Abstract
In the past years, black holes and the fate of their singularity have been heavily studied within loop quantum gravity. Effective spacetime descriptions incorporating quantum geometry corrections are provided by the so-called polymer models. Despite the technical differences, the main common feature shared by these models is that the classical singularity is resolved by a black-to-white hole transition. In a recent paper (Bodendorfer et al 2019 Class. Quantum Grav. 36 195015), we discussed the existence of two Dirac observables in the effective quantum theory respectively corresponding to the black and white hole mass. Physical requirements about the onset of quantum effects then fix the relation between these observables after the bounce, which in turn corresponds to a restriction on the admissible initial conditions for the model. In the present paper, we discuss in detail the role of such observables in black hole polymer models. First, we revisit previous models and analyse the existence of the Dirac observables there. Observables for the horizons or the masses are explicitly constructed. In the classical theory, only one Dirac observable has physical relevance. In the quantum theory, we find a relation between the existence of two physically relevant observables and the scaling behaviour of the polymerisation scales under fiducial cell rescaling. We present then a new model based on polymerisation of new variables which allows to overcome previous restrictions on initial conditions. Quantum effects cause a bound of a unique Kretschmann curvature scale, independently of the relation between the two masses.
Highlights
Understanding the fate of classical gravitational singularities is one of the key questions that any quantum theory of gravity needs to address
In the resulting effective quantum corrected cosmological spacetime, quantum geometry effects induce a natural cutoff for spacetime curvature invariants and the initial big bang singularity is resolved by a quantum bounce interpolating between a contracting and an expanding branch well approximated by classical geometries far from the Planck regime [2, 5]
The effective dynamics can be derived from the loop quantum cosmology (LQC) quantum theory by considering expectation values on suitable semi-classical states peaked on classical phase space points for large volumes [7, 16, 17], showing that the polymerisation procedure is able to capture the relevant features of the quantum theory
Summary
As shown by the detailed analysis of the Dirac observables [1], in order to achieve physical reliable predictions such as a unique mass independent curvature upper bound, certain initial conditions and in turn certain relations between the black hole and white hole masses have to be selected The source of such limitation is rooted in the fact that the on-shell canonical momentum is not exactly proportional to (the square root of ) the Kretschmann scalar unless the integration constant entering the proportionality factor is selected to be independent of the mass. The second part of the paper is devoted to introduce a new effective model for polymer Schwarzschild black holes in which such limitations are resolved and all criteria of physical viability (mass independent Planckian curvature upper bound, see [34]) can be achieved for a large class of initial conditions independently of the relation between the black and white hole masses.
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