Kerr Perihelion Precession via the Laplace-Runge-Lenz Vector Method

TL;DR

This work addresses the Kerr-induced perihelion precession of a test particle in the equatorial plane by applying a perturbative Laplace-Runge-Lenz vector method. By modifying the LRL vector to counteract frame dragging at first order in the spin parameter $a$, the authors preserve the Keplerian orbit form and isolate the frame-dragging contribution, yielding the familiar Lense–Thirring precession. The key result, $\alpha = \frac{6 \pi M}{p} - \frac{8 \pi J}{p^{3/2} M^{1/2}}$ (with $J = Ma$ and $E \approx 1$ for weak-field, far-orbit cases), matches standard GR predictions and connects intuitive LRL dynamics to relativistic frame dragging. The approach offers pedagogical insight into the role of frame dragging within a vector-method framework and can be extended to other spacetime geometries.

Abstract

We calculate, up to the first-order in the black hole spin, the perihelion precession of a test particle in the equatorial plane of a Kerr black hole using the perturbative Laplace-Runge-Lenz (LRL) vector method. To account for the dragging of inertial frames, we modify the LRL vector by incorporating a counteracting term in the angular momentum, which preserves the Keplerian orbit form to first order. We derive the standard Lense-Thirring precession result, leading to a transparent reinterpretation of known results, clarifying the role of frame dragging in LRL-based perturbation methods.