Gravitational radiations from periodic orbits around Einstein-Æther black holes

Technical Summary: Gravitational Radiations from Periodic Orbits around Einstein-Æther Black Holes

Problem Statement
This work investigates the gravitational wave (GW) emission resulting from the periodic orbital motion of a test particle around black holes within the framework of Einstein-Æther theory. Einstein-Æther theory is a generally covariant modification of General Relativity (GR) that introduces a dynamical timelike vector field (the æther), thereby locally breaking Lorentz symmetry. While periodic orbits around such black holes have been previously studied, this paper aims to explore these orbits in greater detail and, crucially, to calculate the corresponding gravitational waveforms. The study focuses on how the æther field modifies orbital dynamics and the resulting radiation signatures, which are relevant for future space-based gravitational wave detectors (e.g., Taiji, Tianqin, LISA) targeting Extreme Mass Ratio Inspirals (EMRIs).

Methodology
The authors employ a multi-step analytical and numerical approach:

1. Theoretical Framework: The study utilizes two specific static, spherically symmetric black hole solutions in Einstein-Æther theory, distinguished by their coupling constants ($c_i$):

   • Solution I: Corresponds to $c_{14} = 0$ but $c_{123} \neq 0$.
   • Solution II: Corresponds to $c_{123} = 0$.
     The metric functions and æther field configurations for both solutions are derived from the field equations.

2. Geodesic Analysis: The motion of a massive test particle is analyzed using the Lagrangian formalism. The authors derive the effective potential and equations of motion for equatorial orbits. To characterize the complex dynamics, they adopt a taxonomy based on three topological integers $(z, w, v)$, representing the zoom, whirl, and vertex behaviors of the orbit. The ratio of radial to azimuthal frequencies ($q$) is calculated to identify periodic orbits where the frequency ratio is rational.

3. Gravitational Waveform Calculation:

   • Polarization Modes: The paper acknowledges that Einstein-Æther theory generally supports six polarization modes (two tensor, two vector, two scalar). However, for the specific static black hole solutions considered, the authors restrict the analysis to the two transverse-traceless tensor modes ($h_+$ and $h_\times$). For Solution II ($c_{123}=0$), the scalar modes are noted to become non-propagating or divergent in the standard expressions, justifying the focus on tensor modes.
   • Adiabatic Approximation: The authors model an EMRI system (a small mass $m$ orbiting a supermassive black hole $M$) using the adiabatic approximation. This assumes the inspiral timescale is much longer than the orbital period, allowing the trajectory to be treated as a sequence of geodesics with constant energy and angular momentum over short intervals.
   • Waveform Generation: Using the quadrupole formula adapted for Einstein-Æther theory, the authors compute the GW waveforms ($h_+$ and $h_\times$) based on the particle's trajectory in Schwarzschild coordinates, transformed into a detector-adapted frame.

Key Contributions and Results

• Orbital Classification: The study successfully generates and visualizes periodic orbits for both black hole solutions using the $(z, w, v)$ taxonomy. The results demonstrate that the coupling constants ($c_{13}$ for Solution I and $c_{14}$ for Solution II) influence the orbital shapes, with higher values of these parameters leading to more complex trajectories.
• Zoom-Whirl Correlation: A direct connection is established between the "zoom-whirl" behavior of the orbits and the emitted GW waveforms.
  • Zoom phases: When the particle is far from the black hole (weak field), the GW signal is smooth with low amplitude and frequency.
  • Whirl phases: As the particle approaches the perihelion and enters the strong-field regime, the trajectory distorts, leading to sharp increases in both the frequency and amplitude of the GW signal.
  • Substructures: The paper finds that higher zoom numbers ($z$) result in increasingly intricate waveform substructures.
• Impact of the Æther Field:
  • Solution I ($c_{14}=0$): The æther parameter $c_{13}$ has a relatively weak effect on the waveforms, with negligible changes in amplitude and phase for the values tested.
  • Solution II ($c_{123}=0$): The parameter $c_{14}$ introduces more significant modifications. As $c_{14}$ increases, the GW amplitudes increase slightly, and the phase of the signal is noticeably altered compared to the Schwarzschild case ($c_{14}=0$).
• Waveform Morphology: The study illustrates that the GW waveform amplitude peaks when the object is in the "whirl" segment of the orbit and decreases during the "zoom" segment. The final stage of the trajectory, near the event horizon, produces a sharp burst in both frequency and amplitude.

Significance and Claims
The paper claims that the calculated waveforms carry significant information regarding the physical properties of the black hole and the surrounding spacetime. Specifically:

• The distinct influence of the æther parameters ($c_{13}$ and $c_{14}$) on the phase and amplitude of the GW signals offers a potential observational signature.
• These signatures could theoretically allow future space-based detectors to distinguish between the two types of Einstein-Æther black holes and the standard Schwarzschild black hole.
• The authors note that while the current analysis focuses on leading-order waveforms and excludes full 2.5 post-Newtonian (PN) radiation-reaction corrections (which are necessary for high-precision long-term phase evolution), the results provide a foundational understanding of how Lorentz-violating effects manifest in periodic orbital radiation.

The work concludes that the interplay between the topological classification of orbits and the specific coupling constants of Einstein-Æther theory provides a framework for testing deviations from General Relativity in the strong-field regime.