Spherically symmetric Einstein–Maxwell theory and loop quantum gravity corrections

Tibrewala, Rakesh

doi:10.1088/0264-9381/29/23/235012

Cited by 44 publications

(64 citation statements)

References 58 publications

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“…otherwise there is an anomaly and the Hamiltonian constraint is no more first class, which means that it does not generate any symmetries. This conclusion is consistent with what has already been found in the literature [62,66,68,87]. Before going further and discuss this condition, notice that in general relativity the Hamiltonian constraint is of the form (3.34) with (3.41) where the functions A and B are explicitly given by 45) which obviously satisfy the closeness condition (3.44).…”

Section: Closeness Of the Deformed Algebrasupporting

confidence: 89%

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Deformed General Relativity and Quantum Black Holes Interior

2020

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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 can not 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 static and spherically symmetric black holes interior) in terms of the functions that govern the deformations of the theory.

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Section: Closeness Of the Deformed Algebrasupporting

confidence: 89%

“…where f 1 and f 2 are functions of π ϕ only. If one imposes the closeness of the constraints algebra, then A and B satisfy the condition (3.52) which translates into 2f 2 = df 1 /dπ ϕ , and one recovers obviously the well-known anomaly-free condition [62,72,87]. To be general, we assume for the moment that f 1 and f 2 are independent.…”

Section: Examplesmentioning

confidence: 99%

Deformed General Relativity and Quantum Black Holes Interior

2020

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“…See [18] for more details. Finally, our regularization is restricted to the µ 0scheme, as in [18], since introducing holonomy corrections within theμ-scheme, i.e K φ → f (K φ , E x ), and requiring at the same time the anomaly freedom of the effective Dirac's algebra generates inconsistencies as shown in [10]. Therefore, the standard improved dynamics used in polymer cosmological models cannot be generalized as it stands to such inhomogeneous spherically symmetric polymer models.…”

Section: Covariant Polymer Phase Space Regularizationmentioning

confidence: 99%

“…However, a main difference is that an outer horizon (at t = r s ) is always present in our geometry, while naked singularities appear for superextreme RN black holes. In the end, this naive extension is not satisfactory since it does not allow recovering Schwarzschild's solution in the classical region (r ≫ r s ), except if the parameter ρ becomes r-dependent and tends to zero, which would drastically modify the equations of motion [10].…”

Section: Inverse Problemmentioning

confidence: 99%

“…Indeed, in this inhomogeneous case, one has to make sure that the effective corrections do no generate anomalies in the algebra of first class constraints, and thus do not spoil covariance. Taking care of this poten-tial covariance issue, one can obtain modified Einstein's equation for polymer black holes [8][9][10]. Their resolution for the simple vacuum modified Schwarzschild interior has not been investigate yet.…”

Section: Introductionmentioning

confidence: 99%

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Polymer Schwarzschild black hole: An effective metric

Achour¹,

Lamy²,

Liu³

et al. 2018

EPL

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We consider the modified Einstein equations obtained in the framework of effective spherically symmetric polymer models inspired by Loop Quantum Gravity. When one takes into account the anomaly free point-wise holonomy quantum corrections, the modification of Einstein equations is parametrized by a function f (x) of one phase space variable. We solve explicitly these equations for a static interior black hole geometry and find the effective metric describing the trapped region, inside the black hole, for any f (x). This general resolution allows to take into account a standard ambiguity inherent to the polymer regularization: namely the choice of the spin j labelling the SU(2)-representation of the holonomy corrections. When j = 1/2, the function f (x) is the usual sine function used in the polymer litterature. For this simple case, the effective exterior metric remains the classical Schwarzschild's one but acquires modifications inside the hole. The interior metric describes a regular trapped region and presents strong similarities with the Reissner-Nordström metric, with a new inner horizon generated by quantum effects. We discuss the gluing of our interior solution to the exterior Schwarzschild metric and the challenge to extend the solution outside the trapped region due to covariance requirement. By starting from the anomaly free polymer regularization for inhomogeneous spherically symmetric geometry, and then reducing to the homogeneous interior problem, we provide an alternative treatment to existing polymer interior black hole models which focus directly on the interior geometry, ignoring covariance issue when introducing the polymer regularization.

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Loop quantum gravity as an effective theory

Bojowald¹

2012

AIP Conference Proceedings

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To descend from a fundamental theory to low energy phenomena one usually makes use of effective equations which capture the main quantum effects in a specified regime. A Hamiltonian procedure to derive effective equations is first illustrated by the example of an anharmonic oscilla-tor and then applied to quantum cosmology. Extending this to full gravity demonstrates in which sense loop quantum gravity provides an emergence scenario for space-time from a microscopic theory.

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Spherically symmetric Einstein–Maxwell theory and loop quantum gravity corrections

Cited by 44 publications

References 58 publications

Deformed General Relativity and Quantum Black Holes Interior

Deformed General Relativity and Quantum Black Holes Interior

Polymer Schwarzschild black hole: An effective metric

Loop quantum gravity as an effective theory

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