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

Leor Barack, Amos Ori, and Norichika Sago

Phys. Rev. D 78, 084021 – Published 16 October 2008

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 $F_{l}^{μ}$ 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 $F_{l}^{μ}$ 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 $F_{l}^{μ}$ , 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 $F_{l}^{μ}$ field at the particle’s worldline. Here we propose a simple and practical method to resolve this problem. The method utilizes the homogeneous modes $l m ω$ of the self-field to construct $F_{l}^{μ}$ (rather than the inhomogeneous modes, as in the standard method), which guarantees an exponentially fast convergence to the correct value of $F_{l}^{μ}$ , 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.

Received 17 August 2008

DOI:https://doi.org/10.1103/PhysRevD.78.084021

Authors & Affiliations

Leor Barack¹, Amos Ori², and Norichika Sago¹

¹School of Mathematics, University of Southampton, Southampton, SO17 1BJ, United Kingdom
²Department of Physics, Technion-Israel Institute of Technology, Haifa, 32000, Israel

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Vol. 78, Iss. 8 — 15 October 2008

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Images

Figure 1
Construction of the scalar-field monopole and its $r$ derivative as a sum over inhomogeneous frequency modes in the standard approach. The numerical solutions shown here correspond to an eccentric geodesic orbit around a Schwarzschild black hole, with semilatus rectum $p = 10$ and eccentricity $e = 0.2$ (see Appendix ). The periastron and apastron for this orbit are at $r_{\min } = (25 / 3) M ≅ 8.333 M$ and $r_{\max } = 12.5 M$ , respectively. We used numerical integration to calculate the inhomogeneous radial functions $R_{n}^{inh} (r)$ through Eq. (c4), and then obtained the partial sums $ϕ (t, r; n_{\max })$ and $ϕ_{, r} (t, r; n_{\max })$ as defined in Eqs. (27, 28). Plotted here (solid lines) are the partial sums (per unit scalar charge) for $n_{\max } = 2$ , 4, 8, 15 as functions of $r$ at a fixed time $t$ when the particle is at $r = 10 M$ (this corresponds to radial phase $χ = π / 2$ ; see Appendix ). The left panel displays the scalar field itself; the right panel shows its $r$ derivative. For comparison, we also display (dashed line) the full scalar monopole solution, which we obtained directly using numerical evolution in the time domain (for our purpose, this latter solution can be taken as an accurate benchmark). It is evident that the $n$ -mode sum converges quickly to the correct value in the regions $r < r_{\min }$ and $r > r_{\max }$ , but the convergence deteriorates inside the domain $r_{\min } < r < r_{\max }$ , where Gibbs waves set in. The partial sum over frequency modes is smooth at the particle, and hence, strictly speaking, cannot recover the true jump discontinuity in the field derivative there. Finding a way around this technical problem is the main goal of this work.Reuse & Permissions
Figure 2
Convergence of the Fourier $n$ -mode sum at the location of the particle (illustrated here based on the numerical data of Fig. 1). Left panel: The deviation (per unit scalar charge) of the partial sum $ϕ (n_{\max })$ from the full field $ϕ$ at the location of the particle, as a function of $n_{\max }$ . The $n$ -mode partial sum appears to converge to the correct value approximately as $1 / n_{\max }$ (note the log-log scale), as expected on theoretical grounds. Right panel: The deviation (per unit scalar charge) of $ϕ_{, r} (n_{\max })$ from the full-field derivative $ϕ_{, r}$ , again evaluated at the particle’s location. Since the true $ϕ_{, r}$ [unlike $ϕ_{, r} (n_{\max })$ ] has two different one-sided values at $r = r_{p}$ , so does the deviation. The two solid lines represent these two one-sided values of the deviation. (These two values actually have opposite signs for each $n_{\max }$ , which is obscured here since only the absolute value of the deviation is shown.) From the essentially horizontal shape of the two solid lines it is evident that the $n$ -mode sum for $ϕ_{, r}$ does not converge to the correct one-sided values. As an aside, we also plot here (dashed line) the difference between $ϕ_{, r} (n_{\max })$ and the two-side average value of the full derivative $ϕ_{, r}$ (per unit scalar charge, at the particle’s location). The graph loosely suggests (referring to its upper envelop and ignoring the seemingly oscillatory deep structure) that $ϕ_{, r} (n_{\max })$ converges, albeit very slowly, to the average value of $ϕ_{, r}$ at the discontinuity. That, indeed, would again be consistent with theoretical prediction.Reuse & Permissions
Figure 3
(To be compared with Fig. 1.) Construction of the scalar-field monopole (left panel) and its $r$ derivative (right panel) using extended homogeneous solutions. The orbital parameters are chosen here as in Fig. 1, i.e., $p = 10$ and $e = 0.2$ , and we again present the various fields as functions of $r$ at a fixed time $t$ when the particle is at $r = 10 M$ . Vertical dotted lines mark the particle’s periastron and apastron radii, $r_{\min } = (25 / 3) M$ and $r_{\max } = 12.5 M$ , respectively. The solid line represents the full scalar monopole solution, obtained using numerical evolution in the time domain. The dashed, dotted, and dash-dotted lines represent partial sums over extended homogeneous $n$ -modes, calculated numerically based on Eqs. (38, 39). For clarity, in both panels we plot the $+$ and $-$ partial sums only in their relevant domains, $r \geq 10 M$ and $r \leq 10 M$ , respectively. We show here the partial sums for $n_{\max } = 0$ , 1, 2 only—the partial sums ${\tilde{ϕ}}^{\pm} (n_{\max } = 3)$ are already indistinguishable from the full solution at the scale of this plot (but see Fig. 4 below). The individual $n$ -modes of the extended fields ${\tilde{ϕ}}^{+}$ and ${\tilde{ϕ}}^{-}$ do not match continuously at the location of the particle, but their sum seems to converge quickly, everywhere, to the true solution (which is continuous). Similar fast convergence is manifest also for the derivative $ϕ_{, r}$ . Gibbs waves, which disrupt the convergence of the actual inhomogeneous $n$ -modes, are altogether avoided within the new scheme.Reuse & Permissions
Figure 4
Convergence of the extended $n$ -mode sum for the fields ${\tilde{ϕ}}^{\pm}$ (left panel) and their derivatives ${\tilde{ϕ}}_{, r}^{\pm}$ (right panel). For the same case shown in Fig. 3, we plot here the fractional differences between the partial sums and the full field (or the full-field derivative), for $n_{\max } = 1 - 5$ . The various graphs are labeled by their corresponding $n_{\max }$ values. The middle vertical dotted line marks the particle’s momentary radius at $r = 10 M$ . Once again, we display the $+$ and $-$ values only in their respective relevant domains $r \geq 10 M$ or $r \leq 10 M$ . Note the exponential scale of the y-axis. (The seemingly odd behavior of the data for $n_{\max } = 2$ and $n_{\max } = 5$ in the right panel is simply due to a change-of-sign which the corresponding fractional differences happen to experience around $r = 14 M$ and $r = 10 M$ , respectively. The tiny wiggly feature, barely visible near $r = 10 M$ for $n_{\max } = 5$ , is due to the numerical error in the time-domain data, which for $ϕ_{, r}$ is estimated at $\sim 10^{- 6}$ in fractional terms.) The exponentially fast convergence of the extended $n$ -mode sum even at $r_{\min } \leq r \leq r_{\max }$ —and particularly at the very location of the particle—is evident from these plots.Reuse & Permissions
Figure 5
Convergence of the extended $n$ -mode sum at the particle’s location. The left and right panels display the values at the particle’s location ( $r = 10 M$ ) of the fractional differences shown in the left and right panels of Fig. 4, correspondingly. The left panel demonstrates the exponential convergence of the extended $n$ -mode sum to the correct value of the field at the particle. In particular, the mismatch between the partial sums for ${\tilde{ϕ}}^{+}$ and ${\tilde{ϕ}}^{-}$ at the particle appears to vanish exponentially with increasing $n_{\max }$ . As we suggest in Sec. 5, one can in fact make a good use of this mismatch in numerical calculations, by recording its residual value and using it to assess the truncation error of the partial $n$ -mode sum. Comparing with Figs. 1, 2 (left panels), it is striking that, in the example considered here, summing up to only $n = 2$ with the new method achieves a better accuracy in the local field than summing over as many as 16 modes using the standard approach. The right panel demonstrates the fast convergence of the derivatives ${\tilde{ϕ}}_{, r}^{+}$ and ${\tilde{ϕ}}_{, r}^{-}$ in the new approach. An exponential convergence, expected form theory, is loosely suggested from the data shown.Reuse & Permissions

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