Frequency-domain calculation of the self-force: The high-frequency problem and its resolution

The mode-sum method provides a practical means for calculating the self-force acting on a small particle orbiting a larger black hole. In this method, one first computes the spherical-harmonic l-mode contributions Flμ of the "full-force" field Fμ, evaluated at the particle's location, and then sums over l subject to a certain regularization scheme. In the frequency-domain variant of this procedure the quantities Flμ are obtained by fully decomposing the particle's self-field into Fourier-harmonic modes lmω, calculating the contribution of each such mode to Flμ, and then summing over ω and m for given l. This procedure has the advantage that one only encounters ordinary differential equations. However, for eccentric orbits, the sum over ω is found to converge badly at the particle's location. This problem (reminiscent of the familiar Gibbs phenomenon of Fourier analysis) results from the discontinuity of the time-domain Flμ field at the particle's worldline. Here we propose a simple and practical method to resolve this problem. The method utilizes the homogeneous modes lmω of the self-field to construct Flμ (rather than the inhomogeneous modes, as in the standard method), which guarantees an exponentially fast convergence to the correct value of Flμ, even at the particle's location. We illustrate the application of the method with the example of the monopole scalar-field perturbation from a scalar charge in an eccentric orbit around a Schwarzschild black hole. Our method, however, should be applicable to a wider range of problems, including the calculation of the gravitational self-force using either Teukolsky's formalism, or a direct integration of the metric perturbation equations.