Effective dynamics of the Schwarzschild black hole interior with inverse triad corrections
Introduction
As one of the most fascinating predictions of general relativity, black holes have been the subject of much analysis and explorations. Particularly their interior, and the singularity located there, has been studied in classical, quantum and semiclassical regimes. The mainstream hope is that the classical singularity will be resolved and replaced by a quantum region. However, there are still many open issues to be answered in a satisfactory way. Within loop quantum gravity (LQG) [1], [2], there have been numerous works about quantum black holes and their singularity resolution in both mini- and midi-superspace models, to mention a few [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15]. One of the most studied models in this context is the Schwarzschild black hole which interior corresponds to a Kantowski–Sachs model, a system with finite degrees of freedom, and hence a mini-superspace, with a singularity at the heart of it [4], [16]. One of the approaches to quantize this model, inspired by LQG, is polymer quantization [17], [18], [19], [20], a technique also used in loop quantum cosmology (LQC) [21], [22], [23]. In this quantization the classical canonical algebra is represented in a way that is unitarily inequivalent to the usual Schrödinger representation even at the kinematical level. The root of this inequivalency is the choice of topology and the form of the inner product of this representation, which renders some of the operators discontinuous in their parameters, resulting in the representation not being weakly continuous. On the other hand, unitary equivalency of a representation to the Schrödinger one is guaranteed by the Stone–von Neumann theorem iff all of its premises, including weak continuity of the representation, are satisfied, and the polymer representation does not. This inequivalency translates into new results that are different from the usual quantization of the system, one of them being the resolution of the singularity of the Kantowski–Sachs model. These results however, are accompanied by some issues that we briefly discuss in what follows.
In one of the earliest attempts in this approach [3], the authors showed that the singularity can be avoided in the quantum regime, but one of the important issues was the dependence of results on auxiliary parameters that define the size of the fiducial cell. The introduction of this fiducial cell, in this case a cylindrical one with topology and volume , where is the area of the 2-sphere and is the cylinder’s height, is necessary to avoid the divergence of some of the spatial integrals in homogeneous models with some non-compact directions. Particularly it is important to be able to define the symplectic structure. Given that the physical results should not depend on these auxiliary parameters, a new proposal, motivated by the “improved quantization” in LQC [24], was put forward that avoided this dependence and yielded bounded expansion and shear scalars [4]. However, this method also leads to some undesired modified behavior at the horizon due to quantum gravitational effects in vacuum, that are manifestation of the coordinate singularity there. There are also some other recent works that take a bit of a different approach to the problem by looking for an effective metric [25], [26].
In [6], a key modification to the quantization was proposed by choosing to fix by a physical scale . This physical permits one to define a Hamiltonian formulation, and in this way is different in nature from the auxiliary scale , which is needed to fix the fiducial cell size to be able to define the symplectic structure. Thus while will be present in physical results in both the classical and quantum theories, these theories should be independent of . The proposal in [6], leads to results that are independent of the auxiliary parameters, and while the theory predicts that the singularity is resolved in the quantum gravity regime, no large quantum gravitational effects appear at low curvatures near the horizon as it should be the case.
It is worth noting that the anisotropic models suffered from an issue: since these models resolve the singularity, they predicted a “bounce” from a black hole to a white hole, with the mass of the resultant white hole not matching with that of the original black hole , but rather . Recent work presented some proposals to deal with it [27] and a different approach was developed in [28], [29] by encompassing the interior region containing the classical singularity with the exterior asymptotic one, which, in the large mass limit, makes the masses of the white and black holes take the same value. See also [30], [31], [32] for a different perspective.
All previous works on black holes ignore inverse triad corrections, to simplify the problem. However, they are important especially at highly quantum regimes and a more complete quantum gravitational analysis of this model should take them into account. In this work we first use a path integral in phase space including inverse triad quantum corrections. These corrections are known to produce severe issues in non compact cosmological models, among them the dependence of the physical quantities on the auxiliary parameters or their rescaling. We put forward two proposals that, in the case of Schwarzschild black hole, yield a physical description with no reference to fiducial parameters. Finally, we study how these proposals modify the “minimum radius at the bounce”.
The structure of this paper is as follows: In Section 2, we present the background, the relation between the Schwarzschild interior and the Kantowski–Sachs model, and their classical Hamiltonian analysis. In Section 3, we briefly review how the quantum Hamiltonian constraint is defined. In Section 4, the path integral analysis is presented and it is shown how a systematic effective Hamiltonian constraint can be derived from the quantum Hamiltonian, including inverse triad corrections. Section 5 is dedicated to presenting some of the important issues that are raised by the presence of the new corrections, and recognizing the root of these issues. In Section 6, we present two proposals to deal with the aforementioned issues, and also show their effect upon some physical quantities. Finally, in Section 7, we conclude the paper by presenting a summary and a discussion about the results.
Section snippets
Background and the classical theory
For Schwarzschild black hole the spacetime metric where is the mass of the black hole, the timelike and spacelike curves switch their causal nature into each other for observers that cross the event horizon. Hence the metric of the interior region is obtained by , with and . This metric is a special case of a Kantowski–Sachs cosmological spacetime that is given by the metric
The quantum Hamiltonian constraint
The next step is to find the classical Hamiltonian in loop variables, and then representing it as an operator on a suitable kinematical Hilbert space. We only briefly go over this, details can be found in previous works [3], [7]. Since in this model, the diffeomorphism constraint is trivially satisfied, after imposing the Gauss constraint, one is left only with the classical Hamiltonian constraint Here the integral is over the fiducial volume, and and
Path integral analysis: effective Hamiltonian and new features
To find the effective version of the constraint we employ path integration. For standard mechanical systems path integrals yield expressions for the matrix elements of the evolution operators. The original derivation by Feynman involved the canonical theory expressing the evolution by composing infinitesimal ones and inserting complete basis between these. Such discrete time path integral gets replaced by the continuum one in the limit . For gravitational models we have two different, but
Issues raised by the new corrections
As mentioned earlier, in this model, a number of differences arise due to the presence of further inverse triad corrections that we have managed to compute through the path integral method. To further highlight these differences, and also to be able to compare our results with some of the previous works, we need to specify a specific lapse , which is needed to write the explicit equations of motion in a certain frame. One such choice takes us to the Hamiltonian in [6] in the limit of not
Proposals to deal with the issues
The above observations together with the detailed forms of and , suggest that some sort of interchanging can fix the problem, either just in , or in both and . At a first glance, it seems that this can be achieved by interchanging , but this has two problems: it is not clear if there are restrictions in doing so, and more importantly, although it fixes the invariance problem in , it makes almost every other terms in and noninvariant. So we should find a way of
Discussion
The quest for a quantum theory of gravity involves looking for physical imprints of the merging of quantum and gravity effects in systems like black holes that classically exhibit singularities according to general relativity. In the case of Schwarzschild black hole, loop quantum gravity predicts the resolution of the classical singularity as well as a bouncing scenario connecting the black hole to a white hole, a phenomenon also predicted for some cosmological models, in such a case connecting
CRediT authorship contribution statement
Hugo A. Morales-Técotl: Analysis and Research, Writing - reviewing & editing. Saeed Rastgoo: Analysis and Research, Writing - reviewing & editing. Juan C. Ruelas: Analysis and Research, Writing - reviewing & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The authors would like to acknowledge the support from the CONACyT, Mexico Grant No. 237351. S.R. would like to thank the CONACyT, Mexico SNI support 59344. J.C.R. acknowledges the support of the grant from UAM, Mexico . H.A.M.T. acknowledges the kind hospitality of the Physics department of the ESFM, IPN, during his sabbatical year
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2022, Progress in Particle and Nuclear PhysicsCitation Excerpt :In LQG, both the interior and exterior/full spacetime of black holes, particularly the Schwarzschild black hole has been studied. The interior corresponds to a minisuperspace model, which is quantized usually using polymer quantization [101–107]. Its effective dynamics leads to the resolution of the singularity since the radius of the infalling 2-spheres never reaches zero and bounces back after reaching a finite value.
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All the Authors have contributed equally to the paper.