To further complicate matters (or, actually, making them more interesting) some parts of the Schwarzschild solution can be faithfully described with minisuperspace models. This is the case for the interior Schwarzschild metric that can be isometrically mapped to a Kantowski–Sachs model. As we will mention later, this fact has been used to discuss issues related to the quantum resolution of black hole singularities by adapting LQC methods. Though this Living Review is devoted to midisuperspace models, for the sake of completeness we deem it necessary to include a discussion of the relevant results obtained in this minisuperspace setting, especially those devoted to black hole singularity resolution, before dealing with the midisuperspace aspects of spherical symmetry.

Early suggestions of singularity resolution coming from the use of LQG inspired techniques go back to the works of Husain, Winkler and Modesto. In [167] an exotic quantization developed in [122] was used to study singularity resolution in a geometrodynamical description of spherical black holes. Later, in [125], the well-known fact that the interior of a Schwarzschild black hole is isometric to the Kantowski–Sachs homogeneous cosmological model is mentioned. This idea, combined with the exotic quantizations considered in [125], leads in a natural way to the claim that black hole singularities are resolved in this scheme. Though some attempts to derive this result within the LQG framework by quantizing the Kantowski–Sachs model appear in the literature [168, 169], the first complete and rigorous account of it is due to Ashtekar and Bojowald [7]. In this paper the authors import some of the results obtained in the framework of LQC to the study of black hole interiors and the issue of singularity resolution. The quantization is performed by first describing the system in real connection variables. This is important because the degenerate triads play a significant role in the quantization process – in particular the phase-space points corresponding to them do not lie on a boundary of the constraint hypersurface. These singular configurations can be naturally accommodated in the Ashtekar variables framework but not in standard geometrodynamics. Then the loop quantization of the Kantowski–Sachs model is carefully carried out to completion. Some important results can be derived from it, in particular it is possible to prove that the physical spacetime corresponding to the interior of a black hole does not end at the classical singularity but can be extended beyond it. On the other hand the quantum fluctuations close to the singularity are such that a classical description breaks down. The issues of matter coupling and the extension of the quantization to the exterior region outside the horizon were not considered in [7]. These problems and the work related to them will be mentioned and described in Section 5.3.3, which is devoted to the loop quantization of spherically-symmetric spacetimes.

Within a semiclassical approach the study of what is the kind of physical object that replaces the classical singularity after quantization has been attempted by several researchers by employing different approximations. In [39] the authors suggest that the original quantization used in [7] may not have the correct semiclassical limit (as it happens when an analogue quantization is performed in LQC). The proposed way to remedy this problem is to allow the parameter (playing the role of the polymeric parameter of LQC) appearing in the construction of the Hamiltonian constraint operator to be a function of the triads (in the same way as is allowed to depend on the scale factor in LQC) and work with an effective Hamiltonian constraint. This generalizes and improves the results of [171] (see also [173] for other different types of suggested improvements) where Modesto uses exactly the approach of [7]. Though the results presented in these works are not rigorous derivations within LQG, they support the idea that the singularity is resolved and replaced by either a wormhole type of solution or a “swollen” singularity described by a spherical surface that is asymptotically approached but never reached by infalling particles. Further refinements on the ideas and approximations presented in these papers can be found in [64] (where a fractal type of spacetime is generated by the creation of a series of smaller and smaller black holes spawned by quantum collapses and bounces).

More general examples have been considered in the minisuperspace framework and using LQG inspired methods. For example, the quantum collapse of a spherical dust cloud is described in [172] both in the ADM and Ashtekar formulations and some issues concerning black hole evaporation are discussed in [174]. Additional references on the minisuperspace treatment of black hole singularities are [170, 187, 188].

In this section, we will review the literature relevant to study the geometrodynamical quantization of spherical midisuperspace models. The upcoming Section 5.3.3 will describe the results that are being obtained these days by using LQG-inspired methods. As a general comment we would like to say that the level of mathematical rigor used in the standard quantization of these systems is sufficient in some cases but, as a rule, the attention payed to functional analytic issues and other mathematical fine points does not reach the level that is standard now in LQG. This should not be taken as a criticism but as a challenge to raise the mathematical standards. In our opinion a rigorous treatment within the geometrodynamical framework could be very useful in order to deepen our understanding of quantum gravity (see [3] for some interesting results in this respect).

A fundamental paper in the study of the classical and quantum behavior of spherically-symmetry reductions of GR is due to Kuchař [143]. This work provides several key results for these systems within the geometrodynamical framework. For example, it gives canonical coordinates leading to a very simple description of the reduced phase space for an eternal black hole and explains along the way how the transformation to this coordinate system can be obtained from the knowledge of the extended Schwarzschild solution (in Kruskal coordinates). This canonical transformation is widely used to analyze the Hamiltonian dynamics of other spherically-symmetric gravitational systems coupled to matter as will be commented on later in this section. This work discusses the quantization of the system in several possible schemes: the direct reduced phase-space quantization and the Dirac approach. Finally, it proves the unitary equivalence of the resulting quantum models. This paper is a basic reference where subtle but important issues are taken into account, in particular those related to the asymptotic behavior of the fields and the introduction of the necessary boundary terms in the Einstein–Hilbert action. It must be said, nonetheless, that the identification of the canonical pair of variables that describe the reduced phase space for eternal black holes was first found by Kastrup and Thiemann in their study of the very same system within the Ashtekar variables approach [201, 130].

The quantization of Schwarzshild black holes was also considered around the same time in [58, 59]. These papers rely on the study of what the authors call the r-dynamics. The reason they have to use some non-standard dynamics for the system to develop a Hamiltonian formalism is that the parametrization of the spherical metrics that they employ involves only functions of a single radial coordinate r (and do not involve an additional “time” variable t). This amounts to introducing from the start the ansatz that the metrics are not only spherical but also static. These papers also rely on the use of a rather formal Wheeler–DeWitt approach to quantization. This means that some well-known consistency problems are present here. It must be said that, at the end of the day, some of the results of [58, 59] are compatible with the ones discussed in [143] and [130] (in particular the unavoidable identification of the black hole mass as the configuration variable for the spherical–black-hole reduced phase space, but in our opinion this approach has been superseded by the treatment provided by Kuchař, Kastrup and Thiemann that considers the full Kruskal spacetime for eternal black holes and discuses the Hamiltonian framework and the identification of the reduced phase space in that setting.

An interesting way to go beyond vacuum spherically-symmetric gravity is by coupling a single particle-like degree of freedom, which in this case corresponds to an infinitely thin spherical matter shell. After the system is quantized, this shell degree of freedom can be conceivably used as a quantum test particle allowing us to extract interesting information about the (spherical) quantized geometry. The paper by Berezin, Boyarsky and Neronov [33] studies this system both from the classical and quantum points of view by using the standard geometrodynamical approach. One of the key ingredients is the use of the canonical transformation introduced by Kuchař in his famous paper on spherically-symmetric gravity [143]. Although the quantization presented in [33] is quite formal, some interesting features are worth comment, in particular the fact that the Schrödinger equation becomes a difference equation (a feature reminiscent of results obtained with LQG methods). By using analyticity arguments the authors identify the quantum numbers characterizing the system (actually a pair of integer numbers that parameterize the mass spectrum). This result, however, does not correspond to the one proposed by Bekenstein and Mukhanov and derived by other authors in different frameworks.

The collapse of a thin null dust shell has been extensively studied by Hájíček in the context of a midisuperspace quantization [107, 108, 109, 110, 111]. Classically this system gives rise to black holes and, hence, its quantization may shed light on the issue of singularity resolution. In this case the unitary evolution of the wave packets representing the collapsing shell degree of freedom suggest that the singularity is resolved because they vanish at the place where the singularity is expected to be (see [108] and references therein).

A series of papers [219, 215, 221, 222, 223, 220, 217, 218, 216, 135, 134] dealing with the quantization of spherically-symmetric models coupled to matter is due to Kiefer, Louko, Vaz, Witten and collaborators. A very nice summary of some of these results appears in the book by Kiefer [132]. We will describe them briefly in the following. The first paper that we will consider is [219]. Here the authors use the Kuchař canonical quantization and the idea of introducing a preferred dust-time variable (the Brown–Kuchař proposal [52]) to quantize eternal black holes. By an appropriate selection of the lapse function it is possible to make the proper dust time to coincide with the proper time of asymptotic observers. This selects a particular time variable that can be used to describe the system (furthermore the total energy can be seen to be the ADM mass of the black hole). An interesting result derived in this reference is the quantization law for the black hole mass of the type (where denotes the Planck mass and a positive integer) for the normalized solutions of the Wheeler–DeWitt equation with definite parity (notice that this is precisely the mass spectrum proposed by Mukhanov and Bekenstein [31, 178]). It is interesting to point out that the authors do not use a purely reduced phase-space quantization, but a Dirac approach. This means that the wave functions for the quantized model do not only depend on the mass (in the vacuum case) but also on some embedding variables that can be used to distinguish between the interior and the exterior of the black hole. The matching conditions for the wave function in these two regions give rise to the quantization of the mass. The fact that the wave function of the system vanishes outside and is different from zero inside can be interpreted by realizing that observers outside the horizon should see a static situation. The results obtained by Kastrup, Thieman [201] and Kuchař [143] have also been used by Kastrup in [129] to get similar results about the black hole mass quantization.

The methods introduced in [219] were used in the midisuperspace approach for the computation of black hole entropy discussed in [215]. Here the authors have shown that it is actually possible to reproduce the Bekenstein–Hawking law for the entropy as a function of the black hole area after introducing a suitable microcanonical ensemble. In order to do this, they first show that the Wheeler–DeWitt wave functional can be written as a direct product of a finite number of harmonic oscillator states that can themselves be thought of as coming from the quantization of a massless scalar field propagating in a flat, 1+1 dimensional background. The finite number of oscillators that originates in the discretization introduced on the spatial hypersurfaces is then estimated by maximizing the density of states. This is proportional to the black hole area and suggests that the degrees of freedom responsible for the entropy reside very close to the horizon. A curious feature of the derivation is the fact that, in order to get the right coefficient for the Bekenstein–Hawking area law, an undetermined constant must be fixed. This is reminiscent of a similar situation in the standard LQG derivation [4], where the value of the Immirzi parameter must be fixed to get the correct coefficient for the entropy. In our opinion there are some arbitrary elements in the construction, such as the introduction of a discretization, that make the entropy computation quite indirect.

Partially successful extensions of the methods used in [219, 215] to the study of other types of black holes appear in [221, 222, 223]. In the first of these papers the authors consider charged black holes. By solving the functional Schrödinger equation it is possible to see that the difference of the areas of the outer and inner horizons is quantized as integer multiples of a single area. This is similar to the Bekenstein area quantization proposal but not exactly the same because the areas themselves are not quantized and the entropy is proportional to this difference of areas. The second paper studies gravitational collapse as described by the LeMaître–Tolman–Bondi models of spherical dust collapse and considers a Dirac quantization of this midisuperspace model. The main technical tool is, again, a generalization of the methods developed by Kuchař, in particular the canonical transformations introduced in [143]. In addition, the authors use the dust as a way to introduce a (natural) time variable following the Brown–Kuchař proposal. The physical consequences of this quantization (for the marginally bound models) are explored in [220, 217, 135, 134] where the authors describe Hawking radiation and show that the Bekenstein area spectrum and the black hole entropy can be understood in a model of collapsing shells of matter. In particular, the mass quantization appears as a consequence of matching the wave function and its derivative at the horizon. This result is compatible with the one mentioned above for the Schwarzschild black hole [219]. Some subtle issues, involving regularization of the Wheeler–DeWitt equation are sidestepped by using the DeWitt regularization , but the whole framework is quite attractive and provides a nice perspective on quantum aspects of gravitational collapse. In [220] dust is modeled as a system consisting of a number of spherical shells. The entropy of the black holes formed after the collapse of these N shells depends, naturally, on N. For black holes in equilibrium the authors estimate this number by maximizing the entropy with respect to N (this is similar to the result mentioned before for the Schwarzschild black hole). As far as Hawking radiation is concerned [217] the authors model this system by taking the dust collapse model as a classical background and quantizing a massless scalar field by using standard techniques of quantum field theory in curved spacetimes. They separately consider the formation of a black hole or a naked singularity. In the first case they find that Hawking radiation is emitted whereas in the second one the breaking of the semiclassical approximation precludes the authors from deriving meaningful results. The treatment provided in this paper is only approximate (scalar products are not exactly conserved) and semiclassical (the WBK approximation is used), but the resulting picture is again quite compelling.

Black holes in AdS backgrounds have been considered in [218, 216] (see also [89]). The most striking result coming from the analysis provided in these papers is the fact that different statistics (Boltzmann and Bose–Einstein) must be used in order to recover the correct behavior of the entropy from the quantization of area in the Schwarzschild and large cosmological constant limits respectively. Finally, we want to mention that the collapse of null dust clouds has been partially discussed in [223].

A number of papers by Louko and collaborators [150, 152, 149] study the Hamiltonian thermodynamics of several types of black holes, in particular of the Schwarzschild, Reissner–Nordström–anti-de Sitter types, and black holes in Lovelock gravity. The idea proposed in [150] is to consider a black hole inside a box and use appropriate boundary conditions to fix the temperature. The black hole thermodynamics can now be described by a canonical ensemble and standard statistical physics methods can be used to compute the entropy. In particular the authors provide a Lorentzian quantum theory and obtain from it a thermodynamical partition function as the trace of the time evolution operator analytically continued to imaginary time. From this partition function it is possible to see that the heat capacity is positive and the canonical ensemble thermodynamically stable. One of the remarkable results presented in [150] is that the partition function thus obtained is the same as the one previously found by standard Euclidean path integral methods [227, 50].

Canonical transformations inspired in the one introduced by Kuchař in [143] play a central role in all these analyses. Also the methods developed in [143] have been extended to study the thermodynamics of spherically-symmetric Einstein–Maxwell spacetimes with a negative cosmological constant [152] and spherically-symmetric spacetimes contained in a one parameter family of five-dimensional Lovelock models [149]. In both cases the canonical transformation is used to find the reduced Hamiltonian describing these systems. The most important conclusion of these works is that the Bekenstein–Hawking entropy is recovered whenever the partition function is dominated by a Euclidean black hole solution. In the Lovelock case [149] the results suggest that the thermodynamics of five-dimensional Einstein gravity is rather robust with regard to the the introduction of Lovelock terms. Another paper where the Kuchař canonical transformation is used is [133], where the authors consider extremal black holes and how their quantization can be obtained as a limit of non-extremal ones. The obtention of the Bekenstein area quantization in this setting (for Schwarzschild and Reissner–Nordström black holes) is described in [148, 155].

We end with a comment about the quantization of spherically-symmetric GR coupled to massless scalar fields. This has been considered by Husain and Winkler in [124]. In this paper the authors study this problem in the geometrodynamical setting by using Painlevé–Gullstrand coordinates (that are especially suitable for this system). They use the non-standard quantization described in [125] that displays some of the features of LQG and suggest that the black hole singularity is resolved. A definition of quantum black hole is proposed in the paper. The key idea is to use operator analogues of the classical null expansions and trapping conditions. As the authors emphasize, their proposal can be used in dynamical situations (at variance with isolated horizons [10]). The construction of semiclassical states in this context and their use is further analyzed in [121].

Historically the first author to consider the treatment of spherically-symmetric gravity within the Ashtekar formalism was Bengtsson in [32]. There he started to develop the formalism needed to describe spherically-symmetric complex SU(2) connections and densitized triads and used it to discuss some classical aspects related to the role of degenerate metrics in the Ashtekar formulation and its connection with Yang–Mills theories. The quantization of spherically-symmetric midisuperspace models written in terms of Ashtekar variables was undertaken rather early in the development of LQG, by Kastrup and Thiemann [201]. At the time the mathematical setting of LQG was at an early stage of development and, hence, the quantization that the authors carried out was not based in the proper type of Hilbert space and made use of the old complex formulation. Fortunately, the reality conditions could be handled in the spherically-symmetric case. A key simplification is due to the fact that the constraints can be written as expressions at most linear in momenta. The resulting quantized model is essentially equivalent to the geometrodynamical one due to Kuchař [143]. An interesting point that Kastrup and Thiemann discuss in [130] is related to the physically-acceptable range of black hole masses (that somehow define the configuration space of the model, at least in the reduced phase-space formulation) and how this should be taken into account when representing the algebra of basic quantum operators. The (unitary) equivalence of the reduced and Dirac quantizations for this system – also found and discussed by Kuchař [143] – can be proven once the right ordering is found for the operators representing the constraints of the system.

The modern approach to spherical-symmetry reductions in LQG starts with [40], where Bojowald carefully introduces the necessary mathematical background to consider the quantization of spherically-symmetric models. In this paper Bojowald constructs the kinematical framework for spherically-symmetric quantum gravity by using the full LQG formalism; in particular he shows how the states and basic operators (holonomies and fluxes) can be derived from those in LQG. An important result of [40] is the realization of the fact that significant simplifications take place that make these symmetry reductions tractable. As expected they are midway between the full theory and the homogeneous models that have been considered in LQC. A very useful feature of the spherically-symmetric case is the commutativity of the flux variables (thus allowing for a flux representation in addition to the connection representation that can be used as in the case of homogeneous models). This work particularizes the general framework developed in [44] to study the quantum symmetry reduction of diff-invariant theories of connections based on the isolation of suitable symmetric states in the full 3+1 dimensional theory and the subsequent restriction to this subspace (defining quantum symmetry reductions).

The study of the volume operator for spherically-symmetric reductions was carried out in [45], where its basic properties were derived. In particular the volume operator was diagonalized and its spectrum explicitly obtained. An important property of the eigenfunctions is that they are not eigenstates of the flux operator (and, in fact, have a complicated dependence on the connection). The fact that on the volume eigenstates the holonomy operators have a complicated dependence makes it quite difficult to study the Hamiltonian constraint because it contains commutators of the volume with holonomies. These are difficult to compute because the volume eigenstates are not eigenstates of the triads (upon which the holonomies act in a simple way). Nonetheless, an explicit construction of the Hamiltonian constraint in the spherically-symmetric case – that makes use of non-standard variables that mix the connection and the extrinsic curvature – has been given in [46]. These new variables cannot be generalized, but are especially tailored for the spherically-symmetric case. The main consequence of using them is the simplification of the volume operator that they provide. This comes about because the volume and flux eigenstates coincide for the new variables. Notice, however, that the hard problem of finding the kernel of the Hamiltonian constraint still has to be solved.

An obvious application of the formalism developed in these papers is the study of singularity resolution for Schwarzschild black holes in LQG. As shown in [42] the structure of the quantized Hamiltonian constraint for spherical-symmetry reductions may allow us to understand the disappearance of spacelike singularities. This issue has also been considered for other spherically-symmetric models, such as the Lemaître–Tolman–Bondi collapse of an inhomogeneous dust cloud [43]. The (approximate) numerical analysis carried out in this paper shows a slow-down of the collapse and suggests that the curvature of naked singularities formed in gravitational collapse can be weakened by effects. This is in agreement with the behavior expected in LQG, where effective repulsive forces of a quantum origin usually make the singularities tamer.

The main problem of this spherically-symmetric approach followed by Bojowald and collaborators – as emphasized by Campiglia, Gambini and Pullin in [54] – is related to the difficulties in finding a particularization of the construction proposed by Thiemann for the Hamiltonian constraint with the appropriate constraint algebra in the diff-invariant space of states. This has led the authors of [54] to explore a different approach to the quantization of spherically-symmetric models in LQG [54, 55, 92, 93, 94] based on a partial gauge fixing of the diffeomorphism invariance.

The quantization of the exterior Schwarzschild geometry has been carried out in [54] where the asymptotic behavior of the fields relevant in this case was carefully considered. This corrected a problem in [46] related to the fall-off of some connection components. The gauge fixing is performed in such a way that the Gauss law is kept so that the reduced system has two sets of constraints – the Gauss law and the Hamiltonian constraint – with a non-trivial gauge algebra. Two approaches are then explored, in the first of them the standard Dirac method is used after abelianizing the constraints. The second is inspired by the fact that generically one does not expect this abelianization to be possible. This leads the authors to use uniform discretizations [82, 83] to deal with the general, non-abelianized constraints. Although the study of the exterior region of a black hole gives no information about singularity resolution, according to the authors there are hints of singularity removal because the discrete equations of the model are similar to those appearing in LQC. The result of the LQG quantization of the exterior of a spherically-symmetric black hole is in agreement with the one obtained by Kuchař in terms of the usual geometrodynamical variables; in particular, the number and interpretation of the quantum degrees of freedom (the mass of the spacetime) are the same in both approaches. This means that the quantum dynamics are trivial: wave functions depend only on the mass and do not evolve.

After studying the exterior problem, the interior problem for a Schwarzschild black hole is considered by these authors in [55] (in a minisuperspace model similar to the one by Ashtekar and Bojowald [7]). Here a suitable gauge fixing leads to a description in terms of a Kantowski–Sachs metric. In this case it is possible to describe the exact quantum evolution as a semi-classical one with quantum corrections. The model is quantized in the connection representation and behaves as a LQC model, where a certain type of bounce replaces the cosmological singularity. When the quantum solution evolves past the singularity it approaches another regime that behaves again in an approximate classical way.

The complete black hole spacetime has been considered in [92]. A key ingredient here is the choice of a gauge fixing such that the radial component of the triad is a function selected in such a way that in the limit when the “polymerization” parameter goes to zero one recovers the Schwarzschild metric in Kruskal coordinates. The authors give a classical metric that represents some of the features of the semiclassical limit for this spherical black-hole system, in which the singularity is effectively avoided. The suggested picture consists of an eternal black hole continued to another spacetime region with a Cauchy horizon. Far away from the singularity, the spacetime resembles the standard Schwarzshild solution.

All these papers deal with partially–gauge-fixed Hamiltonian systems. The issue of the residual diff-invariance in spherically-symmetric models quantized with LQG techniques is discussed in [93]. It is shown there that it is possible to reconstruct spacetime diffeomorphisms in terms of evolving constants of motion (Dirac observables on the physical Hilbert space), but some memory of the polymerization introduced by loop variables remains because only a subset of spacetime diffeomorphisms are effectively implemented. According to Gambini and Pullin this is a reflection of the fact that sub-Planck scales should behave differently from the macroscopic ones.

Finally, it is worth pointing out that the system consisting of spherical gravity coupled to a massless scalar field has been discussed in [94]. The same system was also considered by Husain and Winkler in [125] in the context of geometrodynamical variables. In [94] the authors use the uniform discretization technique to deal with the thorny problem of working with a Lie algebra of constraints with structure functions. Specifically they consider a discrete version of the master constraint and use a variational method to minimize it. It is important to understand that one should find the kernel of this master constraint. The fact that this is not achieved in the present model is interpreted by the authors as a hint that there is no quantum continuum limit. Nevertheless the theory provides a good approximation for GR for small values of the lattice separation introduced in the discretization. The lowest eigenvalue of the master constraint corresponds to a state with a natural physical interpretation, i.e., the tensor product of the Fock vacuum for the scalar field and a Gaussian state centered around a flat spacetime for the gravitational part. The authors argue that it is impossible that LQG regulates the short distance behavior of this model in the gauge that they use. This leads them to conclude that one should face the challenging problem of quantization without gauge fixing (keeping the diffeomorphism and Hamiltonian constraints).

To end this section we want to mention the paper by Husain and Winkler [125] that considers the quantization of the spherically-symmetric gravity plus massless scalar fields after a gauge fixing that reduces the theory to a model with a single constraint generating radial diffeomorphisms. Though they do not work in the framework of LQG proper, they employ a type of polymeric quantization somehow inspired in LQG. In this quantization there are operators corresponding to the curvature that can be used to discuss issues related to singularity resolution (at the dynamical level). From a technical point of view an interesting detail in this work is the use of Painlevé–Gullstrand coordinates that avoid the necessity to consider the interior and exterior problems separately.

Living Rev. Relativity 13, (2010), 6
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