Motion of spinning particles around a polymer black hole in loop quantum gravity

In the curved spacetime background, the trajectory of a spinning test particle will deviate from the geodesic. Using the effective potential method, we study the motion of a spinning test particle on the equatorial plane of a polymer black hole in loop quantum gravity described by the Mathisson-Papapetrou-Dixon equations with minimal spin-gravity interaction. We find that for the bounded orbits in the radial direction, the particle's motion is timelike when its spin is small. The radial range of the orbit and its eccentricity decrease with the loop quantum gravity parameter. However, when the particle takes a large enough spin, we observe an interesting phenomenon that the timelike and spacelike motions alternately appear while are separated by a critical radius. Outside the critical radius, the motion is timelike; however inside it is spacelike, and on the radius r₂ it is null. This property shares similarities with the event horizon radius. However, unlike the event horizon radius, the value of r₂ is related to both the particle's motion and the black hole parameters. To explore more observable effects of the loop quantum gravity parameter on the motion of the spinning particle, we focus our attention on the circular orbits, particularly the innermost stable circular orbits, near the black hole. The result shows that for the same spin, there are two different innermost stable circular orbits, one with a larger radius and the other with a smaller radius. Both the radii decrease as the loop quantum gravity parameter increases. More significantly, with the increase of the spin of the particle, the small innermost stable circular orbit transition from timelike to spacelike, while the one with large radius does not. Instead, it terminates at a certain value of spin. Furthermore, we identify the existence of a minimal spin value, below which the innermost stable circular orbits maintain a timelike motion irrespective of the loop quantum gravity parameter. All the results present the significant influences of the loop quantum gravity parameter on the motion of the spinning particles.