Self-force of a scalar field for circular orbits about a Schwarzschild black hole

TL;DR

This paper develops a practical, high-precision scheme to compute the self-force on a scalar charge moving in a circular orbit around a Schwarzschild black hole. By decomposing the retarded scalar field into a singular part $\psi^{\mathrm{S}}$ and a regular remainder $\psi^{\mathrm{R}}$, and by evaluating regularization parameters derived from a Hadamard/THZ-coordinate analysis, the authors implement a mode-sum approach that yields the self-force from $\mathcal{F}_a^{\mathrm{self}} = q \nabla_a \psi^{\mathrm{R}}$ at the particle’s location. They derive leading and higher-order regularization parameters ($A_r$, $B_r$, $C_r=0$, $D_r$, and $E^k_r$ terms) to accelerate convergence, and demonstrate the method with a numerical application at $r_o=10M$, achieving rapid convergence and agreement with established results. The work provides a scalable framework for accurate self-force computations, with clear applicability to gravitational-wave source modeling andExtensions to gravitational perturbations anticipated to enhance waveform predictions for extreme-mmass-ratio inspirals.

Abstract

The foundations are laid for the numerical computation of the actual worldline for a particle orbiting a black hole and emitting gravitational waves. The essential practicalities of this computation are here illustrated for a scalar particle of infinitesimal size and small but finite scalar charge. This particle deviates from a geodesic because it interacts with its own retarded field $ψ^\ret$. A recently introduced Green's function $G^\SS$ precisely determines the singular part, $ψ^\SS$, of the retarded field. This part exerts no force on the particle. The remainder of the field $ψ^\R = ψ^\ret - ψ^\SS$ is a vacuum solution of the field equation and is entirely responsible for the self-force. A particular, locally inertial coordinate system is used to determine an expansion of $ψ^\SS$ in the vicinity of the particle. For a particle in a circular orbit in the Schwarzschild geometry, the mode-sum decomposition of the difference between $ψ^\ret$ and the dominant terms in the expansion of $ψ^\SS$ provide a mode-sum decomposition of an approximation for $ψ^\R$ from which the self-force is obtained. When more terms are included in the expansion, the approximation for $ψ^\R$ is increasingly differentiable, and the mode-sum for the self-force converges more rapidly.

Figures (1)

• Figure 1: The upper portion of the figure displays $\mathcal{F}^{\text{ret}}_{{\ell} r}$ as a function of ${\ell}$, along with the result of it being regularized by $A_r$, $B_r$, and $D_r$. The lower portion displays the residual after a numerical fit of from 1 to 5 additional parameters, $E^1_r \ldots E^5_r$. A point where the data on a particular curve changes sign from being negative to positive is labeled with $+$, from positive to negative by $\circ$.