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

Abstract

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 ${F}_{l}^{\ensuremath{\mu}}$ of the ``full-force'' field ${F}^{\ensuremath{\mu}}$, 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 ${F}_{l}^{\ensuremath{\mu}}$ are obtained by fully decomposing the particle's self-field into Fourier-harmonic modes $lm\ensuremath{\omega}$, calculating the contribution of each such mode to ${F}_{l}^{\ensuremath{\mu}}$, and then summing over $\ensuremath{\omega}$ and $m$ for given $l$. This procedure has the advantage that one only encounters ordinary differential equations. However, for eccentric orbits, the sum over $\ensuremath{\omega}$ 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 ${F}_{l}^{\ensuremath{\mu}}$ field at the particle's worldline. Here we propose a simple and practical method to resolve this problem. The method utilizes the homogeneous modes $lm\ensuremath{\omega}$ of the self-field to construct ${F}_{l}^{\ensuremath{\mu}}$ (rather than the inhomogeneous modes, as in the standard method), which guarantees an exponentially fast convergence to the correct value of ${F}_{l}^{\ensuremath{\mu}}$, 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.