In the gravitational collapse of matter beyond spherical symmetry, gravitational waves are necessarily present. On the other hand, gravitational waves can collapse to a black hole even without matter. One might therefore wonder whether the interaction and competition between the matter fields and gravitational waves affects critical phenomena at the threshold of black hole formation. As a toy model for this, we study type II critical collapse with two matter fields in spherical symmetry, namely a scalar field and a Yang-Mills field. On their own, both display discrete self-similarity (DSS) in type II critical collapse, and we can take either one of them as a toy model for gravitational waves. To our surprise, in numerical time evolutions we find that, for sufficiently good fine-tuning, the scalar field always dominates on sufficiently small scales. We explain our results by the conjectured existence of a "quasi-discretely self-similar" (QSS) solution shared by the two fields, equal to the known Yang-Mills critical solution at infinitely large scales and the known scalar field critical solution (the Choptuik solution) at infinitely small scales, with a gradual transition from one field to the other. This QSS solution itself has only one unstable mode, and so acts as the critical solution for any mixture of scalar field and Yang-Mills initial data.
The 2+1-dimensional geodesic circularly symmetric solutions of Einstein-massless-scalar field equations with negative cosmological constant are found and their local and global properties are studied. It is found that one of them represents gravitational collapse where black holes are always formed.
We study here the evolution of a massless scalar field in a spacetime, developing from a regular initial spacelike surface. The Einstein equations and regularity and boundary conditions governing the same are specified. Both homogeneous and inhomogeneous collapse models are considered and we analyze when the occurrence of singularity will be simultaneous or otherwise. In the inhomogeneous collapse case, we characterize a wide family of black hole solutions arising in scalar field collapse. We also discuss the possibility of existence of classes of non-singular models where collapse could almos…
We study an analytical solution to the Einstein's equations in 2+1-dimensions, representing the self-similar collapse of a circularly symmetric, minimally coupled, massless, scalar field. Depending on the value of certain parameters, this solution represents the formation of black holes. Since our solution is asymptotically flat, our black holes do not have the BTZ space-time as their long time limit. They represent a new family of black holes in 2+1-dimensions.
Self-similar solutions of a collapsing perfect fluid and a massless scalar field with kinematic self-similarity of the first kind in 2+1 dimensions are obtained. Their local and global properties of the solutions are studied. It is found that some of them represent gravitational collapse, in which black holes are always formed, and some may be interpreted as representing cosmological models.
We present results from the first study of critical behavior in 3-d gravitational collapse. The source of the gravitational field is a massless scalar field. This is a well-studied case for spherically symmetric gravitational collapse, allowing us to understand the reliability and accuracy of the simulations. We study both supercritical and subcritical evolutions to see if one provides more accurate results than the other. We find that even for non-spherical initial data with 35 percent of the power in the $\ell=2$ spherical harmonic, the critical solution is the same as in spherical symmetry.
The phenomena of collapse and dispersal for a massless scalar field has drawn considerable interest in recent years, mainly from a numerical perspective. We give here a sufficient condition for the dispersal to take place for a scalar field that initially begins with a collapse. It is shown that the change of the gradient of the scalar field from a timelike to a spacelike vector must be necessarily accompanied by the dispersal of the scalar field. This result holds independently of any symmetries of the spacetime. We demonstrate the result explicitly by means of an example, which is the scalar…
We report on a numerical study of the spherically symmetric collapse of a self-gravitating massless scalar field. Earlier results of Choptuik(1992, 1994) are confirmed. The field either disperses to infinity or collapses to a black hole, depending on the strength of the initial data. For evolutions where the strength is close to but below the strength required to form a black hole, we argue that there will be a region close to the axis where the scalar curvature and field energy density can reach arbitrarily large levels, and which is visible to distant observers
Several classes of conformally-flat and spherically symmetric exact solutions to the Einstein field equations coupled with either a massless scalar field or a radiation fluid are given, and their main properties are studied. It is found that some represent the formation of black holes due to the gravitational collapse of the matter fields. When the spacetimes have continuous self-similarity (CSS), the masses of black holes take a scaling form $M_{BH} \propto (P - P^{*})^γ$, where $γ= 0.5$ for massless scalar field and $γ= 1$ for radiation fluid. The reasons for the difference between the value…
We study the collapse of a massless scalar field coupled to gravity. A class of blackhole solutions are identified. We also report on a class of solutions where collapse starts from a regular spacelike surface but then the collapsing scalar field freezes. As a result, in these solutions, a black hole does not form, neither is there any singularity in the future.
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…
Gravitational collapse of a massless scalar field with the periodic boundary condition in a cubic box is reported. This system can be regarded as a lattice universe model. We construct the initial data for a Gaussian like profile of the scalar field taking the integrability condition associated with the periodic boundary condition into account. For a large initial amplitude, a black hole is formed after a certain period of time. While the scalar field spreads out in the whole region for a small initial amplitude. It is shown that the expansion law in a late time approaches to that of the radia…
In a recent paper we showed that the collapse to a black hole in one-parameter families of initial data for massless, minimally coupled scalar fields in spherically symmetric semi-classical loop quantum gravity exhibited a universal mass scaling similar to the one in classical general relativity. In particular, no evidence of a mass gap appeared as had been suggested by previous studies. The lack of a mass gap indicated the possible existence of a self-similar critical solution as in general relativity. Here we provide further evidence for its existence. Using an adaptive mesh refinement code,…
This paper studies the perturbations of the continuously self-similar critical solution of the gravitational collapse of a massless scalar field (Roberts solution). The perturbation equations are derived and solved exactly. The perturbation spectrum is found to be not discrete, but occupying continuous region of the complex plane. The renormalization group calculation gives the value of the mass-scaling exponent equal to 1.
We investigate the dynamics of homogeneous gravitational collapse of a massless vector field in the presence of a positive cosmological constant $Λ$. The corresponding density function $ρ(a)$ obtained for the massless vector field is inversely proportional to the fourth power of the scale factor $a (t)$. The variation of the scale factor shows that for $0\, \leΛ< 1$, we obtain the gravitational collapse of the vector fields leading singularity formation in a {\it finite} comoving time resulting in a {\it Blackhole} such that with increasing $Λ$, the singularity formation time, $t_s$ increases.…
In the gravitational collapse of matter beyond spherical symmetry, gravitational waves are necessarily present. On the other hand, gravitational waves can collapse to a black hole even without matter. One might therefore wonder how the interaction and competition between the matter fields and gravitational waves affects critical phenomena at the threshold of black hole formation. As a toy model for this, we study the threshold of black-hole formation in 4+1 dimensions, where we add a massless minimally coupled scalar matter field to the gravitational wave ansatz of Bizón, Chmaj and Schmidt (in…
We examine a one-parameter family of analytical solutions representing spherically symmetric collapse of a nonlinear massless scalar field with self-interaction in an asymptotically flat spacetime. The time evolution exhibits a type of critical behavior. Depending on the scalar charge parameter $q$ as compared to a critical value $q^*$, the incoming scalar wave collapses either to a globally naked central singularity if $q<q^*$ (weak field) or to a scalar-hairy black hole if $q>q^*$ (strong field), both having finite asymptotic masses. Near the critical evolution, the black hole mass follows a…
We study the spherically symmetric gravitational collapse of massless scalar matter field in asymptotic flat spacetime in $f(R)$ gravity. In the Einstein frame of $f(R)$ gravity, an additional scalar field arises due to the conformal transformation. We find that besides the usual competition between gravitational energy and kinetic energy in the process of gravitational collapse, the new scalar field brought by the conformal transformation adds one more competing force in the dynamical system. The dynamical competition can be controlled by tuning the amplitudes of the initial perturbations of …
This paper studies the whole process of gravitational collapse and accretion of a massless scalar field in asymptotically flat spacetime. Two kinds of initial configurations are considered. One is the initial data without black hole, the other contains a black hole. Under suitable initial conditions, we find that multi-horizon will appear, which means that the initial black hole formed by gravitational collapse or existing at the beginning will instantly expand and suddenly grow, rather than grows gradually in the accretion process. A new type of critical behavior is found around the instant e…
We present a new hybrid Cauchy-characteristic evolution method that is particularly suited for the study of gravitational collapse in spherically-symmetric asymptotically (global) Anti-de Sitter (AdS) spacetimes. The Cauchy evolution allows us to track the scalar field through the different bounces off the AdS boundary while the characteristic method can bring us very close to the point of formation of an apparent horizon. Here, we describe all the details of the method, including the transition between the two evolution schemes and the details of the numerical implementation for the case of m…
An analytical model that represents the collapse of a massless scalar wave packet with continuous self-similarity is constructed, and critical phenomena are found. In the supercritical case, the mass of black holes is finite and has the form $M \propto (p - p^{*})^γ$, with $γ= 1/2$.
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.
Four-dimensional cylindrically symmetric spacetimes with homothetic self-similarity are studied in the context of Einstein's Theory of Gravity, and a class of exact solutions to the Einstein-massless scalar field equations is found. Their local and global properties are investigated and found that they represent gravitational collapse of a massless scalar field. In some cases the collapse forms black holes with cylindrical symmetry, while in the other cases it does not. The linear perturbations of these solutions are also studied and given in closed form. From the spectra of the unstable eigen…
This paper studies the non-spherical perturbations of the continuously self-similar critical solution of the gravitational collapse of a massless scalar field (the Roberts solution). The exact analysis of the perturbation equations reveals that there are no growing non-spherical perturbation modes.
We solve the coupled Einstein-Vlasov system in spherical symmetry using direct numerical integration of the Vlasov equation in phase space. Focusing on the case of massless particles we study critical phenomena in the model, finding strong evidence for generic type I behaviour at the black hole threshold that parallels what has previously been observed in the massive sector. For differing families of initial data we find distinct critical solutions, so there is no universality of the critical configuration itself. However we find indications of at least a weak universality in the lifetime scal…
We prove that the quantum stress tensor for a massless scalar field in two dimensional non-selfsimilar Tolman Bondi dust collapse and Vaidya radiation collapse models diverges on the Cauchy horizon, if the latter exists. The two dimensional model is obtained by suppressing angular co-ordinates in the corresponding four dimensional spherical model.
We study the gravitational collapse of a massless scalar field within the effective scenario of loop quantum gravity. Classical singularity is avoided and replaced by a quantum bounce in this model. It is shown that, quantum gravity effects predict a threshold scale below which no horizon can form as the collapse evolves towards the bounce.
We study the dynamics near the central singularity in spherically symmetric collapse of a massless scalar field toward Schwarzschild black hole formation. The equations of motion take different simplified forms in the early and late stages of the singularity curve. We report some fine structures of the analytic solutions and universal features for the metric functions and matter near the singularity.
We study the analytic structure of the S-matrix which is obtained from the reduced Wheeler-DeWitt wave function describing spherically symmetric gravitational collapse of massless scalar fields. The simple poles in the S-matrix occur in the Euclidean spacetime, and the Euclidean Wheeler-DeWitt equation is a variant of the Calogero models, which is discussed in connection with conformal mechanics and a quantum instanton.
We explore numerically the evolution of a collapsing spherical shell of charged, massless scalar field. We obtain an external \RN space-time, and an inner space-time that is bounded by a singularity on the Cauchy Horizon. We compare these results with previous analysis and discuss some of the numerical problems encountered.
Two non-static solutions for three dimensional gravity coupled to matter fields are given. One describes the collapse of radiation that results in a black hole. This is the three dimensional analog of the Vaidya metric, and is used to construct a model for mass inflation. The other describes plane gravitational waves for coupling to a massless scalar field.
We present a self-similar model of spherically symmetric collapse of a massless scalar field with a parameter $p$. The black hole formation is explicitly shown to occur only in the strong-field implosion of $p >1$. The field evolution in the critical limit $ p \rightarrow 1 $ is compared with numerical results found by Choptuik.
Solutions to the mode equation for a quantized massless scalar field outside a black hole that forms from the collapse of a null shell: Late-time behaviors and computation of the stress-energy tensor
The cosmic no hair conjecture is tested in the spherically symmetric Einstein-Maxwell-dilaton~(EMD) system with a positive cosmological constant $Λ$. Firstly, we analytically show that once gravitational collapse occurs in the massless dilaton case, the system of field equations breaks down inevitably in outer communicating regions or at the boundary provided that a future null infinity ${\cal I}^+$ exists. Next we find numerically the static black hole solutions in the massive dilaton case and investigate their properties for comparison with the massless case. It is shown that their Abbott-De…
We obtain a new self-similar solution to the Einstein's equations in four-dimensions, representing the collapse of a spherically symmetric, minimally coupled, massless, scalar field. Depending on the value of certain parameters, this solution represents the formation of naked singularities and black holes. Since the black holes are identified as the Schwarzschild ones, one may naturally see how these black holes are produced as remnants of the scalar field collapse.
We discuss here the issue of regularity of initial data for dynamical spherically symmetric massless scalar field models in a spacetime. Generalizing the known solutions of Einstein equations given in this case by Wyman and Roberts, we examine the issue of regularity on a given spacelike surface, especially when the gradient of the field is spacelike. In particular, we isolate the class of models which would have necessarily a singularity at the center, and therefore these would be unsuitable for studying either gravitational collapse or dynamical evolutions in cosmology.
We present a new open-source, user-friendly toolkit of two codes$-$SFcollapse1D and NRPyCritCol$-$to study critical phenomena in the context of gravitational collapse. SFcollapse1D is a C/C++ tool designed to solve the problem of gravitational collapse of massless, spherically symmetric scalar fields with the ADM formalism in spherical-like coordinates. NRPyCritColis a collection of Python modules that leverage the NRPy+ infrastructure to generate a highly optimized C-code for evolving massless scalar fields within a covariant BSSN formalism. The toolkit was developed with user-friendliness an…
We study numerically the stability of Morris & Thorne's first traversible wormhole, shown previously by Ellis to be a solution for a massless ghost Klein-Gordon field. Our code uses a dual-null formulation for spherically symmetric space-time integration, and the numerical range covers both universes connected by the wormhole. We observe that the wormhole is unstable against Gaussian pulses in either exotic or normal massless Klein-Gordon fields. The wormhole throat suffers a bifurcation of horizons and either explodes to form an inflationary universe or collapses to a black hole, if the total…
We summarize the results of an investigation into the late time behavior of massless scalar fields propagating on spherically symmetric black hole spacetimes with a non-zero cosmological constant. The compatibility of these results with the `minimal requirement' of Brady and Poisson is commented upon.
In the context of gravitational collapse and black hole formation, we reconsider the problem to describe analytically the critical collapse of a massless and minimally coupled scalar field in $2+1$ gravity.
Critical collapse of a massless scalar field in spherical symmetry is systematically studied. We combine numerical simulations and asymptotic analysis, and synthesize critical collapse, spacetime singularities, and complex science. First set of approximate analytic expressions near the center are obtained. We observe that, near the center, the spacetime is nearly conformally flat, the dynamics is not described by the Kasner solution, and the Kreschmann scalar is proportional to r^(-5.30), where r is the areal radius. These features are significantly different from those in black hole singulari…
For the formation of a black hole in the gravitational collapse of a massless scalar field, we calculate a critical exponent that governs the black hole angular momentum for slightly non-spherical initial data near the black hole threshold. We calculate the scaling law by second-order perturbation theory. We then use the numerical results of a previous first-order perturbative analysis to obtain the numerical value mu ~ 0.76 for the angular momentum critical exponent. A quasi-periodic fine structure is superimposed on the overall power law.
In this paper, we present the results on the gravitational collapse of charged massless scalar field in asymptotically flat spacetime with a perfectly reflecting wall. Differing from previous works, we study the system in the double null coordinates, by which we could simulate the system until the black hole forms with higher precision but less performance time. We investigate the influence of charge on the black hole formation and the scaling behavior near the critical collapses. The gapless and gapped critical behaviors for black hole mass and charge are studied numerically. We find that the…
In this paper we study thermalization in a strongly coupled system via AdS/CFT. Initially, the energy is injected into the system by turning on a spatially homogenous scalar source coupled to a marginal composite operator. The thermalization process is studied by numerically solving Einstein's equations coupled to a massless scalar field in the Poincare patch of AdS_5. We define a thermalization time t_T on the AdS side, which has an interpretation in terms of a spacelike Wilson loop <W(l =1/T)> in CFT. Here T is the thermal equilibrium temperature. We study both cases with the source turned o…
This paper investigates the dynamics of thin-shell in the presence of perfect fluid as well as the scalar field. We formulate the equations of motion using Israel thin-shell formalism by taking the interior and exterior regions of Schwarzschild, Kerr as well as Kerr-Newmann black hole. We find numerical solutions of equations of motion and effective potential to analyze the scalar shell for collapse and expansion. It is found that the rate of collapse and expansion of scalar shell through shell's radius depends on charge and rotation parameters. We conclude that the massive scalar shell leads …
A massless scalar field minimally coupled to the gravitational field in a simplified spherical symmetry is discussed. It is shown that, in this case, the solution found by Roberts, describing a scalar field collapse, is in fact the most general one. Taking that solution as departure point, a study of the gravitational collapse for the self-similar conformal case is presented.
Since horizon formation in global anti-de Sitter spacetime is dual to thermalization of a conformal field theory on a compact space, whether generic initial data is stable or unstable against gravitational collapse is of great interest. We argue that all the known stable initial data for massless scalars are dominated by single scalar eigenmodes, specifically providing strong numerical evidence consistent with the interpretation that initial data with equal energies in two modes collapse on time scales of order the inverse square of the amplitude. We further scan the parameter space for massiv…
It has recently been shown that the strong cosmic censorship conjecture can be violated by the massless neutral scalar field in the nearly extremal Reissner-Nordstrom-de Sitter black hole. However, the formation of such a black hole by gravitational collapse necessitates the presence of the charged sector on top of the Einstein-Maxwell system. Thus we numerically calculate the quasi-normal modes for a massless charged scalar field in the Reissner-Nordstrom-de Sitter spacetime by generalizing the characteristic formulation to the charged case. As a result, the strong cosmic censorship turns out…
We consider the quantization of a massless scalar field, using the geometric optics approximation, in the background spacetime of a collapsing spherical self-similar Vaidya star, which forms a black hole or a naked singularity. We show that the outgoing radiation flux of the quantized scalar field diverges on the Cauchy horizon. The spectrum of the produced scalar partcles is non-thermal when the background develops a naked singularity. These results are analogous to those obtained for the scalar quantization on a self-similar dust cloud.
We study the nonlinear gravitational collapse of a charged massless scalar-field. We confirm the existence of oscillatory inverse power-law tails along future timelike infinity, future null infinity and along the future outer-horizon. The nonlinear dumping exponents are in excellent agreement with the analytically predicted ones. Our results prove the analytic conjecture according to which a charged hair decays slower than a neutral one and also suggest the occurrence of mass-inflation along the Cauchy horizon of a dynamically formed charged black-hole.
We develop a numerical method suitable for gravitational collapse based on Cauchy evolution with an ingoing characteristic boundary. Unlike similar methods proposed recently (Ripley; Bieri, Garfinkle & Yau 2019/20), the numerical grid remains fixed during the evolution and no points need to be removed or added. Increasing coordinate refinement of the central region as the field collapses is achieved solely through the choice of spatial gauge and particularly its boundary condition. We apply this method to study critical collapse of a massless scalar field in spherical symmetry using maximal sl…
The gravitational collapse of a massless scalar field with a self-interaction term $λϕ^4$ in anti-de Sitter space is investigated. We numerically investigate the effect of the self-interaction term on the critical amplitudes, forming time of apparent horizon, stable island and energy transformation. The results show that a positive $λ$ suppresses the formation of black hole, while a negative $λ$ enhances the process. We define two susceptibilities to characterize the effect of the self-interaction on the black hole formation, and find that near the critical amplitude, there exists a universal …
We present new analytical self-similar solutions describing a collapse of a massless scalar field in scalar-tensor theories. The solutions exhibit a type of critical behavior. The black hole mass for the near critical evolution is analytically obtained for several scalar-tensor theories and the critical exponent is calculated. Within the framework of the analytical models we consider it is found that the black hole mass law for some scalar-tensor theories is of the form $M_{BH}=f(p-p_{cr})(p-p_{cr})^γ$ which is slightly different from the general relativistic law $M_{BH}=const (p-p_{cr})^γ$.
We investigate the late time behavior of particle creation from an extremal Reissner-Nordstrom (RN) black hole formed by gravitational collapse. We calculate explicitly the particle flux associated with a massless scalar field at late times after the collapse. Our result shows that the expected number of particles in any wave packet spontaneously created from the ``in'' vacuum state approaches zero faster than any inverse power of time. This result confirms the traditional belief that extremal black holes do not emit particles. We also calculate the expectation value of the stress energy tenso…
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 $σ$. For $σ\leq 0.5$ we find collapse of type II with a critical exponent $γ=0.38\pm0.01$ and an echoing period $Δ=3.4\pm0.1$. These values are very similar to the known results for a real massless scalar field. On the other hand, for $σ\geq 2.5$ we obtain collapse of type I. In this case we find that the critical…
We study the time evolution of the Misner-Sharp mass and the apparent horizon for gravitational collapse of a massless scalar field in the $AdS_{5}$ space-time for both cases of narrow and broad waves by numerically solving the Einstein's equations coupled to a massless scalar field. This is done by relying on the full dynamics of the collapse including the concept of the dynamical horizon. It turns out that the Misner-Sharp mass is everywhere constant except for a rapid change across a thin shell defined by the density profile of the collapsing wave. By studying the evolution of the apparent …
We consider a massless quantum scalar field on a two-dimensional space-time describing a thin shell of matter collapsing to form a Schwarzschild-anti-de Sitter black hole. At early times, before the shell starts to collapse, the quantum field is in the vacuum state, corresponding to the Boulware vacuum on an eternal black hole space-time. The scalar field satisfies reflecting boundary conditions on the anti-de Sitter boundary. Using the Davies-Fulling-Unruh prescription for computing the renormalized expectation value of the stress-energy tensor, we find that at late times the black hole is in…
