Gravitational Wave Signatures from Periodic Orbits around a Non--commutative Schwarzschild Black Hole

In this work, we investigate massive particle motion and the gravitational wave emission generated by periodic trajectories around a non–commutative $\mathit{\text{Schwarzschild}}$ black hole sourced by a Lorentzian matter distribution. We analyze the effective potential, the marginally bound orbit, and the innermost stable circular orbit, showing that non–commutative corrections shift these characteristic orbits toward smaller radii and reduce their corresponding angular momenta. The allowed region in the $(E,L)$ plane is also displaced toward lower values, favoring more tightly bound configurations. Periodic trajectories are classified through the rational parameter $q$, which relates the radial and azimuthal frequencies. For a fixed orbital topology, increasing the non–commutative parameter lowers the energy required to produce the orbit and results in more compact zoom–whirl configurations. Small deviations from the periodic energies are also shown to generate precessional drift. From the periastron advance of the S2 star around Sgr~A${}^{\ast }$, we obtain the preliminary bound $\mathrm{\Theta }\mathrm{/}{M}^2<0.014$. Finally, using the adiabatic and numerical kludge approximations, we compute the gravitational wave polarizations and find phase shifts and an overall enhancement of the amplitude.