Gravitational self-force on generic bound geodesics in Kerr spacetime

TL;DR

This work addresses the need for accurate gravitational self-force (GSF) modeling of extreme mass-ratio inspirals in Kerr spacetime by computing the first-order GSF on fully generic bound Kerr geodesics. The authors reconstruct a metric perturbation in the outgoing radiation gauge from the Weyl scalar $\psi_4$, obtained by solving the Teukolsky equation in the frequency domain using the MST formalism, and then extract the local self-force via $l$-mode regularization, including a Kerr-completion to account for perturbations of the background Kerr solution. They implement a comprehensive pipeline that yields GSF components for generic orbits, validate the approach with regularization-parameter checks and flux-balance relations, and present time-series and torus representations to illustrate the bi-periodic structure of the force. The results establish a foundational step toward precise EMRI waveform modeling for LISA, enabling later exploration of resonances, quasi-invariants, and detailed orbital evolution beyond equatorial orbits. The study also highlights computational challenges and charts a path for performance-optimized implementations and broader parameter-space coverage.

Abstract

In this work we present the first calculation of the gravitational self-force on generic bound geodesics in Kerr spacetime to first order in the mass-ratio. That is, the local correction to equations of motion for a compact object orbiting a larger rotating black hole due to its own impact on the gravitational field. This includes both dissipative and conservative effects. Our method builds on and extends earlier methods for calculating the gravitational self-force on equatorial orbits. In particular we reconstruct the local metric perturbation in the outgoing radiation gauge from the Weyl scalar $ψ_4$, which in turn is obtained by solving the Teukolsky equation using semi-analytical frequency domain methods. The gravitational self-force is subsequently obtained using (spherical) $l$-mode regularization. We test our implementation by comparing the large $l$-behaviour against the analytically known regularization parameters. In addition we validate our results be comparing the long-term average changes to the energy, angular momentum, and Carter constant to changes to these constants of motion inferred from the gravitational wave flux to infinity and down the horizon.

Figures (8)

• Figure 1: The $l$-modes of the various components of the GSF on a geodesic with $(a,p,e,z_\mathrm{max}) = (0.9,10,0.1,0.1)$, shown at the point along the orbit identified by $(q_r,q_z)= (\pi/3,\pi/6)$. This point is indicative of the generic behaviour (certain special points such as periapsis and apapsis will show better convergence behaviour). The grey lines are reference lines of $L=l+1/2$. As expected the $\pm$ parts of the $t$ and $r$ components diverge with $L$. The parameters $A_z$ and $A_\phi$ vanish Barack:2009uxBarack:2002mh. Consequently, we see $\pm$ parts of the $r$ and $\phi$ l-modes converge to a constant, just like all the two-side average parts.
• Figure 2: The same $l$ modes as in Fig. \ref{['fig:lmodediv']} after subtracting the Lorenz gauge regularization parameters. At large $l$, all components of the GSF conform with $L^{-2}$ behaviour indicated by the grey reference lines. This is a stringent check on the validity of our method and numerical implementation.
• Figure 3: Time series data of the GSF on an orbits with $(a,p,e,z_\mathrm{max})=(0.9,10,0.1,0.5)$. Because of the bi-periodic nature of the GSF none of the modulation patterns ever repeat.
• Figure 4: GSF as a function on the torus for an orbit with $(a,p,e,z_\mathrm{max})=(0.9,10,0.1,0.1)$. The horizontal axis displays changing $q_r$, while the vertical axis displays $q_z$.
• Figure 5: GSF as a function on the torus for an orbit with $(a,p,e,z_\mathrm{max})=(0.9,10,0.1,0.3)$. The horizontal axis displays changing $q_r$, while the vertical axis displays $q_z$.