Covariant ${\bar{\mu}}$-scheme effective dynamics, mimetic gravity, and non-singular black holes: Applications to spherical symmetric quantum gravity and CGHS model

TL;DR

This paper introduces a covariant $ar{}$-scheme effective dynamics in Loop Quantum Gravity, leading to non-singular black hole solutions and extending to the CGHS model, with implications for quantum gravity and black hole singularity resolution.

Contribution

It develops a covariant $ar{}$-scheme effective dynamics in LQG, applying it to black holes and the CGHS model, and demonstrates singularity resolution and finite curvature in these models.

Findings

Black hole solutions are non-singular and asymptotically approach Nariai geometry.

Effective dynamics reproduces non-singular Loop-Quantum-Cosmology behavior.

Curvature and dilaton derivatives remain finite in the CGHS model with null-shell collapse.

Abstract

We propose a new $\bar{\mu}$-scheme Hamiltonian effective dynamics in the spherical symmetric sector of Loop Quantum Gravity (LQG). The effective dynamics is generally covariant as derived from a covariant Lagrangian. The Lagrangian belongs to the class of extended mimetic gravity Lagrangians in 4 dimensions. We apply the effective dynamics to both cosmology and black hole. The effective dynamics reproduces the non-singular Loop-Quantum-Cosmology (LQC) effective dynamics. From the effective dynamics, we obtain the non-singular black hole solution, which has a killing symmetry in addition to the spherical symmetry and reduces to the Schwarzschild geometry asymptotically near the infinity. The black hole spacetime resolves the classical singularity and approaches asymptotically the Nariai geometry $\mathrm{dS}_2\times S^2$ at the future infinity in the interior of the black hole. The resulting black hole spacetime has the complete future null infinity $\mathscr{I}^+$. Thanks to the general covariance, the effective dynamics can be reformulated in the light-cone gauge. We generalize the covariant $\bar{\mu}$-scheme effective dynamics to the Callan-Giddings-Harvey-Strominger (CGHS) model and apply the light-cone formulation to the CGHS black hole solution with the null-shell collapse. We focus on the effective dynamics projected along the null shell. The result shows that both the 2d scalar curvature and the derivative of dilaton field are finite, in contrast to the divergence in the CGHS model.