Critical Gravitational Collapse Research Articles

Critical gravitational collapse of a massive complex scalar field

We study the critical collapse of a massive complex scalar field coupled minimally to gravity. Taking as initial data a simple gaussian pulse with a shape similar to the harmonic ansatz for boson stars, we obtain critical collapse of type type I and II when varying the gaussian width $\sigma$. For $\sigma \leq 0.5$ we find collapse of type II with a critical exponent $\gamma=0.38\pm0.01$ and an echoing period $\Delta=3.4\pm0.1$. These values are very similar to the known results for a real massless scalar field. On the other hand, for $\sigma \geq 2.5$ we obtain collapse of type I. In this case we find that the critical solutions turn out to be an unstable boson stars in the ground state: all the data obtained from our simulations can be contrasted with the characteristic values for unstable boson stars and their corresponding Lyapunov exponents.

Critical gravitational collapse of a nonminimally coupled scalar field

We study the critical gravitational collapse of a massless scalar field non-minimally coupled to gravity, using a quadratic coupling function with a strength parameter $\xi$. We concentrate on critical phenomena of type II, and determine with an accuracy of at least $10^{-12}$ the value of the critical amplitude for collapse to a black hole, as well as the values of the critical and echoing exponents. Obtaining such high accuracy in the critical amplitude requires us to do a coordinate radial transformation that effectively increases resolution near the central regions by a factor of at least $10^3$. As expected, we find that for the case of small coupling the critical behaviour is very similar to that of a minimally coupled scalar field. On the other hand, for high coupling the dynamics become so violent that we need to introduce a special slicing condition, known as the shock-avoiding slicing condition, in order to avoid gauge pathologies that would otherwise cause our simulations to fail. With this new gauge condition we are able to perform high accuracy simulations even in the strong coupling regime, where we find that the critical and echoing exponents become significantly smaller, and that the echoing behavior is richer and can not be modelled by a single harmonic.

Gravitational field of a pit and maximal mass defects

A general relativistic solution, composed of a Zel'dovich-Letelier interior made of radial strings matched through a spherical thin shell at radius $r_0$ to an exterior Schwarzschild solution with mass $m$, is presented. It is the Zel'dovich-Letelier-Schwarzschild star. When the radius $r_0$ of the star is shrunk to its gravitational radius $2m$, the solutions have interesting properties. There are solutions with $m=0$ and $r_0=0$ that obey $\frac{2m}{r_0}=1$. The solutions have a horizon, but are not black holes, they are quasiblack holes, though atypical. The proper mass $m_p$ of the interior is nonzero and made of one string. Hence, a Minkowski exterior space hides an interior with matter in a pit. These are the pit solutions and show a maximal mass defect. There are two classes of pit solutions, one a finite string and the other a semi-infinite one. These pits are really string pits, that can be seen as Wheeler bags of gold, albeit squashed bags. There is another class, which is a compact stringy star at the $\frac{2m}{r_0}=1$ limit with $m$ nonzero. It is a typical quasiblack hole and it has maximal mass defect. Generically pit solutions with $\frac{2m}{r_0}=1$ and $m=0$ can exist with maximal mass defects. The Zel'dovich-Letelier-Schwarzschild star at the $r_0=2m$ limit is an instance of it. These three classes of static solutions yield the same spectrum of solutions that appear in critical gravitational collapse, there are solutions that yield naked null singularities, which here are the two string pit classes, there are solutions that yield black holes, which here are represented by the compact stringy stars at the quasiblack hole limit, and the solutions that disperse away in critical collapse here are the static Zel'dovich-Letelier-Schwarzschild stars. Thermodynamics of the string pit and stringy star quasiblack hole solutions is provided, and other connections are mentioned.

Gravitational collapse in SYK models and Choptuik-like phenomenon

SYK model is a quantum mechanical model of fermions which is solvable at strong coupling and plays an important role as perhaps the simplest holographic model of quantum gravity and black holes. The present work considers a deformed SYK model and a sudden quantum quench in the deformation parameter. The system, as in the undeformed case, permits a low energy description in terms of pseudo Nambu Goldstone modes. The bulk dual of such a system represents a gravitational collapse, which is characterized by a bulk matter stress tensor whose value near the boundary shows a sudden jump at the time of the quench. The resulting gravitational collapse forms a black hole only if the deformation parameter ∆\U0001d716 exceeds a certain critical value ∆\U0001d716c and forms a horizonless geometry otherwise. In case a black hole does form, the resulting Hawking temperature is given by a fractional power Tbh∝ (∆\U0001d716 − ∆ \U0001d716c)1/2, which is reminiscent of the ‘Choptuik phenomenon’ of critical gravitational collapse.

Critical gravitational collapse with angular momentum. II. Soft equations of state

We study critical phenomena in the collapse of rotating ultrarelativistic perfect fluids, in which the pressure $P$ is related to the total energy density $\rho$ by $P=\kappa\rho$, with $\kappa$ a constant. We generalize earlier results for radiation fluids with $\kappa=1/3$ to other values of $\kappa$, focussing on $\kappa < 1/9$. For $1/9<\kappa \lesssim 0.49$, the critical solution has only one unstable, growing mode, which is spherically symmetric. For supercritical data it controls the black hole mass, while for subcritical data it controls the maximum density. For $\kappa<1/9$, an additional axial $l=1$ mode becomes unstable. This controls either the black hole angular momentum, or the maximum angular velocity. In theory, the additional unstable $l=1$ mode changes the nature of the black hole threshold completely: at sufficiently large initial rotation rates $\Omega$ and sufficient fine-tuning of the initial data to the black hole threshold we expect to observe nontrivial universal scaling functions (familiar from critical phase transitions in thermodynamics) governing the black hole mass and angular momentum, and, with further fine-tuning, eventually a finite black hole mass almost everywhere on the threshold. In practice, however, the second unstable mode grows so slowly that we do not observe this breakdown of scaling at the level of fine-tuning we can achieve, nor systematic deviations from the leading-order power-law scalings of the black hole mass. We do see systematic effects in the black-hole angular momentum, but it is not clear yet if these are due to the predicted non-trivial scaling functions, or to nonlinear effects at sufficiently large initial angular momentum (which we do not account for in our theoretical model).

Mass gap in the critical gravitational collapse of a kink

We study the gravitational collapse of a kink within spherical symmetry and the characteristic formulation of General Relativity. We explore some expected but elusive gravitational collapse issues which have not been studied before in detail, finding new features. The numerical one-parametric solution and the structure of the spacetime are calculated using finite differences, Galerkin collocation techniques, and some scripting for automated grid coverage. We study the threshold of black hole formation and confirm a mass gap in the phase transition. In the supercritical case we find a mass scaling power law $M_{BH}={M^*_{BH}}+K[\lambda-\lambda^*]^{2\gamma}+f(K[\lambda-\lambda^*]^{2\gamma})$, with $\gamma\approx 0.37$ independent of the initial data for the cases considered, and $M^*_{BH}$, $K$ and $\lambda^*$ each depending on the initial datum. The spacetime has a self-similar structure with a period of $\Delta\approx 3.4$. In the subcritical case the Bondi mass at null infinity decays in cascade with $\Delta/2$ interval as expected.

More on critical collapse of axion-dilaton system in dimension four

We complete our previous study of critical gravitational collapse in the axion-dilaton system by analysing thehyperbolic and parabolic ansätze. As could be expected, the corresponding Choptuik exponents in four-dimensionsdiffer from the elliptic case.

Testing modes of exhumation in collisional orogens: Synconvergent channel flow in the southeastern Canadian Cordillera

This study investigates exhumation processes in collisional orogens. A critical test between three modes of exhumation is presented based on a review of quantitative numerical and analogue modeling studies. The test is applied to the large tract of migmatites surrounding the Monashee Complex in the southeastern Canadian Cordillera. It reveals that the extensive and multidisciplinary database of this region is entirely compatible with the synconvergent channel-flow mode but not with critical wedge and gravitational collapse modes. We propose that a partially molten channel decoupled from its lid and base and started to flow in the middle crust toward the foreland at 100–90 Ma. A steady-state channel-flow system was established for the following ∼30 m.y., during which rocks at the front of the channel were exhumed to upper-crustal levels as they flowed above an underthrusting basement ramp. Flow was accommodated by oppositely verging shear zones bounding the channel, by internal ductile deformation, and by shortening in the foreland belt. The locus of flow migrated downward to exhume rocks of the Monashee Complex between 60 and 50 Ma by a similar process. The southeastern Canadian Cordillera thus constitutes an excellent natural analogue for the channel-flow model. In contrast to the commonly held view of large-magnitude extension and core complex formation, the role of extension was limited to the final ∼10–15 km of exhumation after 50 Ma.

Critical gravitational collapse: Towards a holographic understanding of the Regge region

We study the possible holographic connection between the Regge limit in QCD and critical gravitational collapse of a perfect fluid in higher dimensions. We begin by analyzing the problem of critical gravitational collapse of a perfect fluid in any number of dimensions and numerically compute the associated Choptuik exponent in d = 5 , 6 and 7 for a range of values of the speed of sound of the fluid. Using continuous self-similarity as guiding principle, a holographic correspondence between this process and the phenomenon of parton saturation in high-energy scattering in QCD is proposed. This holographic connection relates strong gravitational physics in the bulk with (nonsupersymmetric) QCD at weak coupling in four dimensions.

Choptuik’s scaling in higher dimensions

We extend Choptuik's scaling phenomenon found in general relativistic critical gravitational collapse of a massless scalar field to higher dimensions. We find that in the range $4\ensuremath{\le}D\ensuremath{\le}11$ the behavior is qualitatively similar to that discovered by Choptuik. In each dimension we obtain numerically the universal numbers associated with the critical collapse: the scaling exponent $\ensuremath{\gamma}$ and the echoing period $\ensuremath{\Delta}$. The behavior of these numbers with increasing dimension seems to indicate that $\ensuremath{\gamma}$ reaches a maximum and $\ensuremath{\Delta}$ a minimum value around $11\ensuremath{\le}D\ensuremath{\le}13$. These results and their relation to the black hole--black string system are discussed.

News from critical collapse: Bondi mass, tails, and quasinormal modes

We discuss critical gravitational collapse on the threshold of apparent horizon formation as a model both for the discussion of global aspects of critical collapse and for numerical studies in a compactified context. For our matter model we choose a self-gravitating massless scalar field in spherical symmetry, which has been studied extensively in the critical collapse literature. Our evolution system is based on Bondi coordinates, the mass-function is used as an evolution variable to ensure regularity at null infinity. We compute radiation quantities like the Bondi mass and news-function and find that they reflect the discretely self-similar (DSS) behavior. Surprisingly, the period of radiation at null infinity is related to the formal result for the leading quasinormal mode of a black hole with rapidly decreasing mass. Furthermore, our investigations shed some light on global versus local issues in critical collapse, and the validity and usefulness of the concept of null infinity when predicting detector signals.

Collapse of a scalar field in 2 + 1 gravity

We consider the problem of critical gravitational collapse of a scalar field in 2 + 1 dimensions with spherical (circular) symmetry. After surveying all the analytic, continuously self-similar solutions and considering their global structure, we examine their perturbations with the intent of understanding which are the critical solutions with a single unstable mode. The critical solution which we find is the one which agrees most closely with that found in numerical evolutions. However, the critical exponent which we find does not seem to agree with the numerical result.

Critical collapse of a massive vector field

We perform numerical simulations of the critical gravitational collapse of a massive vector field. The result is that there are two critical solutions. One is equivalent to the Choptuik critical solution for a massless scalar field. The other is periodic.

Critical collapse of the massless scalar field in axisymmetry

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 regions form on the axis, each resembling the spherically symmetric self-similar solution. The existence of a non-spherical unstable mode is in conflict with previous perturbative results, and we therefore discuss whether such a mode exists in the continuum limit, or whether we are instead seeing a marginally stable mode that is rendered unstable by numerical approximation.

New transition between discrete and continuous self-similarity in critical gravitational collapse

We analyze a bifurcation phenomenon associated with critical gravitational collapse in a family of self-gravitating SU(2) $\ensuremath{\sigma}$ models. As the dimensionless coupling constant decreases, the critical solution changes from discretely self-similar (DSS) to continuously self-similar (CSS). Numerical results provide evidence for a bifurcation which is analogous to a heteroclinic loop bifurcation in dynamical systems, where two fixed points (CSS) collide with a limit cycle (DSS) in phase space as the coupling constant tends to a critical value.

Relaxation to Intermediate Attractors in Nonlinear Wave Equations

We study the dynamics of nonlinear wave equations near the threshold of singularity formation, motivated by some unexpected features observed in the critical gravitational collapse. One of the features, the existence of a universal intermediate attractor in the dynamics at the threshold of singularity formation, is studied in more detail. The result is a scheme of the evolution: for all initial data sufficiently near the threshold, the solutions reach one universal static solution, which plays the role of an intermediate attractor. The solutions remain near the attractor for some time, which scales in a simple way with a parameter of the initial data, and then departs in one of two “opposite” directions, eventually forming a singularity or dispersing. The relaxation as well as the departure from the attractor is governed in the linear approximation by quasi-normal modes (QNMs) and one growing mode. Moreover, the profiles of the perturbation modes with their oscillation frequencies and damping or amplifying factors, calculated as eigenvalue problems (linear ODEs) have been observed in numerical evolution (nonlinear PDEs). In particular, the exponent of the growing mode determines the lifetime of the intermediate dynamics (near the attractor), and a few least damped QNMs have been observed as ringing in the relaxation process.

Symmetry-seeking spacetime coordinates

In numerically constructing a spacetime that has an approximate timelike Killing vector, it is useful to choose spacetime coordinates adapted to the symmetry, so that the metric and matter variables vary only slowly with time in these coordinates. In particular, this is a crucial issue in numerically calculating a binary black hole inspiral. An approximate homothetic vector plays a role in critical gravitational collapse. We summarize old and new suggestions for finding such coordinates from a general point of view. We then test some of these in various toy models with spherical symmetry, including critical fluid collapse and critical scalar field collapse.

Bounds from primordial black holes with a near critical collapse initial mass function

Recent numerical evidence suggests that a mass spectrum of primordial black holes (PBHs) is produced as a consequence of near critical gravitational collapse. Assuming that these holes formed from the initial density perturbations seeded by inflation, we calculate model independent upper bounds on the mass variance at the reheating temperature by requiring that the mass density does not exceed the critical density and the photon emission does not exceed current diffuse gamma-ray measurements. We then translate these results into bounds on the spectral index n by utilizing the COBE data to normalize the mass variance at large scales, assuming a constant power law, and scale this result to the reheating temperature. We find that our bounds on n differ substantially $(\ensuremath{\delta}n&gt;0.05)$ from those calculated using initial mass functions derived under the assumption that the black hole mass is proportional to the horizon mass at the collapse epoch. We also find a change in the shape of the diffuse gamma-ray spectrum which results from the Hawking radiation. Finally, we study the impact of a nonzero cosmological constant and find that the bounds on n are strengthened considerably if the universe is indeed vacuum-energy dominated today.

Choptuik scaling in six dimensions

We perform numerical simulations of the critical gravitational collapse of a spherically symmetric scalar field in 6 dimensions. The critical solution has discrete self-similarity. We find the critical exponent \gamma and the self-similarity period \Delta.

Domain wall interacting with a black hole: A new example of critical phenomena

We study a simple system that comprises all main features of critical gravitational collapse, originally discovered by Choptuik and discussed in many subsequent publications. These features include universality of phenomena, mass-scaling relations, self-similarity, symmetry between super-critical and sub-critical solutions, etc. The system we consider is a stationary membrane (representing a domain wall) in a static gravitational field of a black hole. For a membrane that spreads to infinity, the induced 2+1 geometry is asymptotically flat. Besides solutions with Minkowski topology there exists also solutions with the induced metric and topology of a 2+1 dimensional black hole. By changing boundary conditions at infinity, one finds that there is a transition between these two families. This transition is critical and it possesses all the above-mentioned properties of critical gravitational collapse. It is remarkable that characteristics of this transition can be obtained analytically. In particular, we find exact analytical expressions for scaling exponents and wiggle-periods. Our results imply that black hole formation as a critical phenomenon is far more general than one might expect.