[Paper Review] Spinning test particle in four-dimensional Einstein-Gauss-Bonnet Black Hole

Yu‐Peng Zhang, Shao-Wen Wei+1 more|arXiv (Cornell University)|Mar 24, 2020

Astrophysical Phenomena and Observations44 references22 citations

TL;DR

This paper investigates the motion of a spinning test particle in the equatorial plane of a four-dimensional Einstein-Gauss-Bonnet (EGB) black hole, using the Mathisson-Papapetrou-Dixon equations under the pole-dipole approximation. It finds that for a specific range of the Gauss-Bonnet coupling parameter ($-6.1 < \alpha/M^2 < -2$), the effective potential develops two minima, allowing the particle to occupy two distinct stable circular orbits with identical spin and total angular momentum, a novel feature not seen in general relativity.

ABSTRACT

In this paper, we investigate the motion of a classical spinning test particle orbiting around a static spherically symmetric black hole in a novel four-dimensional Einstein-Gauss-Bonnet gravity [D. Glavan and C. Lin, Phys. Rev. Lett. 124, 081301 (2020)]. We find that the effective potential of a spinning test particle in the background of the black hole has two minima when the Gauss-Bonnet coupling parameter $α$ is nearly in a special range $-6.1

Motivation & Objective

• To study the motion of a spinning test particle in the background of a four-dimensional Einstein-Gauss-Bonnet black hole, a recently proposed theory that avoids the Lovelock theorem.
• To investigate how the Gauss-Bonnet coupling parameter α and the particle's spin influence the effective potential and orbital stability.
• To determine the conditions under which the spinning test particle can occupy multiple stable circular orbits with the same spin and total angular momentum.
• To analyze the innermost stable circular orbit (ISCO) of the spinning particle and compare its behavior to that in general relativity and spinless cases.

Proposed method

• The motion of the spinning test particle is modeled using the Mathisson-Papapetrou-Dixon (MPD) equations under the pole-dipole approximation, which accounts for spin-curvature forces.
• The effective potential for the particle's motion is derived from the MPD equations and analyzed in the equatorial plane of the static, spherically symmetric 4D EGB black hole.
• The ISCO is located by solving the condition $ d^2V_{\text{eff}}/dr^2 = 0 $, where the maximum and minimum of the effective potential merge.
• Numerical solutions are used to compute the ISCO radius and angular momentum for varying values of the spin parameter $ \bar{s} $ and GB coupling $ \alpha $.
• The superluminal constraint on the four-velocity is applied to ensure physical validity of the circular orbits.
• The analysis is performed in dimensionless units with $ M = 1 $, $ m = 1 $, and the parameter space $ \alpha/M^2 $, $ \bar{s} $, and $ \bar{l} $ is systematically explored.

Experimental results

Research questions

• RQ1 Can a spinning test particle in a 4D EGB black hole background occupy two distinct stable circular orbits with the same spin and total angular momentum?
• RQ2 How does the Gauss-Bonnet coupling parameter $ \alpha $ affect the effective potential and the existence of multiple minima in the potential well?
• RQ3 How does the spin of the test particle influence the radius and angular momentum of the innermost stable circular orbit (ISCO) in the 4D EGB black hole?
• RQ4 How does the ISCO of the spinning test particle in 4D EGB gravity compare to that in general relativity and in the spinless case?

Key findings

• For $ -6.1 < \alpha/M^2 < -2 $, the effective potential of the spinning test particle develops two distinct minima, enabling two separate stable circular orbits with identical spin and total angular momentum.
• The radius and angular momentum of the ISCO decrease with increasing spin $ \bar{s} $, a behavior consistent with the ISCO dynamics in general relativity.
• The ISCO radius also decreases with increasing Gauss-Bonnet coupling parameter $ \alpha $, mirroring the trend observed in the spinless case in 4D EGB gravity.
• When two stable orbits exist, the ISCO parameters become smaller than those in the spinless EGB case or the spinning case in Schwarzschild spacetime, indicating enhanced orbital stability at smaller radii.
• The jump behavior observed in the ISCO parameter plot for certain $ \alpha $ values is attributed to the merging of two potential minima, indicating a bifurcation in the orbital structure.
• The four-velocity remains timelike in the dual-orbit regime, confirming that the two-orbit solution is physically valid and not superluminal.