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

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Abstract
The self force is computed for a scalar particle in circular (geodesic) orbit about a Schwarzschild black hole. We use a particular Green's function decomposition, which can be exactly specified; however, in a practical computation it can only be given to some finite level of approximation. A special set of coordinates is chosen which is locally inertial in the neighborhood of the orbit and allows direct control over the level of approximation being used in the Green's function decomposition, especially in relation to its singular behavior. A mode sum expansion is used to regularize behavior near the singularity. In conjunction with specific properties which we demonstrate for the mode sum representation and with control we have over the level of approximation, knowledge of the Green's function decomposition is used to improve dramatically the rate of numerical convergence obtained in the computation of the finite self force. Comparison is made between our numerical analysis and that used in previous calculations, which have required a much higher range in mode summation.

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