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Semiclassical loop quantum black hole. (English) Zbl 1200.83003
Summary: We improve the semiclassical analysis of loop quantum black hole (LQBH) in the conservative approach of a constant polymeric parameter. In particular we focus our attention on the space-time structure. We introduce a very simple modification of the spherically symmetric Hamiltonian constraint in terms of holonomies. The new quantum constraint reduces to the classical constraint when the polymeric parameter \(\delta\) goes to zero. Using this modification we obtain a large class of semiclassical solutions parametrized by a generic function \(\sigma (\delta)\). We find that only a particular choice of this function reproduces the Schwarzschild black hole solution outside the black hole with the correct asymptotic flat limit. In \(r=0\) the semiclassical metric is regular and the Kretschmann invariant has a maximum peaked at \(r_{\text{max}} \propto l_{P}\). The radial position of the peak does not depend on the black hole mass and the polymeric parameter \(\delta \). The semiclassical solution is very similar to the Reissner-Nordström metric. We construct the Carter-Penrose diagrams explicitly, giving a causal description of the space-time and its maximal extension. The LQBH metric interpolates between two asymptotically flat regions, the \(r\rightarrow \infty \) region and the \(r\rightarrow 0\) region. We study the thermodynamics of the semiclassical solution. The temperature, entropy and the evaporation process are regular and could be defined independently from the polymeric parameter \(\delta \). We study the particular metric when the polymeric parameter goes towards to zero. This metric is regular in \(r=0\) and has only one event horizon in \(r=2m\). The radial position of the Kretschmann invariant maximum depends only on \(l_{P}\). As such the polymeric parameter \(\delta \) does not play any role in the black hole singularity resolution. The thermodynamics is the same.

MSC:
83-02 Research exposition (monographs, survey articles) pertaining to relativity and gravitational theory
83C57 Black holes
83C45 Quantization of the gravitational field
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
83C22 Einstein-Maxwell equations
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