General orbital perturbation theory in Schwarzschild space-time

TL;DR

We address the problem of modeling timelike orbits in Schwarzschild space-time under generic external forces with high accuracy and computational efficiency. We develop a general-relativistic Gaussian perturbation framework that evolves seven osculating elements, including $E$, $L_z$, $\iota$, $\Omega$, $\bar{\varphi}_0$, and the coordinate/proper anomalies $M_t$, $M_s$, expressed via Weierstrass elliptic functions for the unperturbed geodesics. The method is linearized in perturbation strength and applied to two representative deformations: linear-in-spin Kerr perturbations and linear-in-quadrupole $q$-metric perturbations, yielding secular rates such as the Lense–Thirring nodal precession and quadrupole-induced node shifts, with excellent agreement against exact Kerr results and superior convergence relative to first-order PN in strong-field regimes. The framework enables fast, analytic-like treatment of strong-field dynamics and can underpin kludge waveform models, self-force studies, and extensions to more complex environments around compact objects.

Abstract

We derive general relativistic Gaussian equations for osculating elements for orbits under the influence of a perturbing force without any restrictions in an underlying Schwarzschild space-time. Such a formulation provides a way to describe the evolution of orbital parameters in strong gravity relativistic settings. As examples of external forces we considered Kerr and $q$-metric space-times generated forces, for which we solve equations for osculating elements in linear approximation. For the Kerr space-time in the post-Newtonian limit, our result reproduces the well-known Lense--Thirring precession of the longitude of the ascending node.

Figures (9)

• Figure 1: Two coordinate systems: the general system $\vec{x}$ and the system $\vec{x}'$ in which the orbital plane lies in the equatorial plane, connected by the rotation (\ref{['changing_of_the_variables']}).
• Figure 2: Comparison of orbital trajectories under the influence of the rotation of the central object. The dashed curves represent the unperturbed Schwarzschild geodesic, while the solid curves show the trajectories modified by secular perturbations. We set $r_g=1$ (shown as a black sphere in the plot) together with the initial condition: $\iota=\frac{1}{2}$, $\Omega=1$, $\varphi_0=0$.
• Figure 3: Comparison between the shift of the longitude of the node per revolution given by our approach $\Delta_{\Omega}$ (\ref{['eq::delta:omega']}) (red line), the exact analytical result $\Delta^{FH}_{\Omega}$ (\ref{['eq::kerr::Omega::FH']}) (blue circles), and the first-order post-Newtonian expression $\Delta^{PN}_{\Omega}$ (\ref{['eq::delta::omega::PN']}) (yellow dashed line), shown as functions of the semi-latus rectum $p$, for different values of eccentricity $e$, for the Kerr space-time treated as a perturbation of the Schwarzschild background. We set $r_g=2$, $a=10^{-3}$. Note that this result does not depend on the initial value of the inclination.
• Figure 4: Comparison between the shift of the longitude of the node per revolution obtained from our approach $\Delta_{\Omega}$ (\ref{['eq::delta:omega']}) the exact analytical result $\Delta^{FH}_{\Omega}$ (\ref{['eq::kerr::Omega::FH']}), and the first-order post-Newtonian expression $\Delta^{PN}_{\Omega}$ (\ref{['eq::delta::omega::PN']}), shown as functions of the semi-latus rectum $p$, for different values of eccentricity $e$, for the Kerr space-time treated as a perturbation of the Schwarzschild background. The dashed red line is the reference unity line, the solid yellow and blue lines represent the ratios $\Delta^{FH}_{\Omega}/\Delta^{PN}_{\Omega}$ and $\Delta^{FH}_{\Omega}/\Delta_{\Omega}$, respectively. We set $r_g=2$, $a=10^{-3}$. Note that this result does not depend on the initial value of the inclination.
• Figure 5: The shift of the longitude of the node per revolution. Solid lines correspond to our result $\Delta_{\Omega}$ (\ref{['eq::delta:omega']}) while dashed lines represent the post-Newtonian expression $\Delta^{PN}_{\Omega}$\ref{['eq::delta::omega::PN']}, shown as functions of the eccentricity $e$ (figure \ref{['fig:kerr_delta_omega_e']}) and of the semi-latus rectum $p$ (figure \ref{['fig:kerr_delta_omega_p']}), for the Kerr space-time treated as a perturbation of the Schwarzschild background. The curves are shown in blue, yellow, green, and red, following the order indicated in the set. We set $r_g=1$ and use the initial condition $\varphi_0=\tfrac{9\pi}{5}$. Note that this result does not depend on the initial value of the inclination.