标题： 二阶黑洞扰动：一种计算机代数方法，第一部分 - 施瓦茨希尔德时空 显示英文标题

标题： Second-Order Black Hole Perturbations: A Computer Algebra Approach, I - The Schwarzschild Spacetime

摘要： 本文概述了我们对施瓦茨希尔德黑洞的二阶扰动的推导，强调了我们使用计算机代数以及对其的必要依赖。这里提出的特定扰动情景是线性四极子激发二阶四极子的情况。这个问题相当于找到二阶Zerilli波方程，特别是由于线性四极子引起的有效源项。除了一个次要例外，我们的计算证实了Gleiser等人之前的发现。在得到这些结果的过程中，我们还说明了借助计算机代数，线性施瓦茨希尔德问题可以以非常直接的方式求解（即无需采用通常的函数变换），而这种“直接方法”推动了高阶扰动分析。计算是使用运行在Maple V平台上的GRTensorII计算机代数包进行的，同时还包括我们专门为这类问题编写的几个新的Maple程序。尽管本文中我们只考虑了“四极子-四极子”计算，但通过包含这些新程序的GRTensor环境，可以将此分析应用于更广泛的问题。这些程序以及可重现我们计算的Maple工作表可在GRTensor网站上公开获得：www.astro.queensu.ca/~grtensor。有兴趣的读者被邀请下载并使用它们来重现我们的结果并进行实验。 显示英文摘要

摘要： This article outlines our derivation of the second order perturbations to a Schwarzschild black hole, highlighting our use of, and necessary reliance on, computer algebra. The particular perturbation scenario that is presented here is the case of the linear quadrapole seeding the second order quadrapole. This problem amounts to finding the second order Zerilli wave equation, and in particular the effective source term due to the linear quadrapole. With one minor exception, our calculations confirm the earlier findings of Gleiser, et.al. On route to these results we also illustrate that, with the aid of computer algebra, the linear Schwarzschild problem can be solved in a very direct manner (i.e., without resorting to the usual function transformations), and it is this ``direct method'' that drives the higher order perturbation analysis. The calculations were performed using the GRTensorII computer algebra package, running on the Maple V platform, along with several new Maple routines that we have written specifically for these types of problems. Although we have chosen to consider only the ``quadrapole-quadrapole'' calculation in this article, the GRTensor environment, with the inclusion of these new routines, would allow this analysis to be repeated for a far more general problem. These routines, along with Maple worksheets that reproduce our calculations, are publicly available at the GRTensor website: www.astro.queensu.ca/~grtensor . The interested reader is invited to download and use them to reproduce our results and experiment.