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This article provides a detailed investigation into the motion of thesurrounding particles around a polymer black hole in loop quantum gravity(LQG). Using effective potential, the critical bound orbits and innermoststable circular orbits (ISCO) are analyzed. The study finds that the radii andangular momentum of the critical bound orbits decrease with an increase in theparameter $A_\lambda$ which labels the LQG effects, while the energy andangular momentum of the ISCO also decreases with an increase in $A_\lambda$.Based on these findings, we then explore the periodic orbits of the polymerblack hole in LQG using rational numbers composed of three integers. Ourresults show that the rational numbers increase with the energy of particlesand decrease with the increase of angular momentum based on a classificationscheme. Moreover, compared to a Schwarzschild black hole, the periodic orbitsin a polymer black hole in LQG consistently have lower energy, providing apotential method for distinguishing a polymer black hole in LQG from aSchwarzschild black hole. Finally, we also examine the gravitational waveradiations of the periodic orbits of a test object which orbits a supermassivepolymer black hole in LQG, which generates intricate GW waveforms that can aidin exhibiting the gravitational structure of the system.

Periodic orbits and their gravitational wave radiations in a polymer black hole in loop quantum gravity