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The motion of a small compact object in a curved background spacetimedeviates from a geodesic due to the action of its own field, giving rise to aself-force. This self-force may be calculated by integrating the Green functionfor the wave equation over the past worldline of the small object. We computethe self-force in this way for the case of a scalar charge in Schwarzschildspacetime, making use of the semi-analytic method of matched Green functionexpansions. Inside a local neighbourhood of the compact object, this methoduses the Hadamard form for the Green function in order to render regularizationtrivial. Outside this local neighbourhood, we calculate the Green functionusing a spectral decomposition into poles (quasinormal modes) and a branch cutintegral in the complex-frequency plane. We show that both expansions overlapin a sufficiently large matching region for an accurate calculation of theself-force to be possible. The scalar case studied here is a useful andillustrative toy-model for the gravitational case, which serves to modelastrophysical binary systems in the extreme mass-ratio limit.

Self-Force and Green Function in Schwarzschild spacetime via Quasinormal Modes and Branch Cut