Low multipole contributions to the gravitational self-force

TL;DR

This work analyzes the gravitational self-force on a small mass in a circular Schwarzschild orbit, highlighting the essential role of the nonradiative low multipoles l=0 and l=1. Using Lorenz-gauge perturbation theory and Regge–Wheeler–Zerilli formalisms, the monopole is solved exactly, the odd-parity dipole analytically, and the even-parity dipole numerically with PN checks; all low-modes yield purely radial, conservative self-acceleration. The authors discuss gauge choices, demonstrate near-uniqueness of the Lorenz gauge in this context, and compare PN limits with previous results while explaining horizon-boundary subtleties. Overall, the paper lays groundwork for a complete self-force calculation and its eventual incorporation into gravitational-wave templates for extreme-mass-ratio inspirals.

Abstract

We calculate the unregularized monopole and dipole contributions to the self-force acting on a particle of small mass in a circular orbit around a Schwarzschild black hole. From a self-force point of view, these non-radiating modes are as important as the radiating modes with l greater than 2. In fact, we demonstrate how the dipole self-force contributes to the dynamics even at the Newtonian level. The self-acceleration of a particle is an inherently gauge-dependent concept, but the Lorenz gauge is often preferred because of its hyperbolic wave operator. Our results are in the Lorenz gauge and are also obtained in closed form, except for the even-parity dipole case where we formulate and implement a numerical approach.

Figures (3)

• Figure 1: Internal values of $a[l](R)$, rescaled by a common factor of $3 m/R^2$. For $R \gg M$ we have the following asymptotic behaviors: $a_<[l=0] \sim 3(m/R^2)(M/R)$, $a_<[l=1;\hbox{odd}] \sim -4(m/R^2)(M/R)$, and $a_<[l=1;\hbox{even}] \sim 3(m/R^2)$. An exact expression for $a_<[l=0]$ appears in Eq. (\ref{['3.15']}) below. An exact expression for $a_<[l=1;\hbox{odd}]$ appears in Eq. (\ref{['4.2']}). The values for $a_<[l=1;\hbox{even}]$ are obtained from Eq. (\ref{['5.48']}) and the results listed in Table I.
• Figure 2: External values of $a[l](R)$, rescaled by a common factor of $m/R^2$. For $R \gg M$ we have the following asymptotic behaviors: $a_>[l=0] \sim -m/R^2$, $a_<[l=1;\hbox{odd}] \sim 2(m/R^2)(M/R)$, and $a_<[l=1;\hbox{even}] \sim -3\beta(m/R^2)(M/R)$, where $\beta \simeq 2$ is numerically estimated at the end of Sec. V. An exact expression for $a_>[l=0]$ appears in Eq. (\ref{['3.16']}) below. An exact expression for $a_>[l=1;\hbox{odd}]$ appears in Eq. (\ref{['4.3']}). The values for $a_>[l=1;\hbox{even}]$ are obtained from Eq. (\ref{['5.48']}) and the results listed in Table I.
• Figure 3: Accuracy of the post-Newtonian expressions for the functions $B_<(r)$ and $C_<(r)$, for $R=25M$. The solid curve is a plot of $\hbox{Re}[B_<(r)]$, as given by Eq. (\ref{['5.78']}), divided by the numerical results listed in Table I. The dashed curve is a plot of $\hbox{Re}[C_<(r)]$, as given by Eq. (\ref{['5.79']}), divided by the numerical results listed in Table I. In both cases we have set $\beta = 2$. The error is estimated to be of order $(M/R)^2 \simeq 0.002$. The plots reveal that this estimate is accurate for all values of $r$ except near $r=2M$.