Gauge problem in the gravitational self-force: Harmonic gauge approach in the Schwarzschild background
Gauge problem in the gravitational self-force: Harmonic gauge approach in the Schwarzschild background
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引力自力中的规范问题：史瓦西背景下的谐波规范方法
DOI：
10.1103/physrevd.67.104017
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发表时间：
2002
影响因子：
5
通讯作者：
M. Sasaki
中科院分区：
文献类型：
--
作者：
N. Sago;H. Nakano;M. Sasaki
粒子在Schwarzschild背景中引起的度规扰动通常是在Regge-Wheeler(RW)规范中计算的，而引力自力是由谐和规范中度规扰动的尾部给出的。因此，为了正确地识别特定规范中的引力自力，必须找出连接这两个规范的规范变换。这就是量规问题。作为解决规范问题的直接途径，我们提出了一种计算Schwarzschild背景下谐和规范中度规微扰的方法。我们将傅里叶调和展开应用于度规微扰，并将问题归结为从RW规范到调和规范的傅里叶调和系数(径向函数)的规范变换。我们推导了一组用于规范变换的解耦径向方程。这些方程对于奇宇称部分具有简单的二阶形式，对于偶宇称部分具有自旋$S=0$和1Teukolsky方程的形式。作为副产品，我们更正了Zerilli论文中的打印错误，并在附录A中提出了一组更正后的公式。
The metric perturbation induced by a particle in the Schwarzschild background is usually calculated in the Regge-Wheeler (RW) gauge, whereas the gravitational self-force is known to be given by the tail part of the metric perturbation in the harmonic gauge. Thus, to identify the gravitational self-force correctly in a specified gauge, it is necessary to find out a gauge transformation that connects these two gauges. This is called the gauge problem. As a direct approach to solve the gauge problem, we formulate a method to calculate the metric perturbation in the harmonic gauge on the Schwarzschild background. We apply the Fourier-harmonic expansion to the metric perturbation and reduce the problem to the gauge transformation of the Fourier-harmonic coefficients (radial functions) from the RW gauge to the harmonic gauge. We derive a set of decoupled radial equations for the gauge transformation. These equations are found to have a simple second-order form for the odd parity part and the forms of spin $s=0$ and 1 Teukolsky equations for the even parity part. As a by-product, we correct typographical errors in Zerilli's paper and present a set of corrected equations in Appendix A.