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

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.