The gravitational collapse of a star is an important issue both for general relativity and astrophysics, which is related to the well known "frozen star" paradox. This paradox has been discussed intensively and seems to have been solved in the comoving-like coordinates. However, to a real astrophysical observer within a finite time, this problem should be discussed in the point of view of the distant rest-observer, which is the main purpose of this paper. Following the seminal work of Oppenheimer and Snyder (1939), we present the exact solution for one or two dust shells collapsing towards a pre-existing black hole. We find that the metric of the inner region of the shell is time-dependent and the clock inside the shell becomes slower as the shell collapses towards the pre-existing black hole. This means the inner region of the shell is influenced by the property of the shell, which is contrary to the result in Newtonian theory. It does not contradict the Birkhoff's theorem, since in our case we cannot arbitrarily select the clock inside the shell in order to ensure the continuity of the metric. This result in principle may be tested experimentally if a beam of light travels across the shell, which will take a longer time than without the shell. It can be considered as the generalized Shapiro effect, because this effect is due to the mass outside, but not inside as the case of the standard Shapiro effect. We also found that in real astrophysical settings matter can indeed cross a black hole's horizon according to the clock of an external observer and will not accumulate around the event horizon of a black hole, i.e., no "frozen star" is formed for an external observer as matter falls towards a black hole. (abridged)
We explore in detail the semiclassical environment of collapsing shells of matter, and determine the semiclassical flux measured by a variety of observers. This study is a preliminary step in a broader investigation of thermodynamic properties of the geometry of collapsing objects. Specifically, in this paper we consider spherically symmetric null and timelike collapsing shells which form an event horizon, and calculate the flux measured by observers both inside and outside the shell, and both inside and outside the event horizon, and find nontrivial results in most of the cases. Additionally,…
We consider a collapsing shell of matter to form the Hayward black hole and investigate semiclassically quantum radiation from the shell. Using the Israel's formulation, we obtain the mass relation between the collapsing shell and the Hayward black hole. By using the functional Schrödinger formulation for the massless quantum radiation, the evolution of a vacuum state for a scalar field is shown to be unitary. We find that the number of quanta at a low frequency decreases for a large length parameter characterizing the Hayward black hole. Moreover, in the limit of low frequency, the Hawking te…
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…
This paper deals with the dynamics of scalar field thin shell in the Reissner-Nordstr$\ddot{o}$m geometry. The Israel junction conditions between Reissner-Nordstr$\ddot{o}$m spacetimes are derived, which lead to the equation of motion of scalar field shell and Klien-Gordon equation. These equations are solved numerically by taking scalar field model with the quadratic scalar potential. It is found that solution represents the expanding and collapsing scalar field shell. For the better understanding of this problem, we investigate the case of massless scalar field (by taking the scalar field po…
We construct exact solutions describing the motion of rotating thin shells in a fully backreacted five-dimensional rotating black hole spacetime. The radial equation of motion follows from the Darmois-Israel junction conditions, where both interior and exterior geometries are taken to be equal angular momenta Myers-Perry solutions. We show that rotation generates anisotropic pressures and momentum along the shell. Gravitational collapse scenarios including rotation are analyzed and a new class of stationary solutions is introduced. Energy conditions for the matter shell are briefly discussed.
We provide a detailed analysis of quantum field theory around a collapsing shell and discuss several conceptual issues related to the emission of radiation flux and formation of black holes. Explicit calculations are performed using a model for a collapsing shell which turns out to be analytically solvable. We use the insights gained in this model to draw reliable conclusions regarding more realistic models. We first show that any shell of mass $M$ which collapses to a radius close to $r=2M$ will emit approximately thermal radiation for a period of time. In particular, a shell which collapses …
The quantization of a spherically symmetric null shells is performed and extended to the framework of phase-space noncommutative (NC) quantum mechanics. The encountered properties are investigated making use of the Israel junction conditions on the shell, considering that it is the boundary between two spherically symmetric spacetimes. Using this method, and considering two different Kantowski-Sachs spacetimes as a representation for the Schwarzschild spacetime, the relevant quantities on the shell are computed, such as its stress-energy tensor and the action for the whole spacetime. From the …
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…
In three-dimensional AdS space, we consider the gravitational collapse of dust shell and then investigate the quantum radiation from the collapsing shell by employing the functional Schrödinger formalism. In the formation of the BTZ black hole, the interior geometry of the shell can be chosen as either the massless black hole or the global AdS space. In the incipient black hole limit, we obtain the wave function exactly from the time-dependent Schrödinger equation for a massless scalar field. Then, we show that the occupation number of excited states can be written by analytic expressions, and…
We study the probability that a horizon appears when concentric shells of matter collide, by computing the horizon wave-function of the system. We mostly consider the collision of two ultra-relativistic shells, both shrinking and expanding, at the moment their radii are equal, and find a probability that the system is a black hole which is in qualitative agreement with what one would expect according to the hoop conjecture and the uncertainty principle of quantum physics, and parallels the results obtained for simpler sources. One new feature however emerges, in that this probability shows a m…
We study radiation emitted during the gravitational collapse from two different types of shells. We assume that one shell is made of dark matter and is completely transparent to the test scalar (for simplicity) field which belongs to the standard model, while the other shell is made of the standard model particles and is totally reflecting to the scalar field. These two shells have exactly the same mass, charge and angular momentum (though we set the charge and angular momentum to zero), and therefore follow the same geodesic trajectory. However, we demonstrate that they radiate away different…
We present results of a comprehensive study of collapsing and bouncing thin shells with rotation, framing it in the context of the weak cosmic censorship conjecture. The analysis is based on a formalism developed specifically for higher odd dimensions that is able to describe the dynamics of collapsing rotating shells exactly. We analise and classify a plethora of shell trajectories in asymptotically flat spacetimes. The parameters varied include the shell's mass and angular momentum, its radial velocity at infinity, the (linear) equation-of-state parameter and the spacetime dimensionality. We…
We investigate gravitational collapse of a spherically symmetric thin shell in the Einstein-Gauss-Bonnet (EGB) gravity. Under the recently proposed 4D limit, we find that the collapsing shell will be bounced back at a small radius, without forming a singularity. This bouncing behavior is similar to those of a test particle and a homogeneous spherical dust star, in accordance with the expectation that the Gauss-Bonnet term will modify the small scale behavior of the Einstein gravity. We analyze the causal structure of the dynamic spacetime that represents the bouncing process, finding that the …
Galileon radiation in the collapse of a thin spherical shell of matter is analyzed. In the framework of a cubic Galileon theory, we compute the field profile produced at large distances by a short collapse, finding that the radiated field has two peaks traveling ahead of light fronts. The total energy radiated during the collapse follows a power law scaling with the shell's physical width and results from two competing effects: a Vainshtein suppression of the emission and an enhancement due to the thinness of the shell.
We study the formation of the (noncommutative) Schwarzschild black hole from collapsing shell {of the} generalized matters containing polytropic and Chaplygin gas. We show that this collapsing shell depending on various parameters forms either a black hole or a naked singular shell with the help of the pressure.Furthermore, by considering the smeared gravitational sources, we investigate the noncommutative black holes formation. Though this mild noncommutative correction of matters cannot ultimately resolve the emergence of the naked singularity, we show that in some parameter region the colla…
We study gravitational perturbations sourced by a rotating test shell collapsing into five-dimensional Myers-Perry black holes in anti-de Sitter (AdS). Our attention is restricted to the case in which the two possible angular momenta of the geometry are set equal. In this situation the background is cohomogeneity-1, which amounts to a crucial technical simplification. It is found that the linearized Einstein equations are consistent only when the test shell is corotating with the spacetime. However, it is argued that this is a consequence of the matter on the shell being described by dust or, …
We study the gravitational collapse of two thin shells of matter, in asymptotically flat spacetime or constrained to move within a spherical box. We show that this simple two-body system has surprisingly rich dynamics, which includes prompt collapse to a black hole, perpetually oscillating solutions or black hole formation at arbitrarily large times. Collapse is induced by shell crossing and the black hole mass depends sensitively on the number of shell crossings. At certain critical points, the black hole mass exhibits critical behavior, determined by the change in parity (even or odd) of the…
We investigate the effect of induced massive radiation given off during the time of collapse of a massive spherically symmetric domain wall in the context of the functional Schrödinger formalism. Here we find that the introduction of mass suppresses the occupation number in the infrared regime of the induced radiation during the collapse. The suppression factor is found to be given by $e^{-βm}$, which is in agreement with the expected Planckian distribution of induced radiation. Thus a massive collapsing domain wall will radiate mostly (if not exclusively) massless scalar fields, making it dif…
Hawking radiation explicitly depends only on the black hole's total mass, charge and angular momentum. It is therefore generally believed that one cannot reconstruct the information about the initial mass distribution of an object that made the black hole. However, instead of looking at radiation from a static black hole, we can study the whole time-dependent process of the gravitational collapse, and pre-Hawking radiation which is excited because of the time-dependent metric. We compare radiation emitted by a single collapsing shell with that emitted by two concentric shells of the equivalent…
The problem of how space-time responds to gravitating quantum matter in full quantum gravity has been one of the main questions that any program of quantization of gravity should address. Here we analyze this issue by considering the quantization of a collapsing null shell coupled to spherically symmetric loop quantum gravity. We show that the constraint algebra of canonical gravity is Abelian both classically and when quantized using loop quantum gravity techniques. The Hamiltonian constraint is well defined and suitable Dirac observables characterizing the problem were identified at the quan…
We investigate the fate of the classical singularity in a collapsing dust cloud. For this purpose, we quantize the marginally bound Lemaitre-Tolman-Bondi model for spherically-symmetric dust collapse by considering each dust shell in the cloud individually, taking the outermost shell as a representative. Because the dust naturally provides a preferred notion of time, we can construct a quantum mechanical model for this shell and demand unitary evolution for wave packets. It turns out that the classical singularity can generically be avoided provided the quantization ambiguities fulfill some we…
We study Hawking radiation on the quantum spacetime generated by a quantum ingoing null shell in the 2d theory proposed by Callan-Giddings-Harvey-Strominger (CGHS). The quantum spacetime is a superposition of classical geometries with uncertainty in position and momentum of the collapsing shell. The Hawking radiation spectrum is computed, revealing a non-thermal behavior for finite time as well as a dependence on the shell's physical degrees of freedom. Hawking radiation's dependence on the collapsing shell becomes irrelevant in the late time approximation as we reach $i^+$ since the radiation…
We study the motion of thin-shell bubbles and their tunneling in anti de Sitter (AdS) background. We are interested in the case when the outside of a shell is a Schwarzschild-AdS space (false vacuum) and the inside of it is an AdS space with a lower vacuum energy (true vacuum). If a collapsing true vacuum bubble is created, classically it will form a Schwarzschild-AdS black hole. However, this collapsing bubble can tunnel to a bouncing bubble that moves out to spatial infinity. Then, although the classical causal structure of a collapsing true vacuum bubble has the singularity and the event ho…
We study the harvesting of correlations by two Unruh-DeWitt static detectors from the vacuum state of a massless scalar field in a background Vaidya spacetime consisting of a collapsing null shell that forms a Schwarzschild black hole (hereafter Vaidya spacetime for brevity), and we compare the results with those associated with the three preferred vacua (Boulware, Unruh, Hartle-Hawking-Israel vacua) of the eternal Schwarzschild black hole spacetime. To do this we make use of the explicit Wightman functions for a massless scalar field available in (1+1)-dimensional models of the collapsing spa…
We study time-dependant Hawking-like radiation as seen by an infalling observer during gravitational collapse of a thin shell. We calculate the occupation number of particles whose frequencies are measured in the proper time of an infalling observer in Eddington-Finkelstein coordinates. We solve the equations for the whole process from the beginning of the collapse till the moment when the collapsing shell reaches zero radius. The radiation distribution is not thermal in the whole frequency regime, but it is approximately thermal for the wavelengths of the order of the Schwarzschild radius of …
Hawking showed that the radiation emitted from a classical collapsing shell is thermal. Here, we show that a semiclassical collapsing shell emits radiation that is only approximately thermal, with small but significant deviations. The most important difference is the presence of small off-diagonal elements in the radiation density matrix with a magnitude of order $1/\sqrt{S_{BH}}$, $S_{BH}$ being the Bekenstein-Hawking entropy of the incipient black hole. The off-diagonal elements store the correlations between the collapsing shell and the emitted radiation and allow information to continuousl…
We study the quantum effects of a test Klein-Gordon field in a Vaidya spacetime consisting of a collapsing null shell that forms a Schwazschild black hole, by explicitly obtaining, in a $(1+1)$-dimensional model, the Wightman function, the renormalised stress-energy tensor, and by analysing particle detector rates along stationary orbits in the exterior black hole region, and make a comparison with the folklore that the Unruh state is the state that emerges from black hole formation. In the causal future of the shell, we find a negative ingoing flux at the horizon that agrees precisely with th…
Inhomogeneous dust collapse in five-dimensional AdS-Chern-Simons gravity
Facultad de Ciencias, Universidad Arturo Prat（阿图罗·普拉特大学理学院）；Instituto de Ciencias Exactas y Naturales (ICEN), Universidad Arturo Prat（阿图罗·普拉特大学精确与自然科学研究所）；Departamento de Ciencias Exactas, Facultad de Ingeniería, Universidad San Sebastián（圣塞巴斯蒂安大学工程学院精确科学系）
Using the formalism of the interaction picture we calculate an expression for the Wightman function for only the spherically symmetric modes of a quantum Klein-Gordon scalar field in a general Vaidya spacetime with ingoing null dust. It is demonstrated that particle detectors following time-like trajectories that are in the ground state at some time outside of the collapsing shell will respond independently from the configuration of the ingoing null dust if the response is taken at any time outside of the collapsing shell. For detectors that are taken to be in the ground state at a time interi…
We study Hawking radiation on the quantum space-time of a collapsing null shell. We use the geometric optics approximation as in Hawking's original papers to treat the radiation. The quantum space-time is constructed by superposing the classical geometries associated with collapsing shells with uncertainty in their position and mass. We show that there are departures from thermality in the radiation even though we are not considering back reaction. One recovers the usual profile for the Hawking radiation as a function of frequency in the limit where the space-time is classical. However, when q…
We study Hawking radiation on a Vaidya space-time with a gravitational collapse followed by evaporation. The collapsing body is a null thin-shell and the evaporation is induced by a negative energy collapsing null-shell. This mimics the back-reaction to the Hawking radiation. Using Hawking's original method of Bogoliubov transformations we characterize the radiated spectrum as dominated by a thermal emission with an increasing effective temperature. We compute this time dependent temperature and find numerical agreement with results obtained by other techniques. The known divergences at the ev…
As an attempt to solve the black hole information loss paradox, recently there has been the suggestion that, due to semi-classical effects, a pre-Hawking radiation must exist during the gravitational collapse of matter, which in turn prevents the apparent horizon from forming. Assuming the pre-Hawking radiation does exist, here we argue the opposite. First we note that the stress energy tensor near the horizon for the pre-Hawking radiation is far too small to do anything to the motion of a collapsing shell. Thus the shell will always cross the apparent horizon within a finite proper time. More…
We describe a thermal atmosphere around a black hole as vacuum excitations near to gravitational radius of a contracting thin black shell, i.e., in terms of properties of the physical vacuum of fields around a thin shell of mass $M$ collapsing from infinity to the Schwarzschild radius according to an external stationary observer. A natural explanation is introduced for the necessary cutoff using the equations of motion of the shells. We make a thermodynamic description of a fermionic field near the gravitational radius. Then a solution to the species problem for two fields, scalar one and spin…
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…
We consider a static spherically symmetric thin shell wormhole collides with another thin shell consisting of ordinary matter. By employing the geometrical constraint, which leads to the conservation of energy and momentum, we show that the state after the collision can be solved from the initial data. In the low speed approximation, the solutions are rather simple. The shell may either bounce back or pass through the wormhole. In either case, the wormhole shrinks right after the collision. In the "bouncing" case, a surprising result is that the radial speeds before and after the collision sat…
We consider two spatially flat Friedmann-Robertson-Walker spacetimes divided by a time-like thin shell in the nontrivial case in which the inner region of finite extension contains radiation and the outer region is filled with dust. We will then show that, while the evolution is determined by a large set of constraints, an analytical description for the evolution of the bubble radius can be obtained by formally expanding for short times after the shell attains its minimum size. In particular, we will find that a bubble of radiation, starting out with vanishing expansion speed, can be matched w…
