Gravitational wave signatures from periodic orbits around a Schwarzschild-Bertotti-Robinson black hole

In this paper, we investigate periodic bound orbits and gravitational wave (GW) emission in the Schwarzschild–Bertotti–Robinson (Schwarzschild–BR) spacetime—an exact electrovacuum solution describing a static black hole (BH) immersed in a uniform magnetic field. We explore how the background magnetic field qualitatively alters the BH's gravitational dynamics, affecting timelike geodesics such as the marginally bound orbit (MBO) and the innermost stable circular orbit (ISCO). We then analyze periodic bound orbits using the ratio of the azimuthal and radial frequencies, which characterizes the orbits by their azimuthal and radial motions. Based on the numerical kludge method, we further compute the gravitational waveforms emitted from periodic orbits around a supermassive Schwarzschild–BR BH. We show that the background magnetic field changes the orbital frequencies, resonance conditions, zoom–whirl structures, and the resulting waveforms. Finally, we compute the frequency spectra in the mHz band and estimate the signal-to-noise ratio for LISA, Taiji, and TianQin. Our results show how a magnetic field that is part of the spacetime geometry affects periodic orbits and their gravitational-wave signals.