve the constraint equations as well as elliptic gauge conditions at each time step. We examine two such schemes that have been proposed in the literature and show that some of their elliptic equations are indefinite, thus potentially admitting nonunique solutions and causing numerical solvers based on classical relaxation methods to fail. A new scheme is then presented that does not suffer from these problems. We use our numerical implementation to study the gr…
Using predictions from three-dimensional (3D) hydrodynamics simulations of core-collapse supernovae (CCSNe), we present a coherent network analysis to detection, reconstruction, and the source localization of the gravitational-wave (GW) signals. We use the {\tt RIDGE} pipeline for the analysis, in which the network of LIGO Hanford, LIGO Livingston, VIRGO, and KAGRA is considered. By combining with a GW spectrogram analysis, we show that several important hydrodynamics features in the original waveforms persist in the waveforms of the reconstructed signals. The characteristic excess in the spec…
We perform fully general relativistic simulations of rotating stellar core collapse in three spatial dimension. A parametric equation of state is adopted following Dimmelmeier et al. The early stage of the collapse is followed by an axisymmetric code. When the stellar core becomes compact enough, we start a 3-dimensional simulation adding a bar-mode nonaxisymmetric density perturbation. In the axisymmetric simulations, it is clarified that the maximum value of $β\equiv T/W$ achieved during the stellar collapse and depends sensitively on the velocity profile and total mass of the initial core, …
We study event horizons of non-axisymmetric black holes and show how features found in axisymmetric studies of colliding black holes and of toroidal black holes are non-generic and how new features emerge. Most of the details of black hole formation and black hole merger are known only in the axisymmetric case, in which numerical evolution has successfully produced dynamical space-times. The work that is presented here uses a new approach to construct the geometry of the event horizon, not by locating it in a given spacetime, but by direct construction. In the axisymmetric case, our method pro…
Several interesting astrophysical phenomena are symmetric with respect to the rotation axis, like the head-on collision of compact bodies, the collapse and/or accretion of fields with a large variety of geometries, or some forms of gravitational waves. Most current numerical relativity codes, however, can not take advantage of these symmetries due to the fact that singularities in the adapted coordinates, either at the origin or at the axis of symmetry, rapidly cause the simulation to crash. Because of this regularity problem it has become common practice to use full-blown Cartesian three-dime…
Comparing twist-free axisymmetric gravitational waves near the black hole threshold
Universidade de Lisboa（里斯本大学）；Charles University（查理大学）；University of Cambridge（剑桥大学）；Bowdoin College（鲍登学院）；Friedrich-Schiller-Universität, Jena（耶拿弗里德里希·席勒大学）
We study the neutron star axisymmetric critical solution previously found in the numerical studies of neutron star mergers. Using neutron star-like initial data and performing similar merger simulations, we demonstrate that the solution is indeed a semi-attractor on the threshold plane separating the basin of a neutron star and the basin of a black hole in the solution space of the Einstein equations. In order to explore the extent of the attraction basin of the neutron star semiattractor, we construct initial data phase spaces for these neutron star-like initial data. From these phase spaces,…
This paper presents a quasi-local method of studying the physics of dynamical black holes in numerical simulations. This is done within the dynamical horizon framework, which extends the earlier work on isolated horizons to time-dependent situations. In particular: (i) We locate various kinds of marginal surfaces and study their time evolution. An important ingredient is the calculation of the signature of the horizon, which can be either spacelike, timelike, or null. (ii) We generalize the calculation of the black hole mass and angular momentum, which were previously defined for axisymmetric …
We carry out numerical simulations of the collapse of a complex rotating scalar field of the form $Ψ(t,r,θ)=e^{imθ}Φ(t,r)$, giving rise to an axisymmetric metric, in 2+1 spacetime dimensions with cosmological constant $Λ<0$, for $m=0,1,2$, for four 1-parameter families of initial data. We look for the familiar scaling of black hole mass and maximal Ricci curvature as a power of $|p-p_*|$, where $p$ is the amplitude of our initial data and $p_*$ some threshold. We find evidence of Ricci scaling for all families, and tentative evidence of mass scaling for most families, but the case $m>0$ is ver…
We study the final state of the gravitational collapse of uniformly rotating supramassive neutron stars by axisymmetric simulations in full general relativity. The rotating stars provided as the initial condition are marginally stable against quasiradial gravitational collapse and its equatorial radius rotates with the Kepler velocity (i.e., the star is at the mass-shedding limit). To model the neutron stars, we adopt the polytropic equations of state for a wide range of the polytropic index as $n=2/3$, 4/5, 1, 3/2 and 2. We follow the formation and evolution of the black holes, and show that …
No Kerr-like exact solution has yet been found in Chern-Simons modified gravity. Intrigued by this absence, we study stationary and axisymmetric metrics that could represent the exterior field of spinning black holes. For the standard choice of the background scalar, the modified field equations decouple into the Einstein equations and additional constraints. These constraints eliminate essentially all solutions except for Schwarzschild. For non-canonical choices of the background scalar, we find several exact solutions of the modified field equations, including mathematical black holes and pp…
We investigate the collapse of differentially rotating supermassive stars (SMSs) by means of 3+1 hydrodynamic simulations in general relativity. We particularly focus on the onset of collapse to understand the final outcome of collapsing SMSs. We find that the estimated ratio of the mass between the black hole (BH) and the surrounding disk from the equilibrium star is roughly the same as the results from numerical simulation. This suggests that the picture of axisymmetric collapse is adequate, in the absence of nonaxisymmetric instabilities, to illustrate the final state of the collapse. We al…
In this thesis we present the first numerical study of gravitational collapse in braneworlds within the framework of the single brane model proposed by Randall and Sundrum (RSII). We directly show that the evolutions of sufficiently strong initial data configurations result in black holes (BHs) with finite extension into the bulk. The extension changes from sphere to pancake (or cigar, seen from a different perspective) as the size of BH increases. We find preliminary evidences that BHs of the same size generated from distinct initial data profiles are geometrically indistinguishable. As such,…
Primordial black holes (PBHs) are of fundamental interest in cosmology and astrophysics, and have received much attention as a dark matter candidate and as a potential source of gravitational waves. One possible PBH formation mechanism is the gravitational collapse of cosmic strings. Thus far, the entirety of the literature on PBH production from cosmic strings has focused on the collapse of (quasi)circular cosmic string loops, which make up only a tiny fraction of the cosmic loop population. We demonstrate here a novel PBH formation mechanism: the collapse of a small segment of cosmic string …
We review various approaches to approximating general relativistic effects in hydrodynamic simulations of stellar core collapse and post-bounce evolution. Different formulations of a modified Newtonian gravitational potential are presented. Such an effective relativistic potential can be used in an otherwise standard Newtonian hydrodynamic code. An alternative approximation of general relativity is the assumption of conformal flatness for the three-metric, and its extension by adding second post-Newtonian order terms. Using a code which evolves the coupled system of metric and fluid equations,…
Self-gravitating relativistic disks around black holes can form as transient structures in a number of astrophysical scenarios such as binary neutron star and black hole-neutron star coalescences, as well as the core-collapse of massive stars. We explore the stability of such disks against runaway and non-axisymmetric instabilities using three-dimensional hydrodynamics simulations in full general relativity using the THOR code. We model the disk matter using the ideal fluid approximation with a $Γ$-law equation of state with $Γ=4/3$. We explore three disk models around non-rotating black holes…
We present a new 3D general relativistic hydrodynamics code for simulations of stellar core collapse to a neutron star, as well as pulsations and instabilities of rotating relativistic stars. It uses spectral methods for solving the metric equations, assuming the conformal flatness approximation for the three-metric. The matter equations are solved by high-resolution shock-capturing schemes. We demonstrate that the combination of a finite difference grid and a spectral grid can be successfully accomplished. This "Mariage des Maillages" (French for grid wedding) approach results in high accurac…
An axisymmetric collapse of non-rotating gravitational waves is numerically investigated in the subcritical regime where no black holes form but where curvature attains a maximum and decreases, following the dispersion of the initial wave packet. We focus on a curvature invariant with dimensions of length, and find that near the threshold for black hole formation it reaches a maximum along concentric rings of finite radius around the axis. In this regime the maximal value of the invariant exhibits a power-law scaling with the approximate exponent 0.38, as a function of a parametric distance fr…
We present the first numerical result of fully general relativistic axisymmetric simulations for the collapse of a rotating high-entropy stellar core to a black hole and an accretion disk. The simulations are performed taking into account the relevant microphysics. We adopt as initial condition a spherical core with constant electron fraction ($Y_e = 0.5$) and entropy per baryon $s$ = 8 $k_B$, and angular velocity is superimposed. In the early phase, the core collapses in a homologous manner. Then, it experiences a weak bounce due to the gas pressure of free nucleons. Because the bounce is wea…
