Mode-sum regularization of the scalar self-force: Formulation in terms of a tetrad decomposition of the singular field

We examine the motion in Schwarzschild spacetime of a point particle endowed with a scalar charge. The particle produces a retarded scalar field which interacts with the particle and influences its motion via the action of a self-force. We assume that the magnitude of the scalar charge is small, and that the deviations from geodesic motion produced by the self-force are small. This problem is analogous to that of an electric charge moving under the action of its electromagnetic self-force, and to that of a small mass moving under the action of its gravitational self-force. We exploit the spherical symmetry of the Schwarzschild spacetime and decompose the scalar field in spherical-harmonic modes. Although each mode is bounded at the position of the particle, a mode-sum evaluation of the self-force requires regularization because the sum does not converge: the retarded field is infinite at the position of the particle. The regularization procedure involves the computation of regularization parameters, which are obtained from a mode decomposition of the Detweiler-Whiting singular field; these are subtracted from the modes of the retarded field, and the result is a mode-sum that converges to the actual self-force. We present such a computation in this paper. While regularization parameters have been presented before in the literature, there are two main aspects of our work that are new. First, we define the regularization parameters as scalar quantities by referring them to a tetrad decomposition of the singular field. This is different from standard practice, which is to define regularization parameters as vectorial quantities. The advantage of dealing with tetrad components is that these, unlike vector components, are naturally decomposed in scalar spherical harmonics. Second, we calculate, for any bound orbit around a Schwarzschild black hole, four sets of regularization parameters (denoted schematically by $A$, $B$, $C$, and $D$) instead of the usual three ($A$, $B$, and $C$). While only the first three regularization parameters are needed to produce a convergent mode-sum, the inclusion of a fourth parameter has the practically important consequence of accelerating the convergence. The focus of this paper is entirely on the computation of regularization parameters for the scalar self-force. The techniques that we introduce in this work are not, however, restricted to this context. They will readily be exported to the electromagnetic and gravitational cases, but we leave this generalization for future work. As proof of principle that our methods are reliable, we calculate the self-force acting on a scalar charge in circular motion around a Schwarzschild black hole, and compare our answers with those recorded in the literature. We leave for future work the generalization of this calculation to generic orbits.

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