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

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 illustrated here 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 ${\psi }^{\mathrm{ret}}.$ A recently introduced Green’s function $G^{\mathrm{S}}$ precisely determines the singular part ${\psi }^{\mathrm{S}}$ of the retarded field. This part exerts no force on the particle. The remainder of the field ${\psi }^{\mathrm{R}}={\psi }^{\mathrm{ret}}-{\psi }^{\mathrm{S}}$ 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 ${\psi }^{\mathrm{S}}$ 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 ${\psi }^{\mathrm{ret}}$ and the dominant terms in the expansion of ${\psi }^{\mathrm{S}}$ provide a mode-sum decomposition of an approximation for ${\psi }^{\mathrm{R}}$ from which the self-force is obtained. When more terms are included in the expansion, the approximation for ${\psi }^{\mathrm{R}}$ is increasingly differentiable, and the mode sum for the self-force converges more rapidly.