Periodic orbits and gravitational waveforms of spinning particles in nonlocal Gravity

TL;DR

The paper develops a comprehensive framework for spinning-particle dynamics around DW nonlocal gravity black holes using the MPD equations with a Tulczyjew spin supplementary condition. It analyzes how nonlocal corrections, parameterized by $\zeta$ and $b$, modify the effective potential and ISCO, and it classifies and investigates equatorial periodic orbits whose gravitational-wave signatures exhibit phase shifts relative to Schwarzschild. By employing a numerical-kludge approach, it connects orbital dynamics to GW waveforms and demonstrates that, for plausible EMRI configurations, nonlocal effects can produce a detectable waveform mismatch after about one year of observation. This work provides a pathway to observationally test nonlocal gravity predictions with future space-based GW detectors like LISA, via precise modeling of spinning-body orbits and their GW emissions.

Abstract

In this paper, we investigate the dynamics and gravitational-wave signatures of periodic orbits of spinning test particles moving in the equatorial plane around static, spherically symmetric black holes within the framework of Deser-Woodard nonlocal gravity. Based on the Mathisson-Papapetrou-Dixon equations, combined with the Tulczyjew spin supplementary condition, we derive the orbital dynamic equations for spinning particles moving in the equatorial plane and impose a timelike constraint to exclude unphysical superluminal trajectories. By comparing with the classical Schwarzschild black hole, we systematically analyze the effects of the nonlocal gravitational parameters $ζ$ and $b$ on the effective potential governing the radial motion of particles and the innermost stable circular orbit. In addition, gravitational waveforms exhibit significant phase differences: an increase in $ζ$ induces a phase delay, whereas an increase in $b$ results in a phase advance. A one-year simulation of the orbital evolution of an extreme mass ratio inspiral demonstrates that when $b=2$ and $ζ\approx10^{-6}$, the mismatch between the gravitational waveforms predicted for the nonlocal gravity black hole and those for the Schwarzschild black hole reaches the distinguishable threshold ($\mathcal{M}=0.0125$), providing a basis for observational discrimination between general relativity and nonlocal gravity.

Figures (8)

• Figure 1: Effective potential $V_{\text{eff}}$ as a function of $r$ with $b=2, \zeta=0.2, s=0.3, l=3.5$. The local maximum (blue dashed curve) defines the potential barrier separating bound and plunging trajectories, while the local minimum (blue dots) corresponds to stable circular orbits.
• Figure 2: Allowed parameter region in the $(E,l)$ plane for bound motion. The shaded area corresponds to energies lying between the local maximum and minimum of the effective potential. The vertex of the shaded region marks the ISCO, where the two extrema merge.
• Figure 3: The variation of $V_{\text{eff}}$ with respect to the radial coordinate $r$. (a) Distinct $b$ for fixed $\zeta$,$l$ and $s$; (b) Distinct $\zeta$ for fixed $b$, $s$ and $l$; (c) Distinct positive $s$ for fixed $l$, $b$ and $\zeta$. The red dashed line traces the maxima (peaks) of the effective potential curves, while the dark blue dashed line marks the minima (valleys) across parameter variations.
• Figure 4: Dependence of the ISCO radius (left), energy (center), and orbital angular momentum (right) on the spin parameter $s$, for several values of the parameter $\zeta$, with $b=2$ fixed. Co-rotating ($sl>0$) and counter-rotating ($sl<0$) configurations exhibit distinct trends, illustrating how nonlocal corrections modify spin–orbit coupling and shift the ISCO location.
• Figure 5: ISCO radius (left), energy (center), and orbital angular momentum (right) as functions of the spin parameter $s$ for different values of $b$, with $\zeta=0.1$ fixed. Increasing $b$ shifts the ISCO inward and lowers the corresponding energy, reflecting the influence of the nonlocal parameter on the near-horizon geometry.