We examine the wave-functionals describing the collapse of a self-gravitating dust ball in an exact quantization of the gravity-dust system. We show that ingoing (collapsing) dust shell modes outside the apparent horizon must necessarily be accompanied by outgoing modes inside the apparent horizon, whose amplitude is suppressed by the square root of the Boltzmann factor at the Hawking temperature. Likewise, ingoing modes in the interior must be accompanied by outgoing modes in the exterior, again with an amplitude suppressed by the same factor. A suitable superposition of the two solutions is necessary to conserve the dust probability flux across the apparent horizon, thus each region contains both ingoing and outgoing dust modes. If one restricts oneself to considering only the modes outside the apparent horizon then one should think of the apparent horizon as a partial reflector, the probability for a shell to reflect being given by the Boltzmann factor at the Hawking temperature determined by the mass contained within it. However, if one considers the entire wave function, the outgoing wave in the exterior is seen to be the transmission through the horizon of the interior outgoing wave that accompanies the collapsing shells. This transmission could allow information from the interior to be transferred to the exterior.
We show that the behaviour of outgoing radial null geodesic congruence on the apparent horizon is related to the property of nakedness in spherical dust collapse justifying the difference between the Penrose diagrams in the naked and covered dust collapse scenarios. We provide arguments suggesting that the relationship could be generally valid.
In spherical symmetry, the total energy-momentum tensor near the apparent horizon is identified up to a single function of time from two assumptions: a trapped region forms at a finite time of a distant observer, and values of two curvature scalars are finite at its boundary. In general relativity, this energy-momentum tensor leads to the unique limiting form of the metric. The null energy condition is violated across the apparent horizon and is satisfied in the vicinity of the inner apparent horizon. As a result, homogenous collapse models cannot describe the formation of a black hole. Proper…
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 solve Einstein vacuum equations in a spacetime region up to the "center" of gravitational collapse. Within this region, we construct a sequence of marginally outer trapped surfaces (MOTS) with areas going to zero. These MOTS form a marginally outer trapped tube (apparent horizon). It emerges from a point and is smooth (except at that point) and spacelike. In the proof we employ a scale critical trapped surface formation criterion established by An and Luk and a new type of quasilinear elliptic equation is studied. The main conclusion of this paper proves a conjecture of Ashtekar on black ho…
In this paper we will examine the consequence of a canonical theory of quantum dust collapse in 2+1 dimensions. The solution of the WDW equation describing the collapse indicates that collapsing shells outside the apparent horizon are accompanied by outgoing shells within the apparent horizon during their collapse phase and stop collapsing once they reach the apparent horizon. Taking this picture of quantum collapse seriously, we determine a static solution with energy density corresponding to a dust ball whose collapse has terminated at the apparent horizon. We show that the boundary radius o…
Motivated by its field theory interpretation, we study gravitational collapse of a minimally coupled massless scalar field in Einstein gravity with a negative cosmological constant. After demonstrating the accuracy of the numerical algorithm for the questions we are interested in, we investigate various aspects of the apparent horizon formation. In particular, we study the time and radius of the apparent horizon formed as functions of the initial Gaussian profile for the scalar field. We comment on several aspects of the dual field theory picture.
We extend previous results on the reflection and transmission of self-gravitating dust shells across the apparent horizon during quantum dust collapse to non-marginally-bound dust collapse in arbitrary dimensions with a negative cosmological constant. We show that the Hawking temperature is independent of the energy function and that the wave functional describing the collapse is well behaved at the Hawking-Page transition point. Thermal radiation from the apparent horizon appears as a generic result of non-marginal collapse in AdS space-time owing to the singular structure of the Hamiltonian …
Some experimental implications of the recent progress on wave function collapse are calculated. Exact results are derived for the center-of-mass wave function collapse caused by random scatterings and applied to a range of specific examples. The results show that recently proposed experiments to measure the GRW effect are likely to fail, since the effect of naturally occurring scatterings is of the same form as the GRW effect but generally much stronger. The same goes for attempts to measure the collapse caused by quantum gravity as suggested by Hawking and others. The results also indicate th…
We study responses of the Brans-Dicke field due to gravitational collapses of scalar field pulses using numerical simulations. Double-null formalism is employed to implement the numerical simulations. If we supply a scalar field pulse, it will asymptotically form a black hole via dynamical interactions of the Brans-Dicke field. Hence, we can observe the responses of the Brans-Dicke field by two different regions. First, we observe the late time behaviors after the gravitational collapse, which include formations of a singularity and an apparent horizon. Second, we observe the fully dynamical b…
Classification of Oppenheimer-Snyder Collapse: Singular, Bouncing, and Soft-Landing Scenarios
Center for Joint Quantum Studies, Department of Physics, School of Science, Tianjin University（天津大学理学院物理系联合量子研究中心）；The International Joint Institute of Tianjin University（天津大学国际联合研究院）
We investigate what are the key physical features that cause the development of a naked singularity, rather than a black hole, as the end-state of spherical gravitational collapse. We show that sufficiently strong shearing effects near the singularity delay the formation of the apparent horizon. This exposes the singularity to an external observer, in contrast to a black hole, which is hidden behind an event horizon due to the early formation of an apparent horizon.
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 elaborated the gravitational collapse of a self-gravitating complex charged scalar field in the context of the low-energy limit of the string theory, the so-called dilaton gravity. We begin with the regular spacetime and follow the evolution through the formation of an apparent horizon and the final central singularity.
We study the collapse of spheroidal configurations of collisionless particles in full general relativity. This setup was originally considered by Shapiro and Teukolsky (1991), where it was found that prolate configurations with a sufficiently large semimajor axis gave rise to diverging curvature, but no apparent horizon. This was taken as evidence for the formation of a naked singularity, in violation of cosmic censorship. We revisit such configurations using different coordinates/slicing, and considering a range of values for the semimajor axis and eccentricity of the initial matter distribut…
We model the collapse of a spherically symmetric, constant density, pressureless shell onto a preexisting black hole. Contrary to a recent claim of Liu and Zhang (2009), we show that an observer at infinity never sees any part of the shell cross the event horizon. The entire shell appears to "freeze" outside the black hole. We find that during intermediate stages of the collapse, the change in apparent size of the inner surface is non-monotonic. However, at late times, the apparent sizes of the inner and outer surfaces of the shell both approach 3\sqrt{3}GM/c^2 asymptotically, where M is the c…
We numerically study the gravitational collapse of a massless scalar field with spherical symmetry in Einstein-aether theory, and show that apparent, spin-0 and dynamical universal horizons (dUHs) can be all formed. The spacetime and the aether field are well-behaved and regular, including regions nearby these horizons (but away from the center of spherical symmetry). The spacetime outside the apparent and spin-0 horizons settles down to a static configuration, and some of such resulting static black holes were already found numerically in the literature. On the other hand, the proper distance…
When a black hole forms from collapse in a holographic theory, the information in the black hole interior remains encoded in the boundary. We prove that the area of the black hole's apparent horizon is precisely the entropy associated to coarse graining over the information in its interior, subject to knowing the exterior geometry. This is the maximum holographic entanglement entropy that is compatible with all classical measurements conducted outside of the apparent horizon. We identify the boundary dual to this entropy and explain why it obeys a Second Law of Thermodynamics.
Quantum gravity is treated as a stochastic quantization of the spacetime metric of general relativity. It is found that owing to the stochastic fluctuating behavior of the geometry, the singularity in gravitational collapse of a star has a zero probability density of occurring. Moreover, as a star collapses the probability density of a distant observer seeing an infinite red shift at the Schwarzschild radius of the star is zero. Therefore, there is a vanishing probability density of a black hole event horizon forming during gravitational collapse. A grey apparent event horizon forms during the…
For the study of $3+1$ dimensional Einstein vacuum equations (EVEs), substantial progress has been made recently on the problem of trapped surface formation. However, very limited knowledge of existence and associated properties is acquired on the boundary of the emerged trapped region, i.e., the apparent horizon, which is composed of marginally outer trapped surfaces (MOTSs) and is of great physical importance. In this paper, concerning this emerged apparent horizon we prove a folklore conjecture relating to both cosmic censorship and black hole thermodynamics. In a framework set up by Christ…
In this paper, we consider the gravitational collapse of generalized Vaidya space-time when the matter satisfies the equation of the state either $P=0$ or $P=-αρ$, where $0 < α< 1$. We show that in the case when type I of matter field is dust, then the apparent horizon will never appear, but there is now a family of null radial future-directed geodesics which terminated at the central singularity in the past. Also, we show that in the case of negative pressure, the result of the gravitational collapse might be the naked singularity and the apparent horizon appears and in very short time disapp…
We obtain a general five dimensional (5D) quasispherical solutions of irrotational dust in Einstein gravity with the Gauss-Bonnet combination of quadratic curvature terms. These solutions are generalization, to Einstein-Gauss-Bonnet gravity, of the 5D quasispherical Szkeres like solutions in general relativity. It is found that the collapse proceed in the same way as in the analogous spherical collapse, i.e., there exists a regular initial data such that the collapse proceed to form naked singularities violating cosmic censorship conjecture. The effect of Gauss-Bonnet quadratic curvature terms…
Based on a previously found general class of quantum improved exact solutions composed of non-interacting (dust) particles, we model the gravitational collapse of stars. As the modeled star collapses a closed apparent 3-horizon is generated due to the consideration of quantum effects. The effect of the subsequent emission of Hawking radiation related to this horizon is taken into consideration. Our computations lead us to argue that a total evaporation could be reached. The inferred global picture of the spacetime corresponding to gravitational collapse is devoid of both event horizons and she…
Numerical studies of gravitational collapse in isotropic coordinates have recently shown an interesting connection between the gravitational Lagrangian and black hole thermodynamics. A study of the actual spacetime was not the main focus of this work and in particular, the rich and interesting structure of the interior has not been investigated in much detail and remains largely unknown. We elucidate its features by performing a numerical study of the spacetime in isotropic coordinates during gravitational collapse of a massless scalar field. The most salient feature to emerge is the formation…
Quantum gravity suggests that the paradox recently put forward by Almheiri et. al. (AMPS) can be resolved if matter does not undergo continuous collapse to a singularity but condenses on the apparent horizon. One can then expect a quasi-static object to form even after the gravitational field has overcome any degeneracy pressure of the matter fields. We consider dust collapse. If the collapse terminates on the apparent horizon, the Misner-Sharp mass function of the dust ball is predicted and we construct static solutions with no tangential pressure that would represent such a compact object. T…
A sufficient condition for the validity of Cosmic Censorship in spherical gravitational collapse is formulated and proved. The condition relies on an attractive mathematical property of the apparent horizon, which holds if ''minimal'' requirements of physical reasonableness are satisfied by the matter model.
Gravitational collapse of a massless scalar field in spherically symmetric anti-de Sitter (AdS) spacetimes presents a new phenomenology with a series of critical points whose dynamics is discretely self-similar as in the asymptotically flat case. Each critical point is the limit of a branch of scalar field configurations that have bounced off the AdS boundary a fixed number of times before forming an apparent horizon. We present results from a numerical study that focus on the interfaces between branches. We find that there is a mass gap between branches and that subcritical configurations nea…
Wormholes as red herrings: reflection positivity and the reconstruction of unitary quantum field theories
Simons Center for Geometry and Physics, Stony Brook University（石溪大学西蒙斯几何与物理中心）；Department of Physics, University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校物理系）
This paper proves a theorem about the existence of an apparent horizon in general relativity, which applies equally well to vacuum configurations and matter configurations. The theorem uses the reciprocal of the surface-to-volume ratio of a region on a space slice to measure the radius of the region, and uses the minimum value $K_{\rm min}$ of certain components of the extrinsic curvature to measure the strengh of the gravitational field in the region. The theorem proves that, if the product of the radius times $K_{\rm min}$ is larger than unity, then an apparent horizon must form, signalling …
This paper is devoted to study the gravitational charged perfect fluid collapse in the Friedmann universe models with cosmological constant. For this purpose, we assume that the electromagnetic field is so weak that it does not introduce any distortion into the geometry of the spacetime. The results obtained from the junction conditions between the Friedmann and the Reissner-Nordstr$\ddot{o}$m de-Sitter spacetimes are used to solve the field equations. Further, the singularity structure and mass effects of the collapsing system on time difference between the formation of apparent horizons and …
We consider spherically-symmetric black holes in semiclassical gravity. For a collapsing radiating thin shell we derive a sufficient condition on the exterior geometry that ensures that a black hole is not formed. This is also a sufficient condition for an infalling test particle to avoid the apparent horizon of an existing black hole and approach it only within a certain minimal distance. Taking the presence of a trapped region and its outer boundary --- the apparent horizon--- as the defining feature of black holes, we explore the consequences of their finite time of formation according to a…
In case of spherical symmetry, the assumptions of finite-time formation of a trapped region and regularity of its boundary --- the apparent horizon --- are sufficient to identify the form of the metric and energy-momentum tensor in its vicinity. By comparison with the known results for quasistatic evaporation of black holes, we complete the identification of their parameters. Consistency of the Einstein equations allows only two possible types of higher-order terms in the energy-momentum tensor. By using its local conservation, we provide a method of calculation of the higher-order terms, expl…
The present work investigates the gravitational collapse of a perfect fluid in $f(R)$ gravity models. For a general $f(R)$ theory, it is shown analytically that a collapse is quite possible. The singularity formed as a result of the collapse is found to be a curvature singularity of shell focusing type. The possibility of the formation of an apparent horizon hiding the central singularity depends on the initial conditions.
We study the fate of gravitational collapse in presence of a cosmological constant. The junctions conditions between static and non-static space-times are deduced. Three apparent horizon are formed, but only two have physical significance, one of them being the black hole horizon and the other the cosmological horizon. The cosmological constant term slows down the collapse of matter, limiting also the size of the black hole.
We numerically investigate the gravitational collapse of collisionless particles in spheroidal configurations both in four and five-dimensional (5D) space-time. We repeat the simulation performed by Shapiro and Teukolsky (1991) that announced an appearance of a naked singularity, and also find that the similar results in 5D version. That is, in a collapse of a highly prolate spindle, the Kretschmann invariant blows up outside the matter and no apparent horizon forms. We also find that the collapses in 5D proceed rapidly than in 4D, and the critical prolateness for appearance of apparent horizo…
We examine here in what way the pressures affect the final fate of a continual gravitational collapse. It is shown that the presence of a non-vanishing pressure gradient in the collapsing cloud determines directly the epoch of formation of trapped surfaces and the apparent horizon, thus changing the causal structure in the vicinity of singularity.
The extension of the general relativity theory to higher dimensions, so that the field equations for the metric remain of second order, is done through the Lovelock action. This action can also be interpreted as the dimensionally continued Euler characteristics of lower dimensions. The theory has many constant coefficients apparently without any physical meaning. However, it is possible, in a natural way, to reduce to two (the cosmological and Newton's constant) these several arbitrary coefficients, yielding a restricted Lovelock gravity. In this process one separates theories in even dimensio…
We find analytical solutions describing the collapse of an infinitely long cylindrical shell of counter-rotating dust. We show that--for the classes of solutions discussed herein--from regular initial data a curvature singularity inevitably develops, and no apparent horizons form, thus in accord with the spirit of the hoop conjecture.
A classical solution where the (scalar) field value moves by an ${\cal O}(1)$ range in Planck units is believed to signal the breakdown of Effective Field Theory (EFT). One heuristic argument for this is that such a field will have enough energy to be inside its own Schwarzschild radius, and will result in collapse. In this paper, we consider an inverse problem: what kind of field ranges arise during the gravitational collapse of a classical field? Despite the fact that collapse has been studied for almost a hundred years, most of the discussion is phrased in terms of fluid stress tensors, and…
The thermal duality of E(8) x E(8) and SO(32) heterotic string theories may underpin a mechanism that would convert the kinetic energy of infalling matter during gravitational collapse to form a region of a hot string phase that would expel gravitational gradients. This phase would be the continuation of a Ginzburg-Landau like superconductor in the Euclidean regime. In this scenario, there would be no event horizon or singularity produced in gravitational collapse. Solutions are presented for excitations of the string vacuum that may form during gravitational collapse and drive the transition …
We study here the gravitational collapse of a matter cloud with a non-vanishing tangential pressure in the presence of a non-zero cosmological term. Conditions for bounce and singularity formation are derived for the model. It is also shown that when the tangential pressures vanish, the bounce and singularity conditions reduce to that of the dust case studied earlier. The collapsing interior is matched with an exterior which is asymptotically de Sitter or anti de Sitter, depending on the sign of cosmological constant. The junction conditions for matching the cloud to exterior are specified. Th…
We study the gravitational collapse of a dust dark matter star in a $Λ$-background. We consider two distinct cases: First we do not have a dark matter and dark energy coupling; second, we consider that $Λ$ decay in dark particles. The approach adopted assumes a modified matter expansion rate and we have formation of a black hole, since that, we have the formation of an apparent horizon. A brief comparison of the process of the matter condensation using the gravitational collapse approach and the linear scalar perturbation theory is considered.
Numerical studies of gravitational collapse to black holes make use of apparent horizons, which are intrinsically foliation-dependent. We expose the problem and discuss possible solutions using the Hawking quasilocal mass. In spherical symmetry, we present a physically sensible approach to the problem by restricting to spherically symmetric spacetime slicings. In spherical symmetry the apparent horizons are gauge-independent in any spherically symmetric foliation but physical quantities associated with them, such as surface gravity and temperature, are not. The widely used comoving and Kodama …
We study the inner-structure of a charged black-hole which is formed from the gravitational collapse of a self-gravitating charged scalar-field. Starting with a regular spacetime, we follow the evolution through the formation of an apparent horizon, a Cauchy horizon and a final central singularity. We find a null, weak, mass-inflation singularity along the Cauchy horizon, which is a precursor of a strong, spacelike singularity along the $r=0$ hypersurface.
We present a class of exact solutions of Einstein field equations for a shear-free spherically symmetric anisotropic fluid undergoing radial heat flow. The interior metric fulfilled all the relevant physical and thermodynamic conditions and matched with Vaidya exterior metric over the boundary. Initially the interior solutions represent a static configuration of dissipative fluid which then gradually starts evolving into radiating collapse. The apparent luminosity observed by the distant observer at rest at infinity and the effective surface temperature are zero in remote past at the instance …
The gravitational collapse of a spherical distribution, in a class of f(R) theories of gravity, where f(R) is power function of R, is discussed. The spacetime is assumed to admit a homothetic Killing vector. In the collapsing modes, some of the situations indeed hit a singularity, but they are all covered with an apparent horizon. Some peculiar cases are observed where the collapsing body settles to a constant radius at a given value of the radial coordinate.
In this paper, we perform a detailed investigation on the various geometrical properties of trapped surfaces and the boundaries of trapped region in general relativity. This treatment extends earlier work on LRS II spacetimes to a general 4 dimensional spacetime manifold. Using a semi-tetrad covariant formalism, that provides a set of geometrical and matter variables, we transparently demonstrate the evolution of the trapped region and also extend Hawking's topology theorem to a wider class of spacetimes. In addition, we perform a stability analysis for the apparent horizons in this formalism,…
We derive effective equations with loop quantum gravity corrections for the Lemaître-Tolman-Bondi family of space-times, and use these to study quantum gravity effects in the Oppenheimer-Snyder collapse model. For this model, after the formation of a black hole with an apparent horizon, quantum gravity effects become important in the space-time region where the energy density and space-time curvature scalars become comparable to the Planck scale. These quantum gravity effects first stop the collapse of the dust matter field when its energy density reaches the Planck scale, and then cause the d…
In this paper, the effect of a positive cosmological constant on spherically symmetric collapse with perfect fluid has been investigated. The matching conditions between static exterior and non-static interior spacetimes are given in the presence of a cosmological constant. We also study the apparent horizons and their physical significance. It is concluded that the cosmological constant slows down the collapse of matter and hence limit the size of the black hole. This analysis gives the generalization of the dust case to the perfect fluid. We recover the results of the dust case for $p=0$.
The present paper deals with the gravitational collapse of an inhomogeneous spherical star consisting of dust fluid in the background of dark energy components with linear equation of state. We discussed the development of apparent horizon to investigate the black-hole formation in gravitational collapsing process. The collapsing process is examined first separately for dust cloud and dark energy and then under the combined effect of dust interacting with dark energy. It is obtained that when only dust cloud or dark energy is present the collapse leads to the formation of black-hole under cert…
Gravitational collapse in (n+2) dimensional quasi-spherical space-time is studied for a fluid with non vanishing radial pressure. An exact analytic solution is obtained (ignoring the arbitrary integration function) for the equation of state $p_{r}=(γ-1)ρ.$ The singularity is studied locally by comparing the time of formation of apparent horizon and the central shell focusing singularity while the global nature of the final fate of collapse is characterized by the existence of radial null geodesic. It is revealed that the end state of collapse for D dimension with equation of state $p=-ρ$, for …
We consider a spherical gravitational collapse of inhomogeneous dust (and null dust) in Einstein gravity with the Gauss-Bonnet (GB) combination of quadratic curvature terms. It turns out that the presence of the coupling constant of the GB terms $α> 0$ completely changes the causal structure of the singularities from the analogous general relativistic case. The gravitational collapse of inhomogeneous dust in the five-dimensional GB extended Einstein equations leads to formation of a massive, but weak, timelike singularity which is forbidden in general relativity. The apparent horizons of two d…
We consider here the classic scenario given by Oppenheimer, Snyder, and Datt, for the gravitational collapse of a massive matter cloud, and examine its stability under the introduction of small tangential stresses. We show, by offering an explicit class of physically valid tangential stress perturbations, that an introduction of tangential pressure, however small, can qualitatively change the final fate of collapse from a black hole final state to a naked singularity. This shows instability of black hole formation in collapse and sheds important light on the nature of cosmic censorship hypothe…
We examine the gravitational collapse and black hole formation of multiple non--spherical configurations constructed from Szekeres dust models with positive spatial curvature that smoothly match to a Schwarzschild exterior. These configurations are made of an almost spherical central core region surrounded by a network of "pancake-like" overdensities and voids with spatial positions prescribed through standard initial conditions. We show that a full collapse into a focusing singularity, without shell crossings appearing before the formation of an apparent horizon, is not possible unless the fu…
