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 Fₗ^μ of the ``full-force'' field F^μ, 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 Fₗ^μ are obtained by fully decomposing the particle's self-field into Fourier-harmonic modes lmω, calculating the contribution of each such mode to Fₗ^μ, 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 Fₗ^μ 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 Fₗ^μ (rather than the inhomogeneous modes, as in the standard method), which guarantees an exponentially fast convergence to the correct value of Fₗ^μ, 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.