Massive Particle Motion Around Horndeski Black Holes

TL;DR

The paper analyzes time-like geodesics around a four-dimensional asymptotically flat Horndeski black hole with derivative coupling to the Einstein tensor, described by $f(r)=1-\frac{2M}{r}-\frac{\gamma^{2}}{r^{2}}$. It derives the equations of motion, constructs the effective potential $V_{ m eff}^2(r)=f(r)(1+L^2/r^2)$, and classifies all possible orbits for $L\neq0$ and $L=0$, providing exact analytic solutions in terms of Weierstrass elliptic functions and elementary functions. The analysis covers circular, planetary, scattering, and critical trajectories, including expressions for epicycle frequencies, ISCO, and various turning points, with explicit formulas for orbital precession. By connecting the perihelion shift to Keplerian motion and comparing with Solar System data, the work constrains the Horndeski coupling parameter $\gamma$, demonstrating the viability of using precise orbital measurements to test modified gravity theories.

Abstract

The time-like structure of the four-dimensional asymptotically flat Horndeski black holes is studied in detail. Focusing on the motion of massive neutral test particles, we construct the corresponding effective potential and classify the admissible types of orbits. The equations of motion are solved analytically, yielding trajectories expressed in terms of Weierstrass elliptic functions and elementary functions. As an application, we compute the perihelion precession as a classical test of gravity within the Solar System and use it to place observational constraints on the coupling parameter between the scalar field and gravity.

Figures (17)

• Figure 1: The first light blue solid curve represents the event horizon of the Horndeski black hole, the second black dashed line corresponds to the Schwarzschild black hole, and the third orange solid curve represents the external event horizon of the Reissner--Nordström black hole, with $M=1$ in all three cases. Furthermore, for the last curve we set $Q = \gamma$ in $r_+^{RN}$.
• Figure 2: Three different behaviors of the effective potential are shown for a massive particle. We consider a black hole of mass $M=1$ and $\gamma=2$. The black, purple, and blue curves correspond to $L=0$, $L=5$, and $L=10$, respectively, while the dashed gray line indicates the horizontal asymptotic value $V_{\text{eff}}^{\infty}=1$. Here, $L_C=4.751$ and $L_S=5.330$.
• Figure 3: Effective potentials are shown for a massive particle with angular momentum $L=6$. The black dashed curve corresponds to the Schwarzschild black hole, the curves above it to the Reissner--Nordström black hole, and the curves below it to the Horndeski black hole, the three cases with $M=1$. The gray horizontal dashed line indicates the asymptotic value of the effective potential.
• Figure 4: The epicycle frequency versus the $\gamma$ parameter with $M=1$ and the angular momenta $L=5$, $L=6$ and $L=7$, respectively.
• Figure 5: The behavior of the precession angle $\Phi_P$ for a massive particle, between the energies $E_S<E<E_U$, where $M=1$, $\gamma=2$ and $L=5.3$.