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 $F^μ_l$ of the "full force" field $F^μ$, evaluated at the particle's location, and then sums over $l$ subject to a certain regularization procedure. In the frequency-domain variant of this scheme the quantities $F^μ_l$ are obtained by fully decomposing the particle's self field into Fourier-harmonic modes $l m ω$, calculating the contribution of each such mode to $F^μ_l$, and then summing over $ω$ and $m$ for given $l$. This procedure has the advantage that one only encounters {\it 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) results from the discontinuity of the time-domain $F^μ_l$ field at the particle's worldline. Here we propose a simple and practical method to resolve this problem. The method utilizes the {\it homogeneous} modes $l m ω$ of the self field to construct $F^μ_l$ (rather than the inhomogeneous modes, as in the standard method), which guarantees an exponentially-fast convergence to the correct value of $F^μ_l$, 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.