Classical Collapse to Black Holes and Quantum Bounces: Abstract, Introduction

Abstract

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In the last four decades, different programs have been carried out aiming at understanding the final fate of gravitational collapse of massive bodies once some prescriptions for the behaviour of gravity in the strong field regime are provided. The general picture arising from most of these scenarios is that the classical singularity at the end of collapse is replaced by a bounce. The most striking consequence of the bounce is that the black hole horizon may live for only a finite time. The possible implications for astrophysics are important since, if these models capture the essence of the collapse of a massive star, an observable signature of quantum gravity may be hiding in astrophysical phenomena. One intriguing idea that is implied by these models is the possible existence of exotic compact objects, of high density and finite size, that may not be covered by an horizon. The present article outlines the main features of these collapse models and some of the most relevant open problems. The aim is to provide a comprehensive (as much as possible) overview of the current status of the field from the point of view of astrophysics. As a little extra, a new toy model for collapse leading to the formation of a quasi static compact object is presented.

 

1. Introduction

Our present understanding of the universe and its evolution implies the existence of black holes, bodies whose masses are packed in such small volumes that not even light can escape. We have experimental evidence that such objects do exist from observations of x-ray binaries, which suggests the existence of stellar mass black holes in binary systems, and from the spectral properties of quasars and active galactic nuclei, which suggest that super-massive black holes dwell at the center of most galaxies.
From a theoretical point of view, black holes are a direct consequence of the fact that we must use General Relativity (GR) to describe the late stages of gravitational collapse. For collapsing matter sources satisfying standard energy conditions, Einstein’s field equations imply that eventually collapse must lead to the formation of trapped surfaces and a singularity [1,2,3]. The generic existence of singularities in solutions of Einstein’s field equations is a troublesome issue for classical GR as their presence signals a regime where predictability breaks down [4] and the theory does not hold. The most conservative view on singularities is that they are a consequence of the application of the theory in a regime where quantum effects become important and thus they should not appear in a full theory of quantum gravity. This point of view is usually traced back to Wheeler (see for example [5] for an historical overview).
Our theoretical understanding of how black holes form is rooted in the simplest toy model for spherical collapse that was developed by Oppenheimer and Snyder [6] and independently by Datt [7] in 1939 (from now on referred to as the OSD model). On the one hand, we know that GR works very well in the weak field and we are confident that models of collapse such as OSD are accurate to describe the fundamental features of collapse scenarios far away from the singularity. On the other hand, the behavior of gravity in the strong field is not well understood and even though we do know that GR requires modifications in the regime where the gravitational field becomes strong over very small scales, we still do not know what kind of form such modifications should take. Black hole horizons somehow stand at the crossroad between these two situations. For example, the event horizon for a Schwarzschild black hole sits comfortably far from the strong gravity region but its description is strongly linked to the existence of the singularity and modifications to our classical models in the vicinity of the singularity bear important consequences for the behavior of the horizon itself.
The question of how gravity behaves in the strong field is also tightly connected to the question ‘What happens to matter when it is compressed to volumes small enough that the classical description fails?’ Therefore one could expect that both sides of Einstein’s equations (the geometrical side containing Einstein’s tensor Gμν and the matter side containing the energy-momentum tensor Tμν) will need to be modified in the strong field regime.
The investigation of simple analytical toy models in quantum gravity is motivated by the idea that these scenarios could provide valuable information about the general features that a full theory of quantum gravity should posses. The general view that is taking shape in the last couple of decades is that quantum effects will generate repulsive pressures that are sufficient to counteract the gravitational attraction, thus avoiding the singularity at the end of collapse. The first, immediate consequence, is that it should be impossible to form a Schwarzschild (or Kerr) black hole within a full theory of quantum gravity. Then different possibilities may arise depending on the assumptions made in the model. For example, collapse may stop before the formation of the horizon, leaving behind an exotic compact object. Alternatively collapse may lead to the formation of the horizon and eventually halt at much smaller scales. In this last, case collapsing matter would eventually bounce thus leading to a phase of re-expansion following collapse. The re-expanding phase may not affect the geometry in the exterior, thus leaving an object that looks like a black hole for distant observers, or it may trigger a transition of the black hole horizon to a white hole horizon.
In the present article we focus on modifications to dynamical scenarios where the black hole forms from regular initial data. However, it should be noted that the issue of how the static Schwarzschild black hole solution can be altered by the introduction of quantum effects has been addressed by several authors in different frameworks. For example, solutions within classical GR describe what are now called ‘regular black holes’ (see for example [8,9] for the earliest results, or [10,11] for more recent discussions). These are modifications of the Schwarzschild solution that imply a minimum length scale and a non vanishing energy-momentum tensor. Rotating regular black holes were considered for example in [12], their properties as candidates for astrophysical objects were studied for example in [13,14], while the extension to the case with non vanishing cosmological constant was studied in [15]. Other solutions that modify the Schwarzschild black hole can be obtained via the introduction of quantum effects. These are vacuum solutions within a quantum gravitational description of the black hole space-time. For example, approaches based on Loop Quantum Gravity (LQG) have been studied in [16,17,18,19]. Other approaches have been also considered. For example, an improved Schwarzschild solution based on renormalization group and running gravitational constant was described in [20], while a quantization approach based on canonical formalism was presented in [21] and a discussion of the meaning of the gravitational Schwarzschild radius within a quantum theory was presented in [22]. To this aim a quantum mechanical operator acting on the ‘horizon wave-function’ was introduced. All these models show similar features, like, for example, the appearance of an inner horizon inside the event horizon (see for example [23] for a study on the properties of the inner horizon of the solution presented in [20,24] for a more recent study with cosmological constant) or the possible existence of a massive, dense compact object of finite size smaller than the horizon. However the most important insight coming from the study of quantum corrected black holes is the realization that quantum effects may modify the geometry at large scales thus having implications for the description of the horizon of the black hole.
The focus of the present article is on existing dynamical models of collapse, and on how the introduction of ‘quantum’ effects may alter the standard picture of black hole formation. We will review the current status of our understanding of gravitational collapse when some kind of repulsive effects (that can be interpreted as quantum corrections) are incorporated as the gravitational field becomes strong. In dynamical models the main mechanism that leads to the avoidance of the singularity is when collapse halts at a finite radius, possibly producing a bounce for the infalling matter. As mentioned before, the cloud may halt at a radius larger than the Schwarzschild radius, thus never forming a black hole. In this case it may produce a compact object or it may completely evaporate (see for example, [25] for an example of compact object, [26,27,28] for the effects of black hole evaporation on gravitational collapse and [29,30] for examples of evaporation including 4D Weyl anomaly). Alternatively the cloud may reach a minimum scale and re-expand. If the the repulsive effects close to the bounce influence the geometry in the vacuum exterior then the black hole effectively turns into a white hole (see for example [31,32,33]). On the other hand, if repulsive effects are confined to a small neighborhood of the center then external observers would effectively see a black hole (see for example [34]). Our aim is to point out the main outstanding unresolved issues in theoretical models and how such models might be tested against observations of energetic phenomena in the universe in the near future.
The first proposed model of this kind involved the collapse of a thin light-like shell in a quantum gravity scenario in which the effective Lagrangian was described by the Einstein’s Lagrangian plus the leading higher order terms that describe gravity over short distances [34,35]. There it was shown for the first time that quantum gravitational effects can avoid the occurrence of singularities and, as a consequence, shorten the life of the black hole horizon.
In more recent times the renewed interest for quantum corrections to classical singularities in collapse models was sparked by Loop Quantum Cosmology (LQC). Several authors, based on considerations coming from LQG, showed how quantum corrections near the big bang singularity can remove the initial singularity and produce a bouncing universe (see for example [36,37,38]). Since the simplest relativistic toy models for collapse are essentially the time reversal of big bang models, the same formalism derived from LQC can in principle be applied to collapse (see for example [39]).
However, the collapse of a massive star is very different from the evolution of the universe. Above all, the most notable difference that one has to deal with, when considering collapse, is the matching to an exterior manifold describing the space-time around the collapsing object. In the OSD model, an horizon develops in the exterior once the infalling matter crosses the Schwarzschild radius. How the bounce will affect the horizon in the vacuum exterior is presently still not entirely clear. If quantum effects are confined inside the collapsing matter then the light-cone structure of the space-time must undergo a discontinuous transition from the interior to the exterior. On the other hand if quantum effects propagate in the exterior then the black hole geometry must be altered.
As mentioned above, most dynamical models with ‘quantum’ corrections studied to date result in a bouncing scenario. These models suggest the possibility of the formation of exotic compact remnants as leftovers from collapse and recently there has been a lot of interest in the possible phenomenology of such remnants. One could ask whether a regular black hole can form through a dynamical process (see for example [40]), or what kind of properties such remnants would have (see for example [41]). The idea that compact objects other than neutron stars can form from collapse has been around for a long time. For example, exotic compact objects were proposed by Hawking already in 1971 [42]. Just like the electron degeneracy pressure can stabilize collapse leading to a white dwarf and the neutron degeneracy pressure can stabilize it to produce a neutron star, it seems reasonable to suppose that a further island of stability may exist at densities higher than neutron star’s cores for a yet unknown state of matter. Along these lines several kinds of exotic compact objects have been proposed through the years. For example objects like gravastars (‘gravitational vacuum stars’, obtained from a phase transition of quantum vacuum near the location of the horizon) (see [25,43]) and black stars (see [44]) have a radius slightly larger than the Schwarzschild radius. On the other hand theoretical objects like quark stars (see [45,46]) and boson stars (see [47,48]) have a larger boundary (larger than the photon sphere for black holes). The properties of these proposed objects have been studied in detail. Also, arguments for the existence of compact remnants left over after black hole evaporation were proposed in connection with the information loss problem (see for example [49,50,51] and references therein). In these scenarios, a compact object of Planck scale may be the residue of the complete evaporation of the black hole via Hawking radiation. The question of which kind of dynamical process may lead to the formation of such objects remains open and it is closely connected to how matter and gravity behave at extremely high densities.
The paper is organized as follows: In Section 2 we review the OSD model for dust collapse and Einstein’s equations for the collapse of homogeneous dust and perfect fluids (the reader familiar with such topics can jump directly to the next section). Section 3 is devoted to a review of semi-classical corrections to collapse and their consequences for the OSD model and black hole formation in general. In Section 4 we outline the main open questions related to such modified collapse models. In Section 5we explore the phenomenological consequences that semi-classical collapse bears for astrophysical black holes and we investigate the possibility that exotic compact objects and remnants may occur as leftover from collapse. In Section 5 we also introduce a new dynamical toy model leading to one of such hypothetical remnants (which we call a dark energy star). Finally Section 6 is devoted to a brief discussion of the present and future status of the field.
Bullet points are used throughout the sections in order to highlight the separation of each topic from the next. Finally, throughout the paper we will make use of geometrical units by setting G=c=1 and for simplicity, we will absorb the factor 8π in Einstein’s equations into the definition of the energy momentum tensor.  Academic Editors: Gonzalo J. Olmo and Diego Rubiera-Garcia,  Daniele Malafarina
By:  Dr. TJ Gunn Ph.D. Applied Physics, Ph.M., Masters Philosophy

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