Improved effective dynamics of loop-quantum-gravity black hole and Nariai limit

Abstract

We propose a new model of the spherical symmetric quantum black hole in the reduced phase space formulation. We deparametrize gravity by coupling to the Gaussian dust which provides the material coordinates. The foliation by dust coordinates covers both the interior and exterior of the black hole. After the spherical symmetry reduction, our model is a 1+1 dimensional field theory containing infinitely many degrees of freedom. The effective dynamics of the quantum black hole is generated by an improved physical Hamiltonian ${\bf H}_\Delta$. The holonomy correction in ${\bf H}_\Delta$ is implemented by the $\bar{\mu}$-scheme regularization with a Planckian area scale $\Delta$ (which often chosen as the minimal area gap in Loop Quantum Gravity). The effective dynamics recovers the semiclassical Schwarzschild geometry at low curvature regime and resolves the black hole singularity with Planckian curvature, e.g. $R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\sim 1/{\Delta}^2$. Our model predicts that the evolution of the black hole at late time reaches the charged Nariai geometry ${\rm dS}_2\times S^2$ with Planckian radii $\sim \sqrt{\Delta}$. The Nariai geometry is stable under linear perturbations but may be unstable by nonperturbative quantum effects. Our model suggests the existence of quantum tunneling of the Nariai geometry and a scenario of black-hole-to-white-hole transition.

Dynamically Implementing the μ¯-Scheme in Cosmological and Spherically Symmetric Models in an Extended Phase Space Model

“…which is compatible with the choice of µ-scheme used in [4,28,29,31,37]. We note that there are special cases where one or both f µ x and f µ ϕ are Dirac observables and constants of motion under the effective dynamics, where f µ x and f µ ϕ can still appear as constants, e.g., similar to the cases in [15].…”

“…This completely removes the possibility of the usual µ 0 scheme in the effective dynamics if we have polymerization of K x . If we require, in addition, the scaling invariance of f 1 and f 2 and remove the K dependence in the gauge-fixing function, the only possible ansatz for f µ x and f µ ϕ is the μ scheme used in [4,29,31,41], which is given by…”

“…Going beyond cosmological models, improved dynamics were then proposed for spherically symmetric models [4,9,10,14,[28][29][30][31]37]. We note that there are different implementations of the improved dynamics in these models, unlike in LQC where the choice is unique if we require scale invariance.…”

Quantum Oppenheimer-Snyder and Swiss Cheese models

“…Indeed, the Loop Quantum Cosmology (LQC) models predict a Big Bounce instead of a Big Bang (see, e.g., [2][3][4]). These models also apply to a BH [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. However, in this case, the answers we get do not form a unique picture.…”

Black hole image encoding quantum gravity information

“…A few quantum extensions of the Kruskal spacetime have been proposed in the study of loop quantum gravity (LQG) [17][18][19][20][21][22][23][24][25][26][27][28][29][30]. While the model in [19] will be employed in the following calculation, our analysis is applicable for all these quantum spacetimes where a BH-to-WH transition occurs.…”