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

TL;DR

The paper investigates how an intrinsic magnetic field in the Schwarzschild–Bertotti–Robinson spacetime modifies bound geodesics and the gravitational waves they emit from extreme-mass-ratio inspirals. It develops a geodesic framework in the non-rotating Schwarzschild–BR metric, analyzes ISCO/MBO and the $(E,L)$ parameter space, and classifies periodic orbits using the rational frequency ratio $q = \frac{ω_{\varphi}}{ω_r} - 1$ with topological indices $(z,w,v)$. Gravitational waves are computed via a numerical kludge by solving geodesics and applying the quadrupole formula, yielding detector-frame waveforms and spectra in the $\text{mHz}$ range. The results show that the background magnetic field $B$ shifts orbital frequencies and resonance conditions, alters zoom–whirl dynamics, and produces GW signals that can exceed space-based detector sensitivities (LISA, Taiji, TianQin), providing a potential observational test of magnetized spacetime geometries.

Abstract

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 frequency ratio ${ω_{\varphi}}/{ω_{r}}$, 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 significantly changes orbital frequencies, resonance conditions, zoom-whirl structures, and the resulting waveforms. Finally, we examine the frequency spectra in the mHz range and the detectability of these GW signals by computing the characteristic strain via a discrete Fourier transform on the time-domain waveforms, comparing the results with the sensitivity curves of space-based GW detectors such as LISA, Taiji, and TianQin. Our results show that intrinsically magnetic fields modify spacetime and leave observable imprints on extreme mass-ratio inspiral GWs, which may be tested by future observations.

Figures (8)

• Figure 1: The radial dependence of the effective potential for different values of the magnetic field $B$ and the orbital angular momentum $L$.
• Figure 2: The allowed parameter space of the energy and the angular momentum for the bound orbits around the Schwarzschild-BR BH with different values of the magnetic field $B$
• Figure 3: Left panel: the dependence of the rational number $q$ on the energy of periodic orbits around the Schwarzschild-BR BH for different values of the magnetic field $B$. Here, the orbital angular momentum $L$ is fixed at $L=\frac{1}{2}(L_{MBO}+L_{ISCO})$. Right panel: the dependence of the rational number $q$ on the orbital angular momentum of periodic orbits around the Schwarzschild-BR BH for various combinations of the magnetic field $B$. Here, we set the energy as $E=0.96$.
• Figure 4: The figure demonstrates the periodic orbits for different $(z,w,v)$ around the Schwarzschild-BR BH in the case in which the magnetic field $B = 0.02$ and $L=\frac{1}{2}(L_{MBO}+L_{ISCO})$.
• Figure 5: The figure demonstrates the periodic orbits for different $(z,w,v)$ around the Schwarzschild-BR BH in the case in which the magnetic field $B=0.02$ and $E=0.96$.