Elsevier

Annals of Physics

Volume 426, March 2021, 168401
Annals of Physics

Effective dynamics of the Schwarzschild black hole interior with inverse triad corrections

https://doi.org/10.1016/j.aop.2021.168401Get rights and content

Highlights

  • Systematic derivation of effective Hamiltonian in LQG using path integrals.

  • Inverse triad corrections appear in the effective theory using this systematic method.

  • Prescriptions are introduced to resolve issues when such new corrections are present.

  • Singularity resolution and black-to-white hole bounce hold with new corrections.

  • New larger minimum radius-at-bounce for the interior of the Schwarzschild black hole.

Abstract

We reconsider the study of the interior of the Schwarzschild black hole now including inverse triad quantum corrections within loop quantization. We derive these corrections and show that they are related to two parameters δb,δc associated to the minimum length in the radial and angular directions, that enter Thiemann’s trick for quantum inverse triads. Introduction of such corrections may lead to non-invariance of physical results under rescaling of the fiducial volume needed to compute the dynamics, due to noncompact topology of the model. So, we put forward two prescriptions to resolve this issue. These prescriptions amount to interchange δb,δc in classical computations in Thiemann’s trick. By implementing the inverse triad corrections we found, previous results such as singularity resolution and black-to-white hole bounce hold with different values for the minimum radius-at-bounce, and the mass of the white hole.

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 I×S2 and volume V0=a0L0, where a0 is the area of the 2-sphere S2 and L0 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 a04πr02 by a physical scale r0. This physical r0 permits one to define a Hamiltonian formulation, and in this way is different in nature from the auxiliary scale L0, which is needed to fix the fiducial cell size to be able to define the symplectic structure. Thus while r0 will be present in physical results in both the classical and quantum theories, these theories should be independent of L0. 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 MW of the resultant white hole not matching with that of the original black hole MBMW, but rather MWMB4. 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 ds2=12GMrdt2+12GMr1dr2+r2dθ2+sin2θdϕ2where M 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 rt, ds2=2GMt11dt2+2GMt1dr2+t2dθ2+sin2θdϕ2,with t(0,2GM) and r(,). This metric is a special case of a Kantowski–Sachs cosmological spacetime that is given by the metric ds2=dτ2+A2(τ)dr

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 C=d3xN|detE|ϵijkEaiEbj1γ20FabkΩabk.Here the integral is over the fiducial volume, and Ωabk and 0

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 N infinitesimal ones and inserting complete basis between these. Such discrete time path integral gets replaced by the continuum one in the limit N. 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 N, 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 β1 and β2, suggest that some sort of interchanging pb0pc0 can fix the problem, either just in β2, or in both β1 and β2. At a first glance, it seems that this can be achieved by interchanging δbδc, 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 β2, it makes almost every other terms in θ and σ2 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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