Wave functions for the Schwarschild black hole interior

TL;DR

This paper investigates wave functions inside Schwarzschild black holes using loop quantum gravity, deriving bounds on parameters and providing numerical solutions to the quantum constraint.

Contribution

It derives a bound relating the Immirzi parameter and quantum ambiguity, and presents numerical solutions for the quantum wave functions.

Findings

The derived inequality constrains the Immirzi parameter and quantum ambiguity.

Numerical solutions of the Hamiltonian constraint are provided.

The natural parameter values violate the derived bound.

Abstract

Using the Hamiltonian constraint derived by Ashtekar and Bojowald, we look for pre-classical wave functions in the Schwarzschild interior. In particular, when solving this difference equation by separation of variables, an inequality is obtained relating the Immirzi parameter $\gamma$ to the quantum ambiguity $\delta$ appearing in the model. This bound is violated when we use a natural value for $\delta$ based on loop quantum gravity together with a recent proposal for $\gamma$. We also present numerical solutions of the constraint.