Effective Kerr geometry from loop quantum gravity

Abstract: We derive an effective Kerr metric from an effective Schwarzschild metric inspired by loop quantum gravity through the Newman-Janis algorithm. The resulting spacetime is free from the classical ring singularity and does not allow the existence of closed time-like curves, due to a lower radial bound in its domain inherited by the spherically symmetric seed metric. We give a possible interpretation to this lower bound by casting the metric in generalized Painlev\'e-Gullstrand coordinates. We analyze the horizon structure and the ergoregion pointing out the similarities and the differences with the classical Kerr spacetime. The study of the trajectory of the zero angular momentum observer allows to show two novel effects due purely to quantum gravity related to the frame dragging and the repulsive behaviour in the deep quantum region. We derive an effective separation Carter constant and the geodesic equations. Finally, we specialize the geodesic analysis to equatorial circular trajectories obtaining a modified third Kepler law and the equation defining the photon sphere.