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Loop Quantum Black Hole Extensions Within the Improved Dynamics ORIGINAL RESEARCH published: 11 June 2021 doi: 10.3389/fspas.2021.647241 Loop Quantum Black Hole Extensions Within the Improved Dynamics Rodolfo Gambini 1, Javier Olmedo 2* and Jorge Pullin 3 1Department of Physics, University of the Republic, Montevideo, Uruguay, 2Departamento de Física Teórica y Del Cosmos, Universidad de Granada, Granada, Spain, 3Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA, United States We continue our investigation of an improved quantization scheme for spherically symmetric loop quantum gravity. We find that in the region where the black hole singularity appears in the classical theory, the quantum theory contains semi-classical states that approximate general relativity coupled to an effective anisotropic fluid. The singularity is eliminated and the space-time can be continued into a white hole space-time. This is similar to previously considered scenarios based on a loop quantum gravity quantization. Keywords: loop quantum gravity, black holes, quantum field theory, spin networks, general relativity Edited by: Francesca Vidotto, Western University, Canada 1 INTRODUCTION Reviewed by: Jibril Ben Achour, In a previous paper (Gambini et al., 2020a) we studied an improved quantization for spherically Ludwig-Maximilians-University symmetric loop quantum gravity. Earlier work (Gambini and Pullin, 2013; Gambini et al., 2014; Munich, Germany Gambinietal.,2020b) had considered a constant polymerization parameter, similarly to the Kristina Giesel, “μ ” 0 quantization scheme in loop quantum cosmology, whereas the improved quantization is University of Erlangen Nuremberg, similar to the “μ” quantization scheme (Ashtekar and Singh, 2011). Other approaches involving Germany improved quantizations have also been explored in (Han and Liu, 2020). We observed that the Ana Alonso Serrano, singularity was removed, but we did not analyze in detail what happened to the space-time Max Planck Institute for Gravitational fi Physics (AEI), Germany beyond the region where the singularity used to be. Here we complete that study. We nd that in *Correspondence: that region there exist semi-classical quantum states for which the theory behaves like a fl Javier Olmedo quantum version of general relativity coupled to an effective anisotropic uid (Cho and Kim, [email protected] 2019) that violates the dominant energy condition. In the highest curvature region there is a space-like transition surface, something that was unnoticed in (Gambini et al., 2020a). The Specialty section: space-time continues into a white hole geometry, like in Ref. (Ashtekar et al., 2018). However, This article was submitted in this work we consider a different regularization for the parametrized observable associated to to Cosmology, the shift function. Its very definition requires the choice of a slicing and the new regularization a section of the journal avoids an undesirable dependence on it in the semiclassical limit. Frontiers in Astronomy and The organization of this paper is as follows. In section 2 we discuss the physical sector of the Space Sciences quantum theory, focusing on semiclassical sectors. In section 3 we introduce a horizon penetrating Received: 29 December 2020 slicing based on Painlevé–Gullstrand coordinates and show how it can be used to connect to a white Accepted: 26 April 2021 hole space-time. We end with a discussion. Published: 11 June 2021 Citation: Gambini R, Olmedo J and Pullin J (2021) Loop Quantum Black Hole 2 PHYSICAL SECTOR OF THE QUANTUM THEORY Extensions Within the Improved Dynamics. The physical sector of the theory is obtained after combining Loop Quantum Gravity quantization Front. Astron. Space Sci. 8:647241. techniques and the Dirac quantization program for constrained theories. In summary, we start with a doi: 10.3389/fspas.2021.647241 kinematical Hilbert space in the loop representation adapted to spherically symmetric spacetimes for Frontiers in Astronomy and Space Sciences | www.frontiersin.org1 June 2021 | Volume 8 | Article 647241 Gambini et al. Loop Quantum Black Hole Extensions φ x the geometrical sector [(Kφ, E ), (Kx, E )] together with a emerge from the discrete nature of the spin network treatment standard representation for the spacetime mass and its and are associated with the ki’s, which in turn are associated with conjugate momentum (M, PM). A suitable basis of kinematical the value of the areas of the spheres of symmetry connected with states is the one provided by spherical symmetric spin networks each vertex of the spin network. One can also consider states that tensor product with the standard states for the matter sector in are a superposition of M’s. The analysis will remain the same as the mass representation. Then, we represent the scalar constraint long as the states are peaked around a value of M. as a well-defined operator in the kinemtical Hilbert space (for the In addition to physical states, the physical observables diffeomorphism constraint we rather work with the related finite representing space-time metric components will be defined group of transformations mimicking the full theory). through suitable parametrized observables. They act as local Following the construction of Ref. (Gambini et al., 2020a), operators on each vertex of the spin network. Furthermore, the physical sector of the theory is encoded in physical states they involve point holonomies that are chosen to be (solutions to the scalar constraint) endowed with a suitable compatible with the superselection sectors of the physical inner product and a set of physical observables. This is Hilbert space (see Ref (Gambini et al., 2020a). for more achieved, for instance, by applying group averaging details). Some of the basic parametrized observables are techniques for both the quantum scalar constraint and the → → → fi x | , 〉 | , 〉 ℓ2 | , 〉, group of nite spatial diffeomorphisms (see also Refs E xj M k O z xj M k Plkj(xj) M k (2.4) (Gambini and Pullin, 2013; de Blas et al., 2017)). We focus → → our study to some of the simplest semi-classical states. M|M, k 〉 M|M, k 〉, (2.5) Quantum states consist of spatial spin networks labeled by where z(x) is a suitable gauge function that codifies the freedom the ADM mass M (a Dirac observable) and integer numbers in the choice of radial reparametrizations. that characterize the radii of spheres of symmetry associated For the components of the space-time metric on stationary with each vertex of the network k . The semi-classical states we i slicings we have, for instance, the lapse and shift,1 are going to consider here are given by superpositions in the δ x 2 mass centered at M0 and of width M0 and are therefore ]′ 2 1 E xj associated with a fixed discrete structure in space (see N x : , j 4 φ 2 (Gambini et al., 2020a) for more details). They provide E xj (2.6) excellent approximations to the classical geometry in x 2 ρ x 2 E xj sin Kφ xj regions of small curvature compared to Planck scale. N x j , j φ 2 ρ2 Concretely, we consider the semi-classical states E xj j ​ π( − ) 1 /Z M M0 where |ψ〉 dMeiMP0 cos Θ(M − M + δM )Θ(M + δM − M) δM 2δM 0 0 0 0 0 0 2 x ′ φ 2 E xj 4 ×|M, kS, ..., k0, ..., k−S〉 E xj , (2.7) 2 (2.1) sin ρ Kφ(xj) + j − 2GM 1 ρ2 j x with k0 < kj for all j ≠ 0, namely, j −S, −S + 1, ..., 1, −1, ..., S, |E (xj)| where where we polymerized Kφ with ρ the polymerization parameter of / Δ 2 3 the improved quantization, ⎡⎣2GM0 ⎤⎦, k0 Int πℓ3 (2.2) 4 Pl Δ ρ . (2.8) π x times the Planck length squared determines the smallest area of 4 E the 2-spheres in the theory. This corresponds to the improved We choose the k’s in the one-dimensional spin network in our quantization, where Δ is the area gap. Besides, we choose physical state and the gauge function z(x) such that M0 ≫ mPl and x2 x 2 j E xj ℓ Int , (2.9) 3 2/3 Pl ℓ2 3 4πℓ Pl δM ≤ Pl M1/3. (2.3) 0 2 2GΔ 0 The states in Eq. 2.1 belong to a family of sharply peaked semiclassical states in the mass and with support on a concrete spin network (states with higher dispersion in the mass will require superpositions of different spin networks). This choice 1In Ref (Gambini et al., 2020a). for the shift we adopted the regularization Kφ(x ) → sin(2ρ Kφ(x ))/2ρ , but it introduces an undesirable slicing considerably simplifies the analysis of the effective geometries. As j j j j we discussed in our previous papers, the quantum theory has dependence that is avoided with the present regularization. Besides, the representation that we adopt here for the square of the shift function as a additional observables to the ones encountered in classical parametrized observable is compatible with the superselection rules of the treatments (Kastrup and Thiemann, 1994; Kuchar, 1994) quantum numbers ]j of the kinematical spin networks as it was discussed in which are the ADM mass and the time at infinity. These Ref. (Gambini et al., 2020a). Frontiers in Astronomy and Space Sciences | www.frontiersin.org2 June 2021 | Volume 8 | Article 647241 Gambini et al. Loop Quantum Black Hole Extensions 2 → 2 + δ − 2 → x ℓ ⎢ xj x x ⎥ ′| , 〉 Pl ⎣⎢⎡ j ⎦⎥⎤| , 〉, E xj M k δ Int ℓ2 M k (2.10) x Pl and with xj δx j + x0 and with j ∈ Z, where 2GMΔ 2/3 x Int . (2.11) 0 4π Besides, we will choose δx ℓPl as in the first paper (Gambini et al., 2020a), although we will discuss the consequences of the limiting choices (for a uniform lattice) ℓ2 δx Pl and δx x . The different spacings δx just mentioned 2x0 0 here correspond to different choices of states in the physical space, all of them lead to the same semiclassical behavior but differ in the deep quantum regime close to the singularity, as one would expect.
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