Periodic orbits and gravitational waveforms of black holes in bumblebee gravity

Abstract

In this paper, we investigate the dynamics of massive particles and the associated gravitational waveforms in the spacetime of a black hole within the framework of Einstein-Bumblebee gravity. Our analysis encompasses both charged and uncharged black hole configurations, with a particular focus on the spontaneous Lorentz symmetry breaking mechanism inherent to this model, which is governed by a dimensionless coupling parameter $l$. We analyze the geodesic equations and the effective potential to determine the allowed parameter space for bound orbits, demonstrating that in the charged case, both the Lorentz-violating parameter $l$ and the electric charge $Q$ significantly enhance the confinement capacity of the potential, thereby broadening the energy and angular momentum windows for bound states. A key focus is placed on the classification and properties of periodic orbits, characterized by rational frequency ratios using the whirl, zoom, and vertex taxonomy. We demonstrate that in the uncharged case ($Q=0$), the radial effective potential and standard innermost stable circular orbit (ISCO) properties are degenerate with those of a Schwarzschild black hole. However, despite this degeneracy in static potential properties, the structure of periodic orbits exhibits qualitative differences, providing a possible observational signature that can break this degeneracy. Finally, we compute the corresponding gravitational waveforms extracted from these periodic orbits using the quadrupole formula. The results reveal that $l$ and $Q$ introduce contrasting phase-shifting effects on the waveforms. This suggests that bumblebee gravity leaves measurable imprints on gravitational-wave signals that could be detected by future space-based gravitational-wave observatories.

Figures (8)

• Figure 1: The radial effective potential $V_{\text{eff}}(r)$ for massive particles. In each panel, the red and blue dashed lines trace the loci of the potential maxima and minima, respectively.
• Figure 2: The allowed parameter space $(E, L)$ for bound orbits with different $Q$ or $l$. The shaded regions indicate the energy-angular momentum combinations that permit bound motion. The black dots at the leftmost tip of each region denote the ISCO points.
• Figure 3: The rational frequency ratio $q$ as a function of the angular momentum $L$. In each panel, the vertical dashed lines of corresponding colors denote the allowed ranges of $L$ for bound motion.
• Figure 4: The rational frequency ratio $q$ as a function of the particle's energy $E$. In each panel, the vertical dashed lines of corresponding colors denote the allowed ranges of $E$ for bound motion.
• Figure 5: The rational frequency ratio $q$ as a function of orbital parameters for $Q=0$. In both panels, the vertical dashed lines represent the boundaries of the allowed ranges for bound orbits.