Frequency-domain calculation of the self-force : The high-frequency problem and its resolution
Frequency-domain calculation of the self-force : The high-frequency problem and its resolution
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自力的频域计算：高频问题及其解决方案
DOI：
10.1103/physrevd.78.084021
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发表时间：
2008
影响因子：
5
通讯作者：
N. Sago
中科院分区：
文献类型：
--
作者：
L. Barack;A. Ori;N. Sago
模和法为计算绕大黑洞运行的小粒子的自力提供了一种实用的方法。在该方法中，首先计算‘全力场’场F{sup{u}}的球谐L模贡献F{sup{u}}，在粒子的位置进行计算，然后在一定的正则化方案下求和于L。在这个程序的频域变体中，通过将粒子的自场完全分解成傅里叶调和模lm{omega}，计算每个这样的模对F{subL}{sup{u}}的贡献，然后对给定的L求和{omega}和m，得到量F{SUP{MU}}。该方法的优点是只遇到常微分方程组。然而，对于偏心轨道，发现在粒子的位置处，{omega}上的和收敛得很差。这个问题(让人想起傅里叶分析中熟悉的吉布斯现象)是由粒子世界线上的时间域F{亚L}{超{亩}}场不连续引起的。在这里，我们提出了一种简单实用的方法来解决这个问题。该方法利用自场的齐次模lm{omega}构造F{SUP{MU}}(而不是标准方法中的非齐次模)，从而保证即使在粒子的位置，F{SUB L}{SUP{MU}}的正确值也是指数级快速收敛的。我们以Schwarzschild黑洞偏心轨道上标量电荷引起的单极标量场微扰为例，说明了该方法的应用。然而，我们的方法应该适用于更广泛的问题，包括使用Teukolsky的形式或直接积分度规微扰方程来计算引力自力。
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{sub l}{sup {mu}} of the 'full-force' field F{sup {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{sub l}{sup {mu}} are obtained by fully decomposing the particle's self-field into Fourier-harmonic modes lm{omega}, calculating the contribution of each such mode to F{sub l}{sup {mu}}, and then summing over {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 {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{sub l}{sup {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{omega} of the self-field to construct F{sub l}{sup {mu}} (rather than the inhomogeneous modes, as in the standard method), which guarantees an exponentially fast convergence tomore » the correct value of F{sub l}{sup {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.« less