Loop effective model for Schwarzschild black hole interior: a modified $\bar \mu$ dynamics

TL;DR

This paper develops a new loop quantum gravity effective model for Schwarzschild black hole interiors, showing it resolves singularities and predicts an anti-trapped region, with comparisons to existing models.

Contribution

It introduces a modified $ar $ effective Hamiltonian for Kantowski-Sachs spacetime, incorporating altered Thiemann identities and regulator choices, advancing loop quantum gravity black hole models.

Findings

Classical singularity is resolved in the effective dynamics.

An anti-trapped region bounded by a second Killing horizon emerges.

The model's predictions differ from previous $ar $ schemes and are sensitive to regulator choices.

Abstract

In this article, we introduce a new effective model for the Kantowski-Sachs spacetime in the context of loop quantum gravity, and we use it to evaluate departures from general relativity in the case of Schwarzschild black hole interior. The model is based on an effective Hamiltonian constructed via the regularized Thiemann identities in the $\bar \mu$-scheme. We show that, in contrast with the $\mu_o$-scheme studied in [1], the classical limit imposes certain alterations of Thiemann identities as well as restrictions on the choice of regulators. Once we define the Hamiltonian, we derive the equations of motion for the relevant variables and proceed with the solving using numerical methods, focusing on a specific choice of $\bar \mu$. We establish that for a Schwarzschild black hole interior, the effective dynamics leads to a resolution of the classical singularity and the emergence of an anti-trapped region bounded by a second Killing horizon. We then perform a comparison of the dynamical trajectories and their properties obtained in the new model and some models present in the literature. We finally conclude with few comments on other choices of the regulators and their consequences.