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 ψ^ret. A recently introduced Green’s function G^S precisely determines the singular part ψ^S of the retarded field. This part exerts no force on the particle. The remainder of the field ψ^R=ψ^ret-ψ^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 ψ^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 ψ^ret and the dominant terms in the expansion of ψ^S 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.