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Deformed General Relativity and Quantum Black Holes Interior

1
APC—Astroparticule et Cosmologie, Université Paris Diderot Paris 7, 75013 Paris, France
2
Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
3
Institut Denis Poisson, Université de Tours, Université d’Orléans, Parc de Grandmont, 37200 Tours, France
*
Author to whom correspondence should be addressed.
Universe 2020, 6(3), 39; https://doi.org/10.3390/universe6030039
Received: 4 December 2019 / Revised: 17 February 2020 / Accepted: 26 February 2020 / Published: 4 March 2020
(This article belongs to the Special Issue Probing New Physics with Black Holes)
Effective models of black holes interior have led to several proposals for regular black holes. In the so-called polymer models, based on effective deformations of the phase space of spherically symmetric general relativity in vacuum, one considers a deformed Hamiltonian constraint while keeping a non-deformed vectorial constraint, leading under some conditions to a notion of deformed covariance. In this article, we revisit and study further the question of covariance in these deformed gravity models. In particular, we propose a Lagrangian formulation for these deformed gravity models where polymer-like deformations are introduced at the level of the full theory prior to the symmetry reduction and prior to the Legendre transformation. This enables us to test whether the concept of deformed covariance found in spherically symmetric vacuum gravity can be extended to the full theory, and we show that, in the large class of models we are considering, the deformed covariance cannot be realized beyond spherical symmetry in the sense that the only deformed theory which leads to a closed constraints algebra is general relativity. Hence, we focus on the spherically symmetric sector, where there exist non-trivial deformed but closed constraints algebras. We investigate the possibility to deform the vectorial constraint as well and we prove that non-trivial deformations of the vectorial constraint with the condition that the constraints algebra remains closed do not exist. Then, we compute the most general deformed Hamiltonian constraint which admits a closed constraints algebra and thus leads to a well-defined effective theory associated with a notion of deformed covariance. Finally, we study static solutions of these effective theories and, remarkably, we solve explicitly and in full generality the corresponding modified Einstein equations, even for the effective theories which do not satisfy the closeness condition. In particular, we give the expressions of the components of the effective metric (for spherically symmetric black holes interior) in terms of the functions that govern the deformations of the theory. View Full-Text
Keywords: black holes; loop quantum gravity; modified gravity black holes; loop quantum gravity; modified gravity
MDPI and ACS Style

Arruga, D.; Achour, J.B.; Noui, K. Deformed General Relativity and Quantum Black Holes Interior. Universe 2020, 6, 39.

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