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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 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.