A note on the symmetry reduction of SU(2) on horizons of various topologies

DeBenedictis, Andrew; Kloster, S.; Brännlund, Johan

doi:10.1088/0264-9381/28/10/105023

Cited by 10 publications

(4 citation statements)

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“…6. Horizons with nontrivial topologies and their quantization in loop quantum gravity have been studied in a very interesting series of articles [32][33][34]. This gives a good point of comparison for the results reported in the present work.…”

Section: Discussion and Outlooksupporting

confidence: 60%

Black hole horizons from within loop quantum gravity

Sahlmann

2011

Phys. Rev. D

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In general relativity, the fields on a black hole horizon are obtained from those in the bulk by pullback and restriction. Similarly, in quantum gravity, the quantized horizon degrees of freedom should result from restricting, or pulling-back, the quantized bulk degrees of freedom. This is not yet fully realized in the - otherwise very successful - quantization of isolated horizons in loop quantum gravity. In this work we outline a setting in which the quantum horizon degrees of freedom are simply components of the quantized bulk degrees of freedom. There is no need to quantize them separately. We present evidence that for a horizon of sphere topology, the resulting horizon theory is remarkably similar to what has been found before.Comment: 13 pages, 10 figures, RevTeX 4.1; v3: corrected typos etc.; now virtually identical with published versio

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Section: Discussion and Outlooksupporting

confidence: 60%

Black hole horizons from within loop quantum gravity

Sahlmann

2011

Phys. Rev. D

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“…We briefly sketch the relevant aspects of the black hole entropy calculation in our framework and comment on the outcome. Building on the results of [23,[26][27][28], the entropy can be calculated by counting the horizon states which are in agreement with the macroscopic properties of the black hole prescribed by the invariant area operator instead of the usual LQG area operator.…”

Section: Application To Black Hole Entropymentioning

confidence: 99%

Loop quantum gravity without the Hamiltonian constraint

Bodendorfer¹,

Stottmeister²,

Thurn³

2013

Class. Quantum Grav.

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We show that under certain technical assumptions, including the existence of a constant mean curvature (CMC) slice and strict positivity of the scalar field, general relativity conformally coupled to a scalar field can be quantised on a partially reduced phase space, meaning reduced only with respect to the Hamiltonian constraint and a proper gauge fixing. More precisely, we introduce, in close analogy to shape dynamics, the generator of a local conformal transformation acting on both, the metric and the scalar field, which coincides with the CMC gauge condition. A new metric, which is invariant under this transformation, is constructed and used to define connection variables which can be quantised by standard loop quantum gravity methods. Since this connection is invariant under the local conformal transformation, the generator of which is shown to be a good gauge fixing for the Hamiltonian constraint, the Dirac bracket associated with implementing these constraints coincides with the Poisson bracket for the connection. Thus, the well developed kinematical quantisation techniques for loop quantum gravity are available, while the Hamiltonian constraint has been solved (more precisely, gauge fixed) classically. The physical interpretation of this system is that of general relativity on a fixed spatial CMC slice, the associated "time" of which is given by the constant mean curvature. While it is hard to address dynamical problems in this framework (due to the complicated "time" function), it seems, due to good accessibility properties of the CMC gauge, to be well suited for problems such as the computation of black hole entropy, where actual physical states can be counted and the dynamics is only of indirect importance. The corresponding calculation yields the surprising result that the usual prescription of fixing the Barbero-Immirzi parameter β to a constant value in order to obtain the well-known formula S = a(Φ)A/(4G) does not work for the black holes under consideration, while a recently proposed prescription involving an analytic continuation of β to the case of a self-dual space-time connection yields the correct result. Also, the interpretation of the geometric operators gets an interesting twist, which exemplifies the deep relationship between observables and the choice of a time function and has consequences for loop quantum cosmology.A reduced phase space quantisation of a given theory is generally very problematic due to the complexity of the representation problem resulting from a non-trivial Dirac bracket. When quantising a given classical theory, it is often more practical to perform a Dirac-type quantisation [1] and to represent the constraints of the classical theory on a kinematical Hilbert space, as for example done in loop quantum gravity [2,3]. On the other hand, the quantum equations are generally hard to solve and new technical problems, mostly of functional analytic nature, arise.Concerning general relativity, the Dirac-type quantisation has been performed in the context of loop quantum grav...

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“…At this stage we need to choose an ansatz for the su(2) connection and the densitized triad which is compatible with our geometries. An appropriate ansatz is provided by the following pair [55], [57], [74]:…”

Section: Iiii Black Holes In Connection Variablesmentioning

confidence: 99%

Non-commutative black holes of various genera in the connection formalism

Schneider,

DeBenedictis

2020

Preprint

Self Cite

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We consider black hole interiors of arbitrary genus number within the paradigm of noncommutative geometry. The study is performed in two ways: One way is a simple smearing of a matter distribution within the black hole. The resulting structure is often known in the literature as a "model inspired by non-commutative geometry". The second method involves a more fundamental approach, in which the Hamiltonian formalism is utilized and a non-trivial Poisson bracket is introduced between the configuration degrees of freedom, as well as between the canonical momentum degrees of freedom. This is done in terms of connection variables instead of the more common ADM variables. Connection variables are utilized here since non-commutative effects are usually inspired from the quantum theory, and it is the connection variables that are used in some of the more promising modern theories of quantum gravity. We find that in the first study, the singularity of the black holes can easily be removed. In the second study, we find that introducing a non-trivial bracket between the connections (the configuration variables) may delay the singularity, but not necessarily eliminate it. However, by introducing a non-trivial bracket between the densitized triads (the canonical momentum variables) the singularity can generally be removed. In some cases, new horizons also appear due to the non-commutativity.

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A note on the symmetry reduction of SU(2) on horizons of various topologies

Cited by 10 publications

References 50 publications

Black hole horizons from within loop quantum gravity

Black hole horizons from within loop quantum gravity

Loop quantum gravity without the Hamiltonian constraint

Non-commutative black holes of various genera in the connection formalism

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