Gravitational self force in extreme mass-ratio inspirals

TL;DR

The review surveys gravitational self-force (SF) in extreme mass-ratio inspirals, emphasizing Kerr backgrounds relevant to LISA and the two-step program of computing SF on geodesics before modeling orbital evolution. It covers the fundamental theory (MiSaTaQuWa, Detweiler-Whiting), gauge issues, and practical computation frameworks such as mode-sum and puncture methods, with detailed Schwarzschild demonstrations in Lorenz gauge and emerging Kerr implementations. The text highlights numerical strategies for handling the particle singularity, including 1+1D Lorenz evolution and 2+1D puncture/m-mode approaches, and discusses physical SF effects, energy/angular-momentum dissipation, and gauge-invariant proxies like ISCO shifts. It concludes by outlining cross-validation prospects, Kerr challenges, and future directions for integrating SF results into accurate EMRI waveform templates and strong-field tests of gravity.

Abstract

This review is concerned with the gravitational self-force acting on a mass particle in orbit around a large black hole. Renewed interest in this old problem is driven by the prospects of detecting gravitational waves from strongly gravitating binaries with extreme mass ratios. We begin here with a summary of recent advances in the theory of gravitational self-interaction in curved spacetime, and proceed to survey some of the ideas and computational strategies devised for implementing this theory in the case of a particle orbiting a Kerr black hole. We review in detail two of these methods: (i) the standard mode-sum method, in which the metric perturbation is regularized mode-by-mode in a multipole decomposition, and (ii) $m$-mode regularization, whereby individual azimuthal modes of the metric perturbation are regularized in 2+1 dimensions. We discuss several practical issues that arise, including the choice of gauge, the numerical representation of the particle singularity, and how high-frequency contributions near the particle are dealt with in frequency-domain calculations. As an example of a full end-to-end implementation of the mode-sum method, we discuss the computation of the gravitational self-force for eccentric geodesic orbits in Schwarzschild, using a direct integration of the Lorenz-gauge perturbation equations in the time domain. With the computational framework now in place, researchers have recently turned to explore the physical consequences of the gravitational self force; we will describe some preliminary results in this area. An appendix to this review presents, for the first time, a detailed derivation of the regularization parameters necessary for implementing the mode-sum method in Kerr spacetime.

Figures (4)

• Figure 1: An illustration of the setup described in the text. $z(\tau)$ is a point on the timelike worldline $\Gamma$ (thick solid line) and $x$ is a field point close to $z$, shown with a portion of its past light cone. $\epsilon$ is the spatial geodesic distance from $x$ to $\Gamma$ and $\delta x^{\alpha}\equiv x^{\alpha}-z^{\alpha}$. The metric perturbation at $x$ consists of a direct and a tail contributions, illustrated by the thick dashed and dash-dot lines, respectively. [ Graphics reproduced from Ref. OrleansBook.]
• Figure 2: An illustration of the simple setup described in the text: A particle of mass $\mu$ in flat space is at rest at $\vec{x}_{\rm p}=(r_0,\theta_0,\varphi_0)$. The gravitational field of the particle is decomposed into spherical harmonics, each contributing a finite amount to the full radial force acting on the particle: either $F^l_{r+}$ or $F^l_{r-}$, depending on whether the force is calculated from $r\to r^+$ or $r\to r^-$. $\vec{x}=(r,\theta,\varphi)$ is an arbitrary field point used in the construction described in the text.
• Figure 3: The numerical 1+1D domain in the Barack--Sago code Barack:2007tmBSprep: A staggered mesh based on characteristic (Eddington--Finkelstein) coordinates $v,u$. $t$ and $r^*$ are, respectively, the Schwarzschild time and tortoise radial coordinates. The evolution proceeds from characteristic initial data on two null surfaces. Illustrated are a few sample geodesic orbits (radial plunge, circular, eccentric). [ Graphics reproduced from Ref. OrleansBook.]
• Figure 4: The gravitational SF [in units of $(\mu/M)^2$] along a Schwarzschild geodesic with semi-latus rectum $p=7M$ and eccentricity $e=0.2$. The upper and lower lines show $F_{\rm self}^r$ and $F^{\rm self}_t$, respectively. Integer values on the horizontal axis correspond to periapses ($r=r_{\rm min}$); note the slight retardation manifest in the magnitude of the radial component. The data for these plots were obtained using a direct integration of the metric perturbation equations in the Lorenz gauge, in conjunction with the mode-sum method, as described in Sec. \ref{['sec:Lorenz']}. [ Graphics reproduced from Ref. OrleansBook.]